Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements

Abstract

Given an artin algebra $\Lambda$ with an idempotent element $a$ we compare the algebras $\Lambda$ and $a\Lambda a$ with respect to Gorensteinness, singularity categories and the finite generation condition $\mathsf{Fg}$ for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element $a$ which ensure that $\Lambda$ is Gorenstein if and only if $a\Lambda a$ is Gorenstein, that the singularity categories of $\Lambda$ and $a\Lambda a$ are equivalent and that $\mathsf{Fg}$ holds for $\Lambda$ if and only if $\mathsf{Fg}$ holds for $a\Lambda a$. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorensteinness and singularity categories in this general setting. The results are applied to stable categories of Cohen–Macaulay modules and classes of triangular matrix algebras and quotients of path algebras.

1. Introduction

This paper deals with Gorenstein algebras and categories, singularity categories and a finiteness condition ensuring existence of a useful theory of support for modules over finite dimensional algebras. First we give some background and indicate how these subjects are linked for us. Then we discuss the common framework for our investigations and give a sample of the main results in the paper. Finally we describe the structure of the paper. For related work see Green–Madsen–Marcos 34 and Nagase 47. In Subsection 8.4, we compare our results to those of Nagase.

For a group algebra of a finite group $G$ over a field $k$ there is a theory of support varieties of modules introduced by Jon Carlson in the seminal paper 13. This theory has proven useful and powerful, where the support of a module is defined in terms of the maximal ideal spectrum of the group cohomology ring $H^*(G,k)$. Crucial facts here are that the group cohomology ring is graded commutative and noetherian, and for any finitely generated $kG$-module $M$, the Yoneda algebra $\operatorname {Ext}^*_{kG}(M,M)$ is a finitely generated module over the group cohomology ring (see 293162). For a finitely generated $kG$-module $M$ the support variety is defined as the variety associated to the annihilator ideal of the action of the group cohomology ring $H^*(G,k)$ on $\operatorname {Ext}^*_{kG}(M,M)$. This construction is based on the Hopf algebra structure of the group algebra $kG$, and until recently a theory of support was not available for finite dimensional algebras in general.

Snashall and Solberg 59 have extended the theory of support varieties from group algebras to finite dimensional algebras by replacing the group cohomology $H^*(G,k)$ with the Hochschild cohomology ring of the algebra. Whenever similar properties as for group algebras are satisfied, that is, (i) the Hochschild cohomology ring is noetherian and (ii) all Yoneda algebras $\operatorname {Ext}^*_\Lambda (M,M)$ for a finitely generated $\Lambda$-module $M$ are finitely generated modules over the Hochschild cohomology ring, then many of the same results as for group algebras of finite groups are still true when $\Lambda$ is a selfinjective algebra 26. The above set of conditions is referred to as $\mathsf{Fg}$ (see 2660).

Triangulated categories of singularities or for simplicity singularity categories have been introduced and studied by Buchweitz 12, under the name stable derived categories, and later they have been considered by Orlov 50. For an algebraic variety $\mathbb{X}$, Orlov introduced the singularity category of $\mathbb{X}$, as the Verdier quotient $\mathsf{D}_\mathsf{sg}(\mathbb{X}) = \mathsf{D}^\mathsf{b}(\operatorname {coh}\mathbb{X})/ {\operatorname {perf}}(\mathbb{X})$, where $\mathsf{D}^\mathsf{b}(\operatorname {coh}\mathbb{X})$ is the bounded derived category of coherent sheaves on $\mathbb{X}$ and ${\operatorname {perf}}(\mathbb{X})$ is the full subcategory consisting of perfect complexes on $\mathbb{X}$. The singularity category $\mathsf{D}_\mathsf{sg}(\mathbb{X})$ captures many geometric properties of $\mathbb{X}$. For instance, if the variety $\mathbb{X}$ is smooth, then the singularity category $\mathsf{D}_\mathsf{sg}(\mathbb{X})$ is trivial but this is not true in general 50. It should be noted that the singularity category is not only related to the study of the singularities of a given variety $\mathbb{X}$ but is also related to the Homological Mirror Symmetry Conjecture due to Kontsevich 42. For more information we refer to 505152.

Similarly, the singularity category over a noetherian ring $R$ is defined 12 to be the Verdier quotient of the bounded derived category $\mathsf{D}^\mathsf{b}(\operatorname {mod}{R})$ of the finitely generated $R$-modules by the full subcategory ${\operatorname {perf}}(R)$ of perfect complexes and is denoted by

$$\begin{equation*} \mathsf{D}_\mathsf{sg}(R)=\mathsf{D}^\mathsf{b}(\operatorname {mod}{R})/ {\operatorname {perf}}(R). \end{equation*}$$

In this case the singularity category $\mathsf{D}_\mathsf{sg}(R)$ can be viewed as a categorical measure of the singularities of the spectrum $\operatorname {Spec}(R)$. Moreover, by a fundamental result of Buchweitz 12, and independently by Happel 37, the singularity category of a Gorenstein ring is equivalent to the stable category of (maximal) Cohen–Macaulay modules $\operatorname {\underline{CM}}(R)$, where the latter is well known to be a triangulated category 38. Note that this equivalence generalizes the well known equivalence between the singularity category of a selfinjective algebra and the stable module category, a result due to Rickard 56. If there exists a triangle equivalence between the singularity categories of two rings $R$ and $S$, then such an equivalence is called a singular equivalence between $R$ and $S$. Singular equivalences were introduced by Chen, who studied singularity categories of non-Gorenstein algebras and investigated when there is a singular equivalence between certain extensions of rings 15171920.

Next, from the perspective of support varieties, we describe some links between the above topics. Support varieties for $\mathsf{D}^\mathsf{b}(\operatorname {mod}\Lambda )$ using the Hochschild cohomology ring of $\Lambda$ were considered in 60 for a finite dimensional algebra $\Lambda$ over a field $k$, where all the perfect complexes ${\operatorname {perf}}(\Lambda )$ were shown to have trivial support variety. Hence the theory of support via the Hochschild cohomology ring naturally only says something about the Verdier quotient $\mathsf{D}^\mathsf{b}(\operatorname {mod}\Lambda )/{\operatorname {perf}}(\Lambda )$ – the singularity category. Furthermore, in 65, Hochschild (co)homology is discussed in connection with a particular class of singular equivalences.

To have an interesting theory of support, the finiteness condition $\mathsf{Fg}$ is pivotal. When $\mathsf{Fg}$ is satisfied for an algebra $\Lambda$, then $\Lambda$ is Gorenstein 26, Proposition 1.2, or equivalently, $\operatorname {mod}\Lambda$ is a Gorenstein category.

As we pointed out above, when $\Lambda$ is Gorenstein, then by Buchweitz–Happel the singularity category $\mathsf{D}^\mathsf{b}(\operatorname {mod}\Lambda )/{\operatorname {perf}}(\Lambda )$ is triangle equivalent to $\operatorname {\underline{CM}}(\Lambda )$, the stable category of Cohen–Macaulay modules. When $\Lambda$ is a selfinjective algebra, then $\Lambda ^{\operatorname {e}}$ is selfinjective and $\operatorname {\underline{CM}}(\Lambda ^{\operatorname {e}})=\operatorname {\underline{mod}}\Lambda ^{\operatorname {e}}$ is a tensor triangulated category with $\Lambda$ as a tensor identity. Let $\mathcal{B}$ be the full subcategory of $\operatorname {\underline{CM}}(\Lambda ^{\operatorname {e}})$ consisting of all bimodules which are projective as a left and as a right $\Lambda$-module. Then $\mathcal{B}$ is also a tensor triangulated category with tensor identity $\Lambda$. The strictly positive part of the graded endomorphism ring

$$\begin{equation*} \operatorname {End}^*_{\mathcal{B}}(\Lambda ) = \bigoplus _{i\in \mathbb{Z}} \operatorname {Hom}_{\mathcal{B}}(\Lambda , \Omega ^i_{\Lambda ^{\operatorname {e}}}(\Lambda )), \end{equation*}$$

of the tensor identity $\Lambda$ in $\operatorname {\underline{CM}}(\Lambda ^{\operatorname {e}})$ is isomorphic to the strictly positive part $\operatorname {HH}^{\geqslant 1}(\Lambda )$ of the Hochschild cohomology ring of $\Lambda$. This is the relevant part for the theory of support varieties via the Hochschild cohomology ring. In addition $\mathcal{B}$ is a tensor triangulated category acting on the triangulated category $\operatorname {\underline{CM}}(\Lambda )$, and we can consider a theory of support varieties for $\operatorname {\underline{CM}}(\Lambda )$ based on the framework described in the forthcoming paper 11. Therefore the singularity category of the enveloping algebra $\Lambda ^{\operatorname {e}}$ encodes the geometric object for support varieties of modules and complexes over the algebra $\Lambda$.

Next we describe the categorical framework for our work. There has recently been a lot of interest around recollements of abelian (and triangulated) categories. These are exact sequences of abelian categories

$$\begin{equation*} \vcenter{\img[][154pt][12pt][{$\xymatrix{ 0 \ar[r]^{ } & \A\ar[r]^{i} & \B\ar[r]^{e} & \C\ar[r]^{ } & 0, }$}]{Images/img0556f6063b1eec15ac576d96df2b0fc3.svg}} \end{equation*}$$

where both the inclusion functor $i\colon \mathscr{A}\longrightarrow \mathscr{B}$ and the quotient functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ have left and right adjoints. They have been introduced by Beilinson, Bernstein and Deligne 8 first in the context of triangulated categories in their study of derived categories of sheaves on singular spaces.

Properties of recollements of abelian categories were studied by Franjou and Pirashvili in 32, motivated by the MacPherson–Vilonen construction for the category of perverse sheaves 45, and recently homological properties of recollements of abelian and triangulated categories have also been studied in 54. Recollements of abelian categories were used by Cline, Parshall and Scott in the context of representation theory (see 2453), and later Kuhn used recollements in his study of polynomial functors (see 44). Recently, recollements of triangulated categories have appeared in the work of Angeleri Hügel, Koenig and Liu in connection with tilting theory, homological conjectures and stratifications of derived categories of rings (see 1234). Also, Chen and Xi have investigated recollements in relation with tilting theory 21 and algebraic K-theory 2223. Furthermore, Han 35 has studied the relations between recollements of derived categories of algebras, smoothness and Hochschild cohomology of algebras.

It should be noted that module recollements, i.e. recollements of abelian categories whose terms are categories of modules, appear quite naturally in various settings. For instance any idempotent element $e$ in a ring $R$ induces a recollement situation between the module categories over the rings $R/\langle e \rangle$, $R$ and $eRe$. In fact recollements of module categories are now well understood since every such recollement is equivalent, in an appropriate sense, to one induced by an idempotent element 55.

We want to compare the $\mathsf{Fg}$ condition for Hochschild cohomology, Gorensteinness and the singularity categories of two algebras. Our aim in this paper is to present a common context where we can compare these properties for an algebra $\Lambda$ and $a\Lambda a$, where $a$ is an idempotent of $\Lambda$. This is achieved using recollements of abelian categories. To summarize our main results we introduce the following notion. Given a functor $f\colon \mathscr{B}\longrightarrow \mathscr{C}$ between abelian categories, the functor $f$ is called an eventually homological isomorphism if there is an integer $t$ such that for every pair of objects $B$ and $B'$ in $\mathscr{B}$, and every $j > t$, there is an isomorphism

$$\begin{equation*} \operatorname {Ext}_\mathscr{B}^j(B,B') \cong \operatorname {Ext}_\mathscr{C}^j(f(B),f(B')) \end{equation*}$$

of abelian groups (the isomorphism is not necessarily induced by the functor $f$). Our main results, stated in the context of artin algebras, are summarized in the following theorem. The four parts of the theorem are proved in Corollary 3.12, Corollary 5.4, Corollary 4.7 and Theorem 7.10, respectively. More general versions of the first three parts, in the setting of abelian categories, are given in Corollary 3.6 and Proposition 3.7, Theorem 5.2 and Theorem 4.3.

Main Theorem.

Let $\Lambda$ be an artin algebra over a commutative ring $k$ and let $a$ be an idempotent element of $\Lambda$. Let $e$ be the functor $a-\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{a\Lambda a}$ given by multiplication by $a$. Consider the following conditions:

$$\begin{align*} (\alpha ) \ \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) &< \infty & (\beta ) \ \operatorname {\mathsf{pd}}_{a{\Lambda }a} a{\Lambda } &< \infty \\ (\gamma ) \ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) &< \infty & (\delta ) \ \operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}}{\Lambda }a &< \infty \end{align*}$$

Then the following hold.

(i)

The following are equivalent:

(a)

$(\alpha )$ and $(\beta )$ hold.

(b)

$(\gamma )$ and $(\delta )$ hold.

(c)

The functor $e$ is an eventually homological isomorphism.

(ii)

The functor $e$ induces a singular equivalence between $\Lambda$ and $a\Lambda a$ if and only if conditions $(\beta )$ and $(\gamma )$ hold.

(iii)

Assume that $e$ is an eventually homological isomorphism. Then $\Lambda$ is Gorenstein if and only if $a{\Lambda }a$ is Gorenstein.

(iv)

Assume that $e$ is an eventually homological isomorphism. Assume also that $k$ is a field and that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module (for instance, this is true if $k$ is algebraically closed). Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

Now we describe the contents of the paper section by section. In Section 2, we recall notions and results on recollements of abelian categories and Hochschild cohomology that are used throughout the paper.

In Section 3, we study extension groups in a recollement of abelian categories $(\mathscr{A},\mathscr{B},\mathscr{C})$. More precisely, we investigate when the exact functor $e \colon \mathscr{B}\longrightarrow \mathscr{C}$ is an eventually homological isomorphism. It turns out that the answer to this problem is closely related to the characterization given in 54 of when the functor $e$ induces isomorphisms between extension groups in all degrees below some bound $n$. In Corollary 3.6 and Proposition 3.7 we give sufficient and necessary conditions, respectively, for the functor $e$ to be an eventually homological isomorphism. We specialize these results to recollements of artin algebras and characterize when the functor $e$ is an eventually homological isomorphism in Corollary 3.12. The results of this section are used in Section 4 and Section 7.

In Section 4, we study Gorenstein categories, introduced by Beligiannis and Reiten 9. Assuming that we have an eventually homological isomorphism $f \colon \mathscr{D}\longrightarrow \mathscr{F}$ between abelian categories, we investigate when Gorensteinness is transferred between $\mathscr{D}$ and $\mathscr{F}$. Among other things, we prove that if $f$ is an essentially surjective eventually homological isomorphism, then $\mathscr{D}$ is Gorenstein if and only if $\mathscr{F}$ is (see Theorem 4.3). We apply this to recollements of abelian categories and recollements of module categories.

In Section 5, we investigate singularity categories, in the sense of Buchweitz 12 and Orlov 50, in a recollement $(\mathscr{A},\mathscr{B},\mathscr{C})$ of abelian categories. In fact, we give necessary and sufficient conditions for the quotient functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ to induce a triangle equivalence between the singularity categories of $\mathscr{B}$ and $\mathscr{C}$; see Theorem 5.2. This result generalizes earlier results by Chen 15. We obtain the results of Chen in Corollary 5.4 by applying Theorem 5.2 to rings with idempotents. Finally, for an artin algebra $\Lambda$ with an idempotent element $a$, we give a sufficient condition for the stable categories of Cohen–Macaulay modules of $\Lambda$ and $a\Lambda a$ to be triangle equivalent; see Corollary 5.6.

In Section 6 and Section 7, which form a unit, we investigate the finite generation condition $\mathsf{Fg}$ for the Hochschild cohomology of a finite dimensional algebra over a field. In particular, in Section 6 we show how we can compare the $\mathsf{Fg}$ condition for two different algebras. This is achieved by showing, for two graded rings and graded modules over them, that if we have isomorphisms in all but finitely many degrees, then the noetherian property of the rings and the finite generation of the modules is preserved; see Propositions 6.3 and 6.4. In Section 7, we use this result to show that $\mathsf{Fg}$ holds for a finite dimensional algebra $\Lambda$ over a field if and only if $\mathsf{Fg}$ holds for the algebra $a\Lambda a$, where $a$ is an idempotent element of $\Lambda$ which satisfies certain assumptions (see Theorem 7.10). As part of this, we show that under the same assumptions, the Hochschild cohomology rings of $\Lambda$ and $a{\Lambda }a$ are the same in almost all degrees (Proposition 7.9).

The final Section 8 is devoted to examples and applications of our main results. We first discuss conditions $(\alpha )$–$(\delta )$ of the Main Theorem. Then we consider some special cases where these conditions are related, and provide an interpretation for quotients of path algebras. Then we apply our results to triangular matrix algebras. For a triangular matrix algebra $\Lambda =\bigl (\begin{smallmatrix} \Sigma & 0 \\ {_{\Gamma }M_{\Sigma }} & \Gamma \end{smallmatrix}\bigr )$, we compare $\Lambda$ to the algebras $\Sigma$ and $\Gamma$ with respect to the $\mathsf{Fg}$ condition, Gorensteinness and singularity categories. In particular, we recover a result by Chen 15 concerning the singularity categories of $\Lambda$ and $\Sigma$. Finally, we compare our results to those of Nagase 47.

Conventions and notation

For a ring $R$ we usually work with left $R$-modules and the corresponding category is denoted by $\operatorname {Mod}R$. The full subcategory of finitely presented $R$-modules is denoted by $\operatorname {mod}R$. Our additive categories are assumed to have finite direct sums and our subcategories are assumed to be closed under isomorphisms and direct summands. The Jacobson radical of a ring $R$ is denoted by $\operatorname {rad}R$. By a module over an artin algebra $\Lambda$, we mean a finitely presented (generated) left $\Lambda$-module.

2. Preliminaries

In this section we recall notions and results on recollements of abelian categories and Hochschild cohomology.

2.1. Recollements of abelian categories

In this subsection we recall the definition of a recollement situation in the context of abelian categories (see for instance 323644), we fix notation and recall some well known properties of recollements which are used later in the paper. We also include our basic source of examples of recollements. For an additive functor $F\colon \mathscr{A}\longrightarrow \mathscr{B}$ between additive categories, the essential image $\{B \in \mathscr{B}\mid B \cong F(A) \ \text{for some} \ A \in \mathscr{A}\}$ of $F$ is denoted by $\operatorname {Im}F$ and the kernel $\{A \in \mathscr{A}\mid F(A) = 0\}$ of $F$ is denoted by $\operatorname {Ker}F$.

Definition 2.1.

A recollement situation between abelian categories $\mathscr{A},\mathscr{B}$ and $\mathscr{C}$ is a diagram

$$\begin{equation*} \vcenter{\img[][135pt][53pt][{$\xymatrix@C=5em{ {\A} \ar[r]^{i} & {\B} \ar[r]^{e} \ar@/_1.5pc/[l]_{q} \ar@/^1.5pc/[l]^{p} & {\C} \ar@/_1.5pc/[l]_{l} \ar@/^1.5pc/[l]^{r} }$}]{Images/img3d1bf1741e1d671f5c659fc3e9f4083f.svg}} \end{equation*}$$

henceforth denoted by $(\mathscr{A},\mathscr{B},\mathscr{C})$, satisfying the following conditions:

1.

$(l,e,r)$ is an adjoint triple.

2.

$(q,i,p)$ is an adjoint triple.

3.

The functors $i$, $l$ and $r$ are fully faithful.

4.

$\operatorname {Im}i = \operatorname {Ker}e$.

In the next result we collect some basic properties of a recollement situation of abelian categories that can be derived easily from Definition 2.1. For more details, see 3254.

Proposition 2.2.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories. Then the following hold.

(i)

The functors $i\colon \mathscr{A}\longrightarrow \mathscr{B}$ and $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ are exact.

(ii)

The compositions $ei$, $ql$ and $pr$ are zero.

(iii)

The functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ is essentially surjective.

(iv)

The units of the adjoint pairs $(i,p)$ and $(l,e)$ and the counits of the adjoint pairs $(q,i)$ and $(e,r)$ are isomorphisms:$$\begin{equation*} \operatorname {Id}_\mathscr{A}\xrightarrow {\cong } pi, \qquad \operatorname {Id}_\mathscr{C}\xrightarrow {\cong } el, \qquad qi \xrightarrow {\cong } \operatorname {Id}_\mathscr{A}, \qquad er \xrightarrow {\cong } \operatorname {Id}_\mathscr{C}. \end{equation*}$$

(v)

The functors $l\colon \mathscr{C}\longrightarrow \mathscr{B}$ and $q\colon \mathscr{B}\longrightarrow \mathscr{A}$ preserve projective objects and the functors $r\colon \mathscr{C}\longrightarrow \mathscr{B}$ and $p\colon \mathscr{B}\longrightarrow \mathscr{A}$ preserve injective objects.

(vi)

The functor $i\colon \mathscr{A}\longrightarrow \mathscr{B}$ induces an equivalence between $\mathscr{A}$ and the Serre subcategory $\operatorname {Ker}e = \operatorname {Im}i$ of $\mathscr{B}$. Moreover, $\mathscr{A}$ is a localizing and colocalizing subcategory of $\mathscr{B}$ and there is an equivalence of categories $\mathscr{B}/\mathscr{A}\simeq \mathscr{C}$.

(vii)

For every $B$ in $\mathscr{B}$ there are $A$ and $A'$ in $\mathscr{A}$ such that the units and counits of the adjunctions induce the following exact sequences:$$\begin{equation*} \vcenter{\img[][224pt][10pt][{$\xymatrix{ 0 \ar[r]^{ } & i(A) \ar[r]^{ } & le(B) \ar[r]^{ } & B \ar[r]^{ } & iq(B) \ar[r] & 0 }$}]{Images/imga478f7477b3a872d2cc4bdacd1f648da.svg}} \end{equation*}$$

and$$\begin{equation*} \vcenter{\img[][231pt][10pt][{$\xymatrix{ 0 \ar[r]^{ } & ip(B) \ar[r]^{ } & B \ar[r]^{ } & re(B) \ar[r] & i(A') \ar[r] & 0. }$}]{Images/img433e2d2b7d093b2ff00ad227f0293f25.svg}} \end{equation*}$$

Throughout the paper, we apply our results to recollements of module categories, and in particular to recollements of module categories over artin algebras as described in the following example.

Example 2.3.

Let $\Lambda$ be an artin $k$-algebra, where $k$ is a commutative artin ring, and let $a$ be an idempotent element in $\Lambda$.

(i)

We have the following recollement of abelian categories:$$\begin{equation*} \vcenter{\img[][222pt][56pt][{$\xymatrix@C=5em{ {\fmod\Lambda/\idealgenby{a}} \ar[r]^{\inc} & {\fmod\Lambda} \ar[r]^{e=a(-)} \ar@/_1.5pc/[l]_{\Lambda/\idealgenby{a} \tensor_\Lambda-} \ar@/^1.5pc/[l]^{\Hom_\Lambda(\Lambda/\idealgenby{a},-)} & {\fmod a \Lambda a} \ar@/_1.5pc/[l]_{\Lambda a \tensor_{a \Lambda a} -} \ar@/^1.5pc/[l]^{\Hom_{a \Lambda a}(a \Lambda,-)} }$}]{Images/imgbe6d6ab0ea606e3091085c191bffe4f4.svg}} \end{equation*}$$

The functor $e\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{a\Lambda a}$ can also be described as follows: $e=a(-)\cong \operatorname {Hom}_{\Lambda }(\Lambda a,-)\cong a\Lambda \otimes _{\Lambda }-$. We write $\langle a \rangle$ for the ideal of $\Lambda$ generated by the idempotent element $a$. Then every left $\Lambda /\langle a \rangle$-module is annihilated by $\langle a \rangle$ and thus the category $\operatorname {mod}{\Lambda /\langle a \rangle }$ is the kernel of the functor $a(-)$.

(ii)

Let $\Lambda ^{\operatorname {e}}=\Lambda \otimes _k \Lambda ^{\operatorname {op}}$ be the enveloping algebra of $\Lambda$. The element $\varepsilon =a\otimes a^{\operatorname {op}}$ is an idempotent element of $\Lambda ^{\operatorname {e}}$. Therefore as above we have the following recollement of abelian categories:$$\begin{equation*} \vcenter{\img[][237pt][61pt][{$\xymatrix@C=5em{ {\fmod\envalg{\Lambda}/\idealgenby{\varepsilon}} \ar[r]^{\inc} & {\fmod\envalg{\Lambda}} \ar[r]^{E = \varepsilon(-)} \ar@/_1.5pc/[l]_{\envalg{\Lambda}/\idealgenby{\varepsilon} \tensor_{\envalg{\Lambda}} -} \ar@/^1.5pc/[l]^{\Hom_{\envalg{\Lambda}}(\envalg{\Lambda}/\idealgenby{\varepsilon},-)} & {\fmod\envalg{(a \Lambda a)}} \ar@/_1.5pc/[l]_{\envalg{\Lambda}\varepsilon\tensor_{\envalg{(a \Lambda a)}} -} \ar@/^1.5pc/[l]^{\Hom_{\envalg{(a \Lambda a)}}(\varepsilon\envalg{\Lambda},-)} }$}]{Images/img140f8d6c9f58fc3cfe7775e54101b98d.svg}} \end{equation*}$$

Note that $(a \Lambda a)^{\operatorname {e}} \cong \varepsilon \Lambda ^{\operatorname {e}}\varepsilon$ as $k$-algebras.

Remark 2.4.

As in Example 2.3, any idempotent element $e$ in a ring $R$ induces a recollement situation between the module categories over the rings $R/\langle e \rangle$, $R$ and $eRe$. This should be considered as the universal example for recollements of abelian categories whose terms are categories of modules. Indeed, in 55 it is proved that any recollement of module categories is equivalent, in an appropriate sense, to one induced by an idempotent element.

2.2. Hochschild cohomology rings

We briefly explain the terminology we need regarding Hochschild cohomology and the finite generation condition $\mathsf{Fg}$, and recall some important results. For a more detailed exposition of these topics, see sections 2–5 of 60.

Let $\Lambda$ be an artin algebra over a commutative ring $k$. We define the Hochschild cohomology ring $\operatorname {HH}^{*}(\Lambda )$ of $\Lambda$ by

$$\begin{equation*} \operatorname {HH}^{*}(\Lambda ) = \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^*(\Lambda , \Lambda ) = \bigoplus _{i=0}^\infty \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^i(\Lambda , \Lambda ). \end{equation*}$$

This is a graded $k$-algebra with multiplication given by Yoneda product. Hochschild cohomology was originally defined by Hochschild in 39, using the bar resolution. It was shown in 14, IX, §6 that our definition coincides with the original definition when $\Lambda$ is projective over $k$.

Gerstenhaber showed in 33 that the Hochschild cohomology ring as originally defined is graded commutative. This implies that the Hochschild cohomology ring as defined above is graded commutative when $\Lambda$ is projective over $k$. The following more general result was shown in 59, Theorem 1.1 (see also 61, which proves graded commutativity of several cohomology theories in a uniform way).

Theorem 2.5.

Let $\Lambda$ be an algebra over a commutative ring $k$ such that $\Lambda$ is flat as a module over $k$. Then the Hochschild cohomology ring $\operatorname {HH}^{*}(\Lambda )$ is graded commutative.

To describe the finite generation condition $\mathsf{Fg}$, we first need to define a $\operatorname {HH}^{*}(\Lambda )$-module structure on the direct sum of all extension groups of a $\Lambda$-module with itself (for more details about this module structure, see 59). Assume that $\Lambda$ is flat as a $k$-module, and let $M$ be a $\Lambda$-module. The direct sum

$$\begin{equation*} \operatorname {Ext}_\Lambda ^*(M, M) = \bigoplus _{i=0}^\infty \operatorname {Ext}_\Lambda ^i(M, M) \end{equation*}$$

of all extension groups of $M$ with itself is a graded $k$-algebra with multiplication given by Yoneda product. We give it a graded $\operatorname {HH}^{*}(\Lambda )$-module structure by the graded ring homomorphism

$$\begin{equation*} \varphi _M \colon \operatorname {HH}^{*}(\Lambda ) \longrightarrow \operatorname {Ext}_\Lambda ^*(M, M), \end{equation*}$$

which is defined in the following way. Any homogeneous element of positive degree in $\operatorname {HH}^{*}(\Lambda )$ can be represented by an exact sequence

$$\begin{equation*} \eta \colon 0 \longrightarrow \Lambda \longrightarrow X \longrightarrow P_n \longrightarrow \cdots \longrightarrow P_1 \longrightarrow P_0 \longrightarrow \Lambda \longrightarrow 0 \end{equation*}$$

of $\Lambda ^{\operatorname {e}}$-modules, where every $P_i$ is projective. Tensoring this sequence throughout with $M$ gives an exact sequence

$$\begin{multline} 0 \longrightarrow \Lambda \otimes _\Lambda M \longrightarrow X \otimes _\Lambda M \longrightarrow P_n \otimes _\Lambda M \longrightarrow \cdots \\ \longrightarrow P_1 \otimes _\Lambda M \longrightarrow P_0 \otimes _\Lambda M \longrightarrow \Lambda \otimes _\Lambda M \longrightarrow 0 \end{multline}$$

of $\Lambda$-modules (the exactness of this sequence follows from the facts that $\Lambda$ is flat as a $k$-module and that the modules $P_i$ are projective $\Lambda ^{\operatorname {e}}$-modules). Using the isomorphism $\Lambda \otimes _\Lambda M \cong M$, we get an exact sequence of $\Lambda$-modules starting and ending in $M$; we define $\varphi _M([\eta ])$ to be the element of $\operatorname {Ext}_\Lambda ^*(M, M)$ represented by this sequence. For elements of degree zero in $\operatorname {HH}^{*}(\Lambda )$, the map $\varphi _M$ is defined by tensoring with $M$ and using the identification $\Lambda \otimes _\Lambda M \cong M$.

In 26, Erdmann–Holloway–Snashall–Solberg–Taillefer identified certain assumptions about an algebra $\Lambda$, which are sufficient in order for the theory of support varieties to have good properties. They called these assumptions Fg1 and Fg2. We say that an algebra satisfies $\mathsf{Fg}$ if it satisfies both Fg1 and Fg2. We use the following definition of $\mathsf{Fg}$, which is equivalent (by 60, Proposition 5.7) to the definition of Fg1 and Fg2 given in 26.

Definition 2.6.

Let $\Lambda$ be an algebra over a commutative ring $k$ such that $\Lambda$ is flat as a module over $k$. We say that $\Lambda$ satisfies the $\mathsf{Fg}$ condition if the following is true:

(i)

The ring $\operatorname {HH}^{*}(\Lambda )$ is noetherian.

(ii)

The $\operatorname {HH}^{*}(\Lambda )$-module $\operatorname {Ext}_\Lambda ^*(\Lambda /\operatorname {rad}\Lambda , \Lambda /\operatorname {rad}\Lambda )$ is finitely generated.

The following result states that in our definition of $\mathsf{Fg}$ we could have replaced part (ii) by the same requirement for all $\Lambda$-modules. It can be proved in a similar way as 26, Proposition 1.4.

Theorem 2.7.

If an artin algebra $\Lambda$ satisfies the $\mathsf{Fg}$ condition, then $\operatorname {Ext}_\Lambda ^*(M, M)$ is a finitely generated $\operatorname {HH}^{*}(\Lambda )$-module for every $\Lambda$-module $M$.

We end this section by describing a connection between the $\mathsf{Fg}$ condition and Gorensteinness.

Theorem 2.8 (26, Theorem 1.5 (a)).

If an artin algebra $\Lambda$ satisfies the $\mathsf{Fg}$ condition, then $\Lambda$ is Gorenstein.

3. Eventually homological isomorphisms in recollements

Given a functor $f\colon \mathscr{D}\longrightarrow \mathscr{F}$ between abelian categories and an integer $t$, the functor $f$ is called a $t$-homological isomorphism if there is a group isomorphism

$$\begin{equation*} \operatorname {Ext}_\mathscr{B}^j(B,B') \cong \operatorname {Ext}_\mathscr{C}^j(f(B),f(B')) \end{equation*}$$

for every pair of objects $B$ and $B'$ in $\mathscr{B}$, and every $j > t$. Note that we do not require these isomorphisms to be induced by the functor $f$. If $f$ is a $t$-homological isomorphism for some $t$, then it is an eventually homological isomorphism. In this section, we investigate when the functor $e$ in a recollement

$$\begin{equation*} \vcenter{\img[][135pt][53pt][{$\xymatrix@C=5em{ {\A} \ar[r]^{i} & {\B} \ar[r]^{e} \ar@/_1.5pc/[l]_{q} \ar@/^1.5pc/[l]^{p} & {\C} \ar@/_1.5pc/[l]_{l} \ar@/^1.5pc/[l]^{r} }$}]{Images/img3d1bf1741e1d671f5c659fc3e9f4083f.svg}} \end{equation*}$$

of abelian categories is an eventually homological isomorphism.

The functor $e$ induces maps

$$\begin{equation} \operatorname {Ext}^j_\mathscr{B}(X,Y) \ \longrightarrow \ \operatorname {Ext}^j_\mathscr{C}(e(X),e(Y)) {\tag{3.1}} \end{equation}$$

of extension groups for all objects $X$ and $Y$ in $\mathscr{B}$ and for every $j \ge 0$. With one argument fixed and the other one varying over all objects we study when these maps are isomorphisms in almost all degrees, that is, for every degree $j$ greater than some bound $n$ (see Theorem 3.4 and Theorem 3.5). We use this to find two sets of sufficient conditions for the functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ to be an eventually homological isomorphism (Corollary 3.6), and we find a partial converse (Proposition 3.7). Finally, we specialize these results to artin algebras, using the recollement $(\operatorname {mod}\Lambda /\langle a \rangle , \operatorname {mod}\Lambda , \operatorname {mod}a{\Lambda }a)$ of Example 2.3 (i). In particular, we characterize when the functor $e\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism (Corollary 3.12).

These results are used in Section 4 for comparing Gorensteinness of the categories in a recollement, and in Section 7 for comparing the $\mathsf{Fg}$ condition of the algebras $\Lambda$ and $a{\Lambda }a$, where $a$ is an idempotent in $\Lambda$.

We start by fixing some notation. For an injective coresolution $0 \longrightarrow B \longrightarrow I^0 \longrightarrow I^1{\longrightarrow }\cdots$ of $B$ in $\mathscr{B}$, we say that the image of the morphism $I^{n-1}\longrightarrow I^n$ is an $n$-th cosyzygy of $B$, and we denote it by $\Sigma ^n(B)$. Dually, if $\cdots \longrightarrow P_1 \longrightarrow P_0 \longrightarrow B\longrightarrow 0$ is a projective resolution of $B$ in $\mathscr{B}$, then we say that the kernel of the morphism $P^{n-1}\longrightarrow P^{n-2}$ is an $n$-th syzygy of $B$, and we denote it by $\Omega ^n(B)$. Also, if $\mathcal{X}$ is a class of objects in $\mathscr{B}$, then the right orthogonal subcategory $\{B\in \mathscr{B}\mid \operatorname {Hom}_{\mathscr{B}}(\mathcal{X},B)=0\}$ of $\mathcal{X}$ is denoted by $\mathcal{X}^{\bot }$ and the left orthogonal subcategory $\{B\in \mathscr{B}\mid \operatorname {Hom}_{\mathscr{B}}(B,\mathcal{X})=0\}$ of $\mathcal{X}$ is denoted by ${^{\bot }\mathcal{X}}$.

We now describe precisely how the maps 3.1 induced by the functor $e$ in a recollement are defined. Let $\mathscr{D}$ and $\mathscr{F}$ be abelian categories and let $f\colon \mathscr{D}\longrightarrow \mathscr{F}$ be an additive exact functor. If

$$\begin{equation*} \xi \colon \vcenter{\img[][253pt][15pt][{$\xymatrix{ 0 \ar[r]^{} & X_n \ar[r]^{d_n \ \ } & X_{n-1} \ar[r]^{} & \cdots\ar[r]^{} & X_{1} \ar[r]^{d_1 \ } & X_0 \ar[r]^{} & 0 }$}]{Images/img6dee9cd81f73f6dd5a5514a4400a7f6d.svg}} \end{equation*}$$

is an exact sequence in $\mathscr{D}$, then we denote by $f(\xi )$ the exact sequence

$$\begin{equation*} f(\xi )\colon \vcenter{\img[][305pt][15pt][{$\xymatrix{ 0 \ar[r]^{} & f(X_n) \ar[r]^{f(d_n) \ \ \ } & f(X_{n-1}) \ar[r]^{} & \cdots\ar[r]^{} & f(X_{1}) \ar[r]^{f(d_1) \ } & f(X_0) \ar[r]^{} & 0 }$}]{Images/imgfba7a10ec47d08b38050885e03dbe201.svg}} \end{equation*}$$

in $\mathscr{F}$. It is clear that this operation commutes with Yoneda product; that is, if $\xi$ and $\zeta$ are composable exact sequences in $\mathscr{D}$, then $f(\xi \zeta ) = f(\xi ) \cdot f(\zeta )$. For every pair of objects $X$ and $Y$ in $\mathscr{D}$ and every nonnegative integer $j$, we define a group homomorphism

$$\begin{equation*} f_{X,Y}^j \ \colon \ \operatorname {Ext}_{\mathscr{D}}^j(X,Y) \ \longrightarrow \ \operatorname {Ext}_{\mathscr{F}}^j(f(X),f(Y)) \end{equation*}$$

by

$$\begin{align*} f_{X,Y}^0(d) &= f(d) &&\text{for a morphism $d \colon X \longrightarrow Y$;} \\ f_{X,Y}^j([\eta ]) &= [f(\eta )] &&\text{for a $j$-fold extension $\eta $ of $X$ by $Y$, where $j>0$.} \end{align*}$$

For an object $X$ in $\mathscr{D}$, the direct sum $\operatorname {Ext}^*_\mathscr{D}(X,X) = \bigoplus _{j=0}^\infty \operatorname {Ext}^j_\mathscr{D}(X,X)$ is a graded ring with multiplication given by Yoneda product, and taking the maps $f^j_{X,X}$ in all degrees $j$ gives a graded ring homomorphism

$$\begin{equation*} f^*_{X,X}\colon \operatorname {Ext}^*_\mathscr{D}(X,X) \longrightarrow \operatorname {Ext}^*_\mathscr{F}(f(X),f(X)). \end{equation*}$$

Remark 3.1.

We explain briefly why the maps $f_{X,Y}^j$ and $f^*_{X,X}$ defined above are homomorphisms.

(i)

Since the functor $f$ is exact, it preserves pullbacks and pushouts. Thus the map $f_{X,Y}^j$ preserves the Baer sum between two extensions.

(ii)

For checking that the map $f_{X,X}^*$ is a graded ring homomorphism, the only nontrivial case to consider is the product of a morphism and an extension. For this case, we again use that the functor $f$ preserves pullbacks and pushouts.

We now consider the maps

$$\begin{equation*} e_{B,B'}^j \colon \operatorname {Ext}_\mathscr{B}^j(B, B') \longrightarrow \operatorname {Ext}_\mathscr{C}^j(e(B), e(B')) \end{equation*}$$

induced by the functor $e \colon \mathscr{B}\longrightarrow \mathscr{C}$ in a recollement, where we let one argument be fixed and the other vary over all objects of $\mathscr{B}$. In 54, the first author studied when these maps are isomorphisms for all degrees up to some bound $n$, that is, for $0 \le j \le n$. This immediately leads to a description of when these maps are isomorphisms in all degrees, which we state as the following theorem.

Theorem 3.2 (54, Propositions 3.3 and 3.4, Theorem 3.10).

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories and assume that $\mathscr{B}$ and $\mathscr{C}$ have enough projective and injective objects. Let $B$ be an object in $\mathscr{B}$.

(i)

The following statements are equivalent:

(a)

The map $e_{B,B'}^{j} \colon \operatorname {Ext}_{\mathscr{B}}^{j}(B,B') \longrightarrow \operatorname {Ext}_{\mathscr{C}}^{j}(e(B),e(B'))$ is an isomorphism for every object $B'$ in $\mathscr{B}$ and every nonnegative integer $j$.

(b)

The object $B$ has a projective resolution of the form$$\begin{equation*} \vcenter{\img[][226pt][10pt][{$\xymatrix{ \cdots\ar[r]^{} & l(P_2) \ar[r]^{} & l(P_1) \ar[r]^{} & l(P_0) \ar[r]^{} & B \ar[r]^{} & 0, }$}]{Images/img9173c4212df62413ed52915202bbe6ba.svg}} \end{equation*}$$

where $P_j$ is a projective object in $\mathscr{C}$.

(c)

$\operatorname {Ext}_{\mathscr{B}}^j(B,i(A))=0$ for every $A\in \mathscr{A}$ and $j\geq 0$.

(d)

$\operatorname {Ext}_{\mathscr{B}}^j(B,i(I))=0$ for every $I\in \operatorname {Inj}{\mathscr{A}}$ and $j\geq 0$.

(ii)

The following statements are equivalent:

(a)

The map $e_{B',B}^{j} \colon \operatorname {Ext}_{\mathscr{B}}^{j}(B',B) \longrightarrow \operatorname {Ext}_{\mathscr{C}}^{j}(e(B'),e(B))$ is an isomorphism for every object $B'$ in $\mathscr{B}$ and every nonnegative integer $j$.

(b)

The object $B$ has an injective coresolution of the form$$\begin{equation*} \vcenter{\img[][233pt][11pt][{$\xymatrix{ 0 \ar[r]^{} & B \ar[r]^{} & r(I^0) \ar[r]^{} & r(I^1) \ar[r]^{} & r(I^2) \ar[r]^{} & \cdots, }$}]{Images/img67ef84e104ce4220c53a2ce3daa4fedd.svg}} \end{equation*}$$

where $I^j$ is an injective object in $\mathscr{C}$.

(c)

$\operatorname {Ext}_{\mathscr{B}}^j(i(A),B)=0$ for every $A\in \mathscr{A}$ and $j\geq 0$.

(d)

$\operatorname {Ext}_{\mathscr{B}}^j(i(P),B)=0$ for every $P\in \operatorname {Proj}{\mathscr{A}}$ and $j\geq 0$.

The above theorem describes when the maps $e_{B,B'}^j$ induced by the functor $e$ are isomorphisms in all degrees $j$. Our aim in this section is to give a similar description of when these maps are isomorphisms in almost all degrees. The basic idea is to translate the conditions in the above theorem to similar conditions stated for almost all degrees, and show the equivalence of these conditions by using the above theorem and dimension shifting. In order for this to work, however, we need to modify the conditions somewhat. We obtain Theorem 3.4 which is stated below and generalizes parts of Theorem 3.2 (i) (and the dual Theorem 3.5 which generalizes parts of Theorem 3.2 (ii)). In order to prove the theorem, we need a general version of dimension shifting as stated in the following lemma.

Lemma 3.3.

Let $\mathscr{A}$ be an abelian category, let $n$ be an integer, and let

$$\begin{equation*} \epsilon \colon \vcenter{\img[][246pt][10pt][{$\xymatrix{ 0 \ar[r]^{} & X \ar[r]^{} & E_{m-1} \ar[r]^{} & \cdots\ar[r]^{} & E_{0} \ar[r]^{} & Y \ar[r]^{} & 0 }$}]{Images/img2ea0ae603feb55fc646c0a7349088610.svg}} \end{equation*}$$

be an exact sequence in $\mathscr{A}$ with $\operatorname {\mathsf{pd}}_{\mathscr{A}}E_i \le n$ for every $i$. Then for every $i > n$ and $Z\in \mathscr{A}$, the map given by $\epsilon ^*([\eta ]) = [\eta \epsilon ]$ is an isomorphism:

$$\begin{equation*} \vcenter{\img[][169pt][13pt][{$\xymatrix@C=0.5cm{ \epsilon^* \ \colon\ {\Ext}_{\A}^{i}(X,Z) \ \ar[rr] && \ \Ext^{i+m}_{\A}(Y,Z).}$}]{Images/imgb18947895c57708eee087689390861ee.svg}} \end{equation*}$$

Now we are ready to show our characterization of when the functor $e$ in a recollement induces isomorphisms of extension groups in almost all degrees.

Theorem 3.4.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories and assume that $\mathscr{B}$ and $\mathscr{C}$ have enough projective and injective objects. Consider the following statements for an object $B$ of $\mathscr{B}$ and two integers $n$ and $m$:

(a)

The map $e_{B,B'}^{j} \colon \operatorname {Ext}_{\mathscr{B}}^{j}(B,B') \longrightarrow \operatorname {Ext}_{\mathscr{C}}^{j}(e(B),e(B'))$ is an isomorphism for every object $B'$ in $\mathscr{B}$ and every integer $j > m + n$.

(b)

The object $B$ has a projective resolution of the form$$\begin{equation*} \vcenter{\img[][310pt][10pt][{$\xymatrix{ \cdots\ar[r] & l(Q_1) \ar[r] & l(Q_0) \ar[r] & P_{n-1} \ar[r] & \cdots\ar[r] & P_0 \ar[r] & B \ar[r] & 0, }$}]{Images/imgd416e1406ddfeedb990835c0605887be.svg}} \end{equation*}$$

where each $Q_j$ is a projective object in $\mathscr{C}$.

(c)

$\operatorname {Ext}_{\mathscr{B}}^j(B,i(A))=0$ for every $A \in \mathscr{A}$ and $j > n$, and there exists an $n$-th syzygy of $B$ lying in ${^{\bot }i(\mathscr{A})}$.

(d)

$\operatorname {Ext}_{\mathscr{B}}^j(B,i(I))=0$ for every $I\in \operatorname {Inj}{\mathscr{A}}$ and $j > n$, and there exists an $n$-th syzygy of $B$ lying in ${^{\bot }i(\operatorname {Inj}{\mathscr{A}})}$.

We have the following relations between these statements:

(i)

$\text{(b)} \iff \text{(c)} \iff \text{(d)}$.

(ii)

If $\operatorname {\mathsf{pd}}_{\mathscr{C}} e(P) \le m$ for every projective object $P$ in $\mathscr{B}$, then $\text{(b)} \implies \text{(a)}$.

Proof.

(i) By dimension shift, statement (c) is equivalent to

$$\begin{equation*} \operatorname {Ext}_\mathscr{B}^j(\Omega ^n(B), i(A)) = 0 \qquad \text{for every $j \ge 0$ and every $A \in \mathscr{A}$,} \end{equation*}$$

and statement (d) is equivalent to

$$\begin{equation*} \operatorname {Ext}_\mathscr{B}^j(\Omega ^n(B), i(I)) = 0 \qquad \text{for every $j \ge 0$ and every $I \in \operatorname {Inj}\mathscr{A}$,} \end{equation*}$$

where in both cases $\Omega ^n(B)$ is a suitably chosen $n$-th syzygy of $B$. The equivalence of statements (b), (c) and (d) now follows from the equivalence of (b), (c) and (d) in Theorem 3.2 (i).

(ii) Let

$$\begin{equation*} \pi \colon \vcenter{\img[][276pt][10pt][{$\xymatrix{ 0 \ar[r] & K \ar[r] & P_{n-1} \ar[r] & \cdots\ar[r] & P_1 \ar[r] & P_0 \ar[r] & B \ar[r] & 0 }$}]{Images/imgefb45dbd1c5491377f6083df60c1d56a.svg}} \end{equation*}$$

be the beginning of the chosen projective resolution of $B$, where $K = \Omega ^n(B)$ is the $n$-th syzygy of $B$. Consider the following group homomorphisms:

$$\begin{multline} \operatorname {Ext}_\mathscr{B}^j(B,B') \xleftarrow {\pi ^*} \operatorname {Ext}_\mathscr{B}^{j-n}(K,B') \xrightarrow {e_{K,B'}^{j-n}} \operatorname {Ext}_\mathscr{C}^{j-n}(e(K),e(B'))\\ \xrightarrow {(e(\pi ))^*} \operatorname {Ext}_\mathscr{C}^j(e(B),e(B')). {\tag{3.2}} \end{multline}$$

Here, the maps $\pi ^*$ and $(e(\pi ))^*$ are isomorphisms by Lemma 3.3. Note that for $(e(\pi ))^*$ we use the fact that $\operatorname {\mathsf{pd}}_{\mathscr{C}} e(P) \le m$ for every projective object $P$ in $\mathscr{B}$. The map $e_{K,B'}^{j-n}$ is an isomorphism by Theorem 3.2 (i). Thus, we have an isomorphism

$$\begin{equation*} (e(\pi ))^* \circ e_{K,B'}^{j-n} \circ (\pi ^*)^{-1} \colon \operatorname {Ext}_\mathscr{B}^j(B,B') \longrightarrow \operatorname {Ext}_\mathscr{C}^j(e(B),e(B')) \end{equation*}$$

for every $j\geq m+n+1$ and $B'\in \mathscr{B}$. We want to show that this is the same map as $e_{B,B'}^j$. We consider an element $[\eta ] \in \operatorname {Ext}_\mathscr{B}^{j-n}(K,B')$, and follow it through the maps 3.2. We then get the following elements:

$$\begin{equation*} \vcenter{\img[][325pt][74pt][{$\xymatrix@R=1em@C=2em{ {\!\Ext_\B^j(B,B')} & {\Ext_\B^{j-n}\!(K,B')} \ar[l]_{\pi^*}^\iso\ar[r]^-{e_{K,B'}^{j-n}}_-\iso& {\Ext_\C^{j-n}\!(e(K),e(B'))} \ar[r]^-{(e(\pi))^*}_-\iso& {\Ext_\C^j(e(B),e(B'))} \\ {[\eta\pi]} & {[\eta]} \ar@{|->}[l] \ar@{|->}[r] & {[e(\eta)]} \ar@{|->}[r] & {[e(\eta) \cdot e(\pi)]} \ar@{=}[d] \\ &&& {[e(\eta\pi)]} }$}]{Images/img0ecbaabdefc4d029f106dc607a2896ca.svg}} \end{equation*}$$

This shows that our isomorphism takes any element $[\zeta ] \in \operatorname {Ext}_\mathscr{B}^j(B,B')$ to the element $[e(\zeta )] \in \operatorname {Ext}_\mathscr{C}^j(e(B),e(B'))$. Thus, our isomorphism is $e_{B,B'}^j$.

■

Dually to the above theorem, we have the following generalization of some of the implications in Theorem 3.2 (ii).

Theorem 3.5.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories and assume that $\mathscr{B}$ and $\mathscr{C}$ have enough projective and injective objects. Consider the following statements for an object $B$ of $\mathscr{B}$ and two integers $n$ and $m$:

(a)

The map $e_{B',B}^{j} \colon \operatorname {Ext}_{\mathscr{B}}^{j}(B',B) \longrightarrow \operatorname {Ext}_{\mathscr{C}}^{j}(e(B'),e(B))$ is an isomorphism for every object $B'$ in $\mathscr{B}$ and every integer $j > m + n$.

(b)

The object $B$ has an injective coresolution of the form$$\begin{equation*} \vcenter{\img[][311pt][11pt][{$\xymatrix{ 0 \ar[r] & B \ar[r] & I^0 \ar[r] & \cdots\ar[r] & I^{n-1} \ar[r] & r(J^0) \ar[r] & r(J^1) \ar[r] & \cdots, }$}]{Images/imgba1d373b793b349f5745a7665a8a0c8b.svg}} \end{equation*}$$

where each $J^j$ is a projective object in $\mathscr{C}$.

(c)

$\operatorname {Ext}_{\mathscr{B}}^j(i(A),B)=0$ for every $A \in \mathscr{A}$ and $j > n$, and there exists an $n$-th cosyzygy of $B$ lying in $i(\mathscr{A})^{\bot }$.

(d)

$\operatorname {Ext}_{\mathscr{B}}^j(i(P),B)=0$ for every $P\in \operatorname {Proj}{\mathscr{A}}$ and $j > n$, and there exists an $n$-th cosyzygy of $B$ lying in $i(\operatorname {Proj}{\mathscr{A}})^{\bot }$.

We have the following relations between these statements:

(i)

$\text{(b)} \iff \text{(c)} \iff \text{(d)}$.

(ii)

If $\operatorname {\mathsf{id}}_{\mathscr{C}} e(I) \le m$ for every injective object $I$ in $\mathscr{B}$, then $\text{(b)} \implies \text{(a)}$.

In the above results, we fixed an object $B$ of the category $\mathscr{B}$, and considered the maps $e_{B,B'}^j$ or $e_{B',B}^j$ for all objects $B'$ in $\mathscr{B}$. With certain conditions on the object $B$, we found that these maps are isomorphisms for almost all degrees $j$. We now describe some conditions on the recollement which are sufficient to ensure that the maps $e_{B,B'}^j$ are isomorphisms in almost all degrees $j$ for all objects $B$ and $B'$ of $\mathscr{B}$, in other words, that the functor $e$ is an eventually homological isomorphism. These conditions are given in the following corollary, which follows directly from Theorem 3.4 and Theorem 3.5.

Corollary 3.6.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement and assume that $\mathscr{B}$ and $\mathscr{C}$ have enough projective and injective objects. Let $m$ and $n$ be two integers. Assume that one of the following conditions holds:

(i)

$(\alpha ')$ $\sup \{\operatorname {\mathsf{id}}_{\mathscr{B}}i(I) \mid I\in \operatorname {Inj}{\mathscr{A}} \}<m$.

$(\epsilon )$ Every object of $\mathscr{B}$ has an $m$-th syzygy which lies in ${^{\bot }i(\operatorname {Inj}{\mathscr{A}})}$.

$(\beta )$ $\sup \{\operatorname {\mathsf{pd}}_{\mathscr{C}}e(P) \mid P\in \operatorname {Proj}{\mathscr{B}} \}\leq n$.

(ii)

$(\gamma ')$ $\sup \{\operatorname {\mathsf{pd}}_{\mathscr{B}}i(P) \mid P\in \operatorname {Proj}{\mathscr{A}} \}<m$.

$(\epsilon ^{\operatorname {op}})$ Every object of $\mathscr{B}$ has an $m$-th cosyzygy which lies in $i(\operatorname {Proj}{\mathscr{A}})^{\bot }$.

$(\delta )$ $\sup \{\operatorname {\mathsf{id}}_{\mathscr{C}}e(I) \mid I\in \operatorname {Inj}{\mathscr{B}} \}\leq n$.

Then the functor $e$ is an $(m+n)$-homological isomorphism, and in particular the map

$$\begin{equation*} \vcenter{\img[][193pt][15pt][{$\xymatrix@C=0.5cm{ e^j_{B,B'} \ \colon\ {\Ext}_{\B}^j(B,B') \ \ar[rr]^{ \iso} && \ \Ext^j_{\C}(e(B),e(B'))}$}]{Images/img56c910556084a94821cb84cfa904da8d.svg}} \end{equation*}$$

is an isomorphism for all objects $B$ and $B'$ of $\mathscr{B}$ and for every $j>m+n$.

We now show a partial converse of the above result.

Proposition 3.7.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement and assume that $\mathscr{B}$ and $\mathscr{C}$ have enough projective and injective objects. Assume that the functor $e$ is an eventually homological isomorphism. Then the following hold:

$(\alpha )$

$\sup \{ \operatorname {\mathsf{id}}_\mathscr{B}i(A) \mid A \in \mathscr{A}\} < \infty$.

$(\beta )$

$\sup \{ \operatorname {\mathsf{pd}}_\mathscr{C}e(P) \mid P \in \operatorname {Proj}\mathscr{B}\} < \infty$.

$(\gamma )$

$\sup \{ \operatorname {\mathsf{pd}}_\mathscr{B}i(A) \mid A \in \mathscr{A}\} < \infty$.

$(\delta )$

$\sup \{ \operatorname {\mathsf{id}}_\mathscr{C}e(I) \mid I \in \operatorname {Inj}\mathscr{B}\} < \infty$.

In particular, if $e$ is a $t$-homological isomorphism for a nonnegative integer $t$, then each of the above dimensions is bounded by $t$.

Proof.

$(\alpha )$ Let $A$ be an object of $\mathscr{A}$. For every $B$ in $\mathscr{B}$ and $j > t$, we get

$$\begin{equation*} \operatorname {Ext}_\mathscr{B}^j(B, i(A)) \cong \operatorname {Ext}_\mathscr{C}^j(e(B), ei(A)) \cong \operatorname {Ext}_\mathscr{C}^j(e(B), 0) = 0, \end{equation*}$$

since $ei = 0$ by Proposition 2.2, and thus $\operatorname {\mathsf{id}}_\mathscr{B}i(A) \le t$. The proof of $(\gamma )$ is similar.

$(\beta )$ Let $P$ be a projective object of $\mathscr{B}$. For every $C$ in $\mathscr{C}$ and $j > t$, we get

$$\begin{equation*} \operatorname {Ext}_\mathscr{C}^j(e(P), C) \cong \operatorname {Ext}_\mathscr{C}^j(e(P), el(C)) \cong \operatorname {Ext}_\mathscr{B}^j(P, l(C)) = 0, \end{equation*}$$

since $el \cong \operatorname {Id}_\mathscr{C}$ by Proposition 2.2, and thus $\operatorname {\mathsf{pd}}_\mathscr{C}e(P) \le t$. The proof of $(\delta )$ is similar.

■

Remark 3.8.

Recall from 54 that $\sup \{ \operatorname {\mathsf{pd}}_\mathscr{B}i(A) \mid A \in \mathscr{A}\} < \infty$, which appears in statement $(\gamma )$ above, is called the $\mathscr{A}$-relative global dimension of $\mathscr{B}$, and denoted by $\operatorname *{\mathsf{gl.dim}}_\mathscr{A}\mathscr{B}$.

We close this section by interpreting Theorem 3.4, Theorem 3.5 and Corollary 3.6 for artin algebras. To this end, for an artin algebra $\Lambda$ and $a\in \Lambda$ an idempotent element, we denote by

$$\begin{equation*} e = (a\Lambda \otimes _\Lambda -) \colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a \Lambda a \end{equation*}$$

the quotient functor of the recollement $(\operatorname {mod}{\Lambda /\langle a \rangle }, \operatorname {mod}{\Lambda }, \operatorname {mod}{a \Lambda a})$; see Example 2.3.

We first need the following well known observation.

Lemma 3.9.

Let $\Lambda$ be an artin algebra, let $M$ be a $\Lambda$-module and let $S$ be a simple $\Lambda$-module. Then for every $n\geq 1$ we have:

$$\begin{equation*} \operatorname {Ext}_{\Lambda }^n(M,S)\cong \operatorname {Hom}_{\Lambda }(\Omega ^n(M),S) \qquad \text{and}\qquad \operatorname {Ext}_{\Lambda }^n(S,M)\cong \operatorname {Hom}_{\Lambda }(S,\Sigma ^n(M)), \end{equation*}$$

where $\Omega ^n(M)$ is the $n$-th syzygy in a minimal projective resolution of $M$, and $\Sigma ^n(M)$ is the $n$-th cosyzygy in a minimal injective coresolution of $M$.

We also need the next easy result whose proof is left to the reader.

Lemma 3.10.

Let $\Lambda$ be an artin algebra and let $a$ be an idempotent element of $\Lambda$.

Then the following inequalities hold:

(i)

$\operatorname {\mathsf{pd}}_{a \Lambda a} e(P) \le \operatorname {\mathsf{pd}}_{a \Lambda a} a\Lambda$, for every $P\in \operatorname {proj}{\Lambda }$.

(ii)

$\operatorname {\mathsf{id}}_{a \Lambda a} e(I) \le \operatorname {\mathsf{pd}}_{(a \Lambda a)^{\operatorname {op}}} \Lambda a$, for every $I\in \operatorname {inj}{\Lambda }$.

The following is a consequence of Theorem 3.4 and Theorem 3.5 for artin algebras.

Corollary 3.11.

Let $\Lambda$ be an artin algebra, let $a$ be an idempotent element in $\Lambda$ and let $m$ and $n$ be integers.

(i)

Let $M$ be a $\Lambda$-module such that $\operatorname {Ext}^j_{\Lambda }\big (M,(\Lambda /\langle a \rangle )/(\operatorname {rad}{\Lambda / \langle a \rangle })\big )=0$ for every $j\geq m$. Assume that $\operatorname {\mathsf{pd}}_{a \Lambda a} a \Lambda \le n$. Then the map$$\begin{equation*} \vcenter{\img[][206pt][15pt][{$\xymatrix@C=0.5cm{ e^j_{M,N} \ \colon\ {\Ext}_{\Lambda}^j(M,N) \ \ar[rr]^{ \iso\ } && \ \Ext^j_{a\Lambda a}(e(M),e(N)) }$}]{Images/imgcee34d9b72ee568138b8277f78cfc1a4.svg}} \end{equation*}$$

is an isomorphism for every $\Lambda$-module $N$, and for every integer $j > m + n$.

(ii)

Let $M$ be a $\Lambda$-module such that $\operatorname {Ext}^j_{\Lambda }\big ((\Lambda /\langle a \rangle )/(\operatorname {rad}{\Lambda / \langle a \rangle }),M\big )=0$ for every $j\geq m$. Assume that $\operatorname {\mathsf{pd}}_{(a \Lambda a)^{\operatorname {op}}} \Lambda a \le n$. Then the map$$\begin{equation*} \vcenter{\img[][206pt][15pt][{$\xymatrix@C=0.5cm{ e^j_{N,M} \ \colon\ {\Ext}_{\Lambda}^j(N,M) \ \ar[rr]^{ \iso\ } && \ \Ext^j_{a\Lambda a}(e(N),e(M)) }$}]{Images/img0c574ab2b509680c31d21bb2a9d39dda.svg}} \end{equation*}$$

is an isomorphism for every $\Lambda$-module $N$, and for every integer $j > m + n$.

Proof.

(i) Consider the recollement $(\operatorname {mod}{\Lambda /\langle a \rangle },\operatorname {mod}{\Lambda },\operatorname {mod}{a\Lambda a})$ of Example 2.3. Since every simple $\Lambda / \langle a \rangle$-module is also simple as a $\Lambda$-module it follows from Lemma 3.9 that

$$\begin{equation*} \operatorname {Hom}_{\Lambda } \big (\Omega ^m(M), (\Lambda /\langle a \rangle )/(\operatorname {rad}{\Lambda /\langle a \rangle }) \big ) = 0. \end{equation*}$$

This implies that $\operatorname {Hom}_{\Lambda }(\Omega ^m(M),N)=0$ for every $\Lambda /\langle a \rangle$-module $N$ since every module has a finite composition series. Then the result is a consequence of Theorem 3.4.

(ii) The result follows similarly as in (i), using Theorem 3.5 and the second isomorphism of Lemma 3.9.

■

As an immediate consequence of the above results we have the following characterization of when the functor $e \colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism. This constitutes the first part of the Main Theorem presented in the Introduction.

Corollary 3.12.

Let $\Lambda$ be an artin algebra and let $a$ be an idempotent element in $\Lambda$. The following are equivalent:

(i)

There is an integer $s$ such that for every pair of $\Lambda$-modules $M$ and $N$, and every $j > s$, the map$$\begin{equation*} \vcenter{\img[][206pt][15pt][{$\xymatrix@C=0.5cm{ e^j_{M,N} \ \colon\ {\Ext}_{\Lambda}^j(M,N) \ \ar[rr]^{ \iso\ } && \ \Ext^j_{a\Lambda a}(e(M),e(N)) }$}]{Images/imgcee34d9b72ee568138b8277f78cfc1a4.svg}} \end{equation*}$$

is an isomorphism.

(ii)

The functor $e$ is an eventually homological isomorphism.

(iii)

$(\alpha )$ $\operatorname {\mathsf{id}}_{\Lambda }\big ((\Lambda /\langle a \rangle )/(\operatorname {rad}{\Lambda / \langle a \rangle })\big ) < \infty$ and $(\beta )$ $\operatorname {\mathsf{pd}}_{a \Lambda a} a \Lambda < \infty$.

(iv)

$(\gamma )$ $\operatorname {\mathsf{pd}}_{\Lambda }\big ((\Lambda /\langle a \rangle )/(\operatorname {rad}{\Lambda / \langle a \rangle })\big ) < \infty$ and $(\delta )$ $\operatorname {\mathsf{pd}}_{(a \Lambda a)^{\operatorname {op}}} \Lambda a < \infty$.

In particular, if the functor $e$ is a $t$-homological isomorphism, then each of the dimensions in (iii) and (iv) are at most $t$. The bound $s$ in (i) is bounded by the sum of the dimensions occurring in (iii), and also bounded by the sum of the dimensions occurring in (iv).

Proof.

The implications (ii) $\implies$ (iii) and (ii) $\implies$ (iv) follow from Proposition 3.7. The implications (iii) $\implies$ (i) and (iv) $\implies$ (i) follow from Corollary 3.11.

■

The above corollary shows that $e$ being an eventually homological isomorphism is equivalent to the two conditions $(\alpha )$ and $(\beta )$, and also equivalent to the two conditions $(\gamma )$ and $(\delta )$. In Subsection 8.1, we give examples showing that no other pair of the four conditions $(\alpha )$–$(\delta )$ implies that $e$ is an eventually homological isomorphism.

4. Gorenstein categories and eventually homological isomorphisms

Our aim in this section is to study Gorenstein categories, introduced by Beligiannis–Reiten 9. The main objective is to study when a functor $f\colon \mathscr{D}\longrightarrow \mathscr{F}$ between abelian categories preserves Gorensteinness. A central property here is whether the functor $f$ is an eventually homological isomorphism. We prove that for an essentially surjective eventually homological isomorphism $f \colon \mathscr{D}\longrightarrow \mathscr{F}$, then $\mathscr{D}$ is Gorenstein if and only if $\mathscr{F}$ is. The results are applied to recollements of abelian categories, and recollements of module categories.

We start by briefly recalling the notion of Gorenstein categories introduced in 9. Let $\mathscr{A}$ be an abelian category with enough projective and injective objects. We consider the following invariants associated to $\mathscr{A}\colon$

$$\begin{equation*} \operatorname *{\mathsf{spli}}{\mathscr{A}} = \sup \{\operatorname {\mathsf{pd}}_{\mathscr{A}}I \mid I\in \operatorname {Inj}{\mathscr{A}} \} \ \ \text{and} \ \ \operatorname *{\mathsf{silp}}{\mathscr{A}} = \sup \{\operatorname {\mathsf{id}}_{\mathscr{A}}P \mid P\in \operatorname {Proj}{\mathscr{A}} \}. \end{equation*}$$

Then we have the following notion of Gorensteinness for abelian categories.

Definition 4.1 (9).

An abelian category $\mathscr{A}$ with enough projective and injective objects is called Gorenstein if $\operatorname *{\mathsf{spli}}{\mathscr{A}}<\infty$ and $\operatorname *{\mathsf{silp}}{\mathscr{A}}<\infty$.

Note that the above notion is a common generalization of Gorensteinness in the commutative and in the noncommutative setting. We refer to 9, Chapter VII for a thorough discussion on Gorenstein categories and connections with Cohen–Macaulay objects and cotorsion pairs.

We start with the following useful observation whose direct proof is left to the reader.

Lemma 4.2.

Let $\mathscr{A}$ be an abelian category with enough projective and injective objects and let $X$ be an object of $\mathscr{A}$.

(i)

If $\operatorname {\mathsf{pd}}_{\mathscr{A}}X<\infty$, then $\operatorname {\mathsf{id}}_{\mathscr{A}}X\le \operatorname *{\mathsf{silp}}{\mathscr{A}}$.

(ii)

If $\operatorname {\mathsf{id}}_{\mathscr{A}}X<\infty$, then $\operatorname {\mathsf{pd}}_{\mathscr{A}}X\le \operatorname *{\mathsf{spli}}{\mathscr{A}}$.

In the main result of this section we study eventually homological isomorphisms between abelian categories with enough projective and injective objects. In particular we show that an essentially surjective eventually homological isomorphism preserves Gorensteinness. This is a general version of the third part of the Main Theorem presented in the Introduction.

Theorem 4.3.

Let $f \colon \mathscr{D}\longrightarrow \mathscr{F}$ be a functor, where $\mathscr{D}$ and $\mathscr{F}$ are abelian categories with enough projective and injective objects, and let $t$ be a nonnegative integer. Consider the following four statements:

$$\begin{alignat*}{2} \text{\textrm{(a)} For every $D$ in $\mathscr{D}$\textrm{:}} &\left\{ \begin{aligned} \operatorname {\mathsf{pd}}_\mathscr{D}D &\le \sup \{ \operatorname {\mathsf{pd}}_\mathscr{F}f(D), t \} \\ \operatorname {\mathsf{id}}_\mathscr{D}D &\le \sup \{ \operatorname {\mathsf{id}}_\mathscr{F}f(D), t \} \end{aligned} \right. \quad & \text{(c)} &\left\{ \begin{aligned} \operatorname *{\mathsf{spli}}\mathscr{D}&\le \sup \{ \operatorname *{\mathsf{spli}}\mathscr{F}, t \} \\ \operatorname *{\mathsf{silp}}\mathscr{D}&\le \sup \{ \operatorname *{\mathsf{silp}}\mathscr{F}, t \} \end{aligned} \right. \\ \text{\textrm{(b)} For every $D$ in $\mathscr{D}$\textrm{:}} &\left\{ \begin{aligned} \operatorname {\mathsf{pd}}_\mathscr{F}f(D) &\le \sup \{ \operatorname {\mathsf{pd}}_\mathscr{D}D, t \} \\ \operatorname {\mathsf{id}}_\mathscr{F}f(D) &\le \sup \{ \operatorname {\mathsf{id}}_\mathscr{D}D, t \} \end{aligned} \right. & \text{(d)} &\left\{ \begin{aligned} \operatorname *{\mathsf{spli}}\mathscr{F}&\le \sup \{ \operatorname *{\mathsf{spli}}\mathscr{D}, t \} \\ \operatorname *{\mathsf{silp}}\mathscr{F}&\le \sup \{ \operatorname *{\mathsf{silp}}\mathscr{D}, t \} \end{aligned} \right. \end{alignat*}$$

We have the following.

(i)

If $f$ is a $t$-homological isomorphism, then (a) holds.

(ii)

If $f$ is an essentially surjective $t$-homological isomorphism, then (a) and (b) hold.

(iii)

If (a) and (b) hold, then (c) holds.

(iv)

If (a) and (b) hold and $f$ is essentially surjective, then (c) and (d) hold.

In particular, we obtain the following.

(v)

If $f$ is an essentially surjective eventually homological isomorphism, then $\mathscr{D}$ is Gorenstein if and only if $\mathscr{F}$ is Gorenstein.

(vi)

If $f$ is an eventually homological isomorphism and (b) holds, then $\mathscr{F}$ being Gorenstein implies that $\mathscr{D}$ is Gorenstein.

Proof.

We first assume that $f$ is an essentially surjective $t$-homological isomorphism and show the inequality $\operatorname {\mathsf{pd}}_\mathscr{F}f(D) \le \sup \{ \operatorname {\mathsf{pd}}_\mathscr{D}D, t \}$; the other inequalities in parts (i) and (ii) are proved similarly. The inequality clearly holds if $D$ has infinite projective dimension. Assume that $D$ has finite projective dimension, and let $n = \max \{ \operatorname {\mathsf{pd}}_\mathscr{D}D, t \} + 1$. For any object $X$ in $\mathscr{F}$, there is an object $X'$ in $\mathscr{D}$ with $f(X') \cong X$, since the functor $f$ is essentially surjective. By using that $f$ is a $t$-homological isomorphism, we get

$$\begin{equation*} \operatorname {Ext}_\mathscr{F}^n(f(D), X) \cong \operatorname {Ext}_\mathscr{F}^n(f(D), f(X')) \cong \operatorname {Ext}_\mathscr{D}^n(D, X') = 0. \end{equation*}$$

This means that we have $\operatorname {\mathsf{pd}}_\mathscr{F}f(D) < n$, and therefore $\operatorname {\mathsf{pd}}_\mathscr{F}f(D) \le \sup \{ \operatorname {\mathsf{pd}}_\mathscr{D}D, t \}$.

We now assume that (a) and (b) hold and $f$ is essentially surjective, and show the inequality $\operatorname *{\mathsf{spli}}\mathscr{F}\le \sup \{ \operatorname *{\mathsf{spli}}\mathscr{D}, t \}$; the other inequalities in parts (iii) and (iv) are proved similarly. Let $I$ be an injective object of $\mathscr{F}$. Since $f$ is essentially surjective, we can choose an object $D$ in $\mathscr{D}$ such that $f(D) \cong I$. By (a), the object $D$ has finite injective dimension, and then by Lemma 4.2, its projective dimension is at most $\operatorname *{\mathsf{spli}}\mathscr{D}$. Using (b), we get

$$\begin{equation*} \operatorname {\mathsf{pd}}_\mathscr{F}I \le \sup \{ \operatorname {\mathsf{pd}}_\mathscr{D}D, t \} \le \sup \{ \operatorname *{\mathsf{spli}}\mathscr{D}, t \}. \end{equation*}$$

Since this holds for any injective object $I$ in $\mathscr{F}$, we have $\operatorname *{\mathsf{spli}}\mathscr{F}\le \sup \{ \operatorname *{\mathsf{spli}}\mathscr{D}, t \}$.

Parts (v) and (vi) follow by combining parts (i)–(iv).

■

Now we return to the setting of a recollement $(\mathscr{A}, \mathscr{B}, \mathscr{C})$. We use Theorem 4.3 to study the functors $i\colon \mathscr{A}\longrightarrow \mathscr{B}$ and $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ with respect to Gorensteinness.

Corollary 4.4.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories.

(i)

Assume that the categories $\mathscr{B}$ and $\mathscr{C}$ have enough projective and injective objects, and that the functor $e$ is an eventually homological isomorphism. Then $\mathscr{B}$ is Gorenstein if and only if $\mathscr{C}$ is Gorenstein.

(ii)

Assume that the category $\mathscr{B}$ has enough projective and injective objects, and that we have either$$\begin{equation*} \left. \begin{aligned} \sup \{\operatorname {\mathsf{pd}}_{\mathscr{B}}i(P) \mid P\in \operatorname {Proj}{\mathscr{A}}\} &\le 1 \\ \sup \{\operatorname {\mathsf{id}}_{\mathscr{B}}i(I) \mid I\in \operatorname {Inj}{\mathscr{A}}\} &< \infty \end{aligned} \right\} \text{ or } \left\{ \begin{aligned} \sup \{\operatorname {\mathsf{pd}}_{\mathscr{B}}i(P) \mid P\in \operatorname {Proj}{\mathscr{A}}\} &< \infty \\ \sup \{\operatorname {\mathsf{id}}_{\mathscr{B}}i(I) \mid I\in \operatorname {Inj}{\mathscr{A}}\} &\le 1 \end{aligned} \right. \end{equation*}$$

If $\mathscr{B}$ is Gorenstein, then $\mathscr{A}$ is Gorenstein.

Proof.

Part (i) follows directly from Theorem 4.3 (v), noting that $e$ is essentially surjective by Proposition 2.2.

We now show part (ii). By Proposition 2.2 (iv) and (v), $\mathscr{A}$ has enough projective and injective objects since $\mathscr{B}$ does (see 54, Remark 2.5).

It follows from 54, Proposition 4.15 (or its dual) that the functor $i\colon \mathscr{A}\longrightarrow \mathscr{B}$ is a homological embedding, i.e. the map $i_{X,Y}^n$ is an isomorphism for all objects $X$ and $Y$ in $\mathscr{A}$ and every $n \ge 0$. In particular, this means that $i$ is a $0$-homological isomorphism. By Theorem 4.3 (i), we have

$$\begin{equation} \operatorname {\mathsf{pd}}_\mathscr{A}A \le \operatorname {\mathsf{pd}}_\mathscr{B}i(A) \qquad \text{and}\qquad \operatorname {\mathsf{id}}_\mathscr{A}A \le \operatorname {\mathsf{id}}_\mathscr{B}i(A) {\tag{4.1}} \end{equation}$$

for every object $A$ in $\mathscr{A}$.

We show that $\operatorname *{\mathsf{spli}}\mathscr{A}\le \operatorname *{\mathsf{spli}}\mathscr{B}$. Let $I$ be an injective object in $A$. By assumption, we have $\operatorname {\mathsf{id}}_\mathscr{B}i(I) < \infty$, and then by the first inequality in 4.1 and Lemma 4.2, we have

$$\begin{equation*} \operatorname {\mathsf{pd}}_\mathscr{A}I \le \operatorname {\mathsf{pd}}_\mathscr{B}i(I) \le \operatorname *{\mathsf{spli}}\mathscr{B}. \end{equation*}$$

Hence we have $\operatorname *{\mathsf{spli}}\mathscr{A}\le \operatorname *{\mathsf{spli}}\mathscr{B}$. By a similar argument, we have $\operatorname *{\mathsf{silp}}\mathscr{A}\le \operatorname *{\mathsf{silp}}\mathscr{B}$. The result follows.

■

In a recollement $(\mathscr{A}, \mathscr{B}, \mathscr{C})$ we have seen that the implications (i) $\mathscr{B}$ Gorenstein if and only if $\mathscr{C}$ Gorenstein and (ii) $\mathscr{B}$ Gorenstein implies $\mathscr{A}$ Gorenstein hold under various additional assumptions. It is then natural to ask if the categories $\mathscr{A}$ and $\mathscr{C}$ being Gorenstein could imply that $\mathscr{B}$ is Gorenstein. The next example shows that this is not true in general.

Example 4.5.

Let $k$ be a field and consider the algebra $k[x]/\langle x^2 \rangle$. Then from the triangular matrix algebra

$$\begin{equation*} \Lambda = \begin{pmatrix} k & k \\ 0 & k[x]/\langle x^2 \rangle \\ \end{pmatrix} \end{equation*}$$

we have the recollement of module categories $(\operatorname {mod}{k[x]/\langle x^2 \rangle }, \operatorname {mod}{\Lambda }, \operatorname {mod}{k})$, where $\operatorname {mod}{k[x]/\langle x^2 \rangle }$ and $\operatorname {mod}{k}$ are Gorenstein categories but $\operatorname {mod}{\Lambda }$ is not Gorenstein. We refer to 15, Example 4.3 (2) for more details about the algebra $\Lambda$.

Recall from 9 that a ring $R$ is called left Gorenstein if the category $\operatorname {Mod}{R}$ of left $R$-modules is a Gorenstein category. Applying Corollary 4.4 to the module recollement $(\operatorname {Mod}{R/\langle e \rangle },\operatorname {Mod}{R},\operatorname {Mod}{eRe})$ from Example 2.3, we have the following result.

Corollary 4.6.

Let $R$ be a ring and $e$ an idempotent element of $R$.

(i)

If the functor $e-\colon \operatorname {Mod}R \longrightarrow \operatorname {Mod}eRe$ is an eventually homological isomorphism, then the ring $R$ is left Gorenstein if and only if the ring $eRe$ is left Gorenstein.

(ii)

Assume that we have either$$\begin{equation*} \left. \begin{aligned} \operatorname {\mathsf{pd}}_R R/\langle e \rangle &\le 1 \\ \sup \{\operatorname {\mathsf{id}}_{\mathscr{B}}i(I) \mid I\in \operatorname {Inj}{R/\langle e \rangle } \} &< \infty \end{aligned} \right\} \text{ or } \left\{ \begin{aligned} \operatorname {\mathsf{pd}}_R R/\langle e \rangle &< \infty \\ \sup \{\operatorname {\mathsf{id}}_{\mathscr{B}}i(I) \mid I\in \operatorname {Inj}{R/\langle e \rangle } \} &\le 1 \end{aligned} \right. \end{equation*}$$

If the ring $R$ is left Gorenstein, then the ring $R/\langle e \rangle$ is left Gorenstein.

Recall that an artin algebra $\Lambda$ is called Gorenstein if $\operatorname {\mathsf{id}}{_{\Lambda }\Lambda }<\infty$ and $\operatorname {\mathsf{id}}\Lambda _{\Lambda }<\infty$ (see 56). Note that $\operatorname {mod}{\Lambda }$ is a Gorenstein category if and only if $\Lambda$ is a Gorenstein algebra. We close this section with the following consequence for artin algebras, whose first part constitutes the third part of the Main Theorem presented in the Introduction.

Corollary 4.7.

Let $\Lambda$ be an artin algebra and $a$ an idempotent element of $\Lambda$.

(i)

Assume that the functor $a-\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism. Then the algebra $\Lambda$ is Gorenstein if and only if the algebra $a\Lambda a$ is Gorenstein.

(ii)

Assume that we have either$$\begin{equation*} \left. \begin{aligned} \operatorname {\mathsf{pd}}_\Lambda \Lambda /\langle a \rangle &\le 1 \\ \operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {op}}} \Lambda /\langle a \rangle &< \infty \end{aligned} \right\} \qquad \text{or}\qquad \left\{ \begin{aligned} \operatorname {\mathsf{pd}}_\Lambda \Lambda /\langle a \rangle &< \infty \\ \operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {op}}} \Lambda /\langle a \rangle &\le 1 \end{aligned} \right. \end{equation*}$$

If the algebra $\Lambda$ is Gorenstein, then the algebra $\Lambda /\langle a \rangle$ is Gorenstein.

5. Singular equivalences in recollements

Our purpose in this section is to study singularity categories, in the sense of Buchweitz 12 and Orlov 50, in a recollement of abelian categories $(\mathscr{A},\mathscr{B},\mathscr{C})$. In particular we are interested in finding necessary and sufficient conditions such that the singularity categories of $\mathscr{B}$ and $\mathscr{C}$ are triangle equivalent. We start by recalling some well known facts about singularity categories.

Let $\mathscr{B}$ be an abelian category with enough projective objects. We denote by $\mathsf{D}^\mathsf{b}(\mathscr{B})$ the derived category of bounded complexes of objects of $\mathscr{B}$ and by $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ the homotopy category of bounded complexes of projective objects of $\mathscr{B}$. Then the singularity category of $\mathscr{B}$ (1250) is defined to be the Verdier quotient:

$$\begin{equation*} \mathsf{D}_\mathsf{sg}(\mathscr{B}) = \mathsf{D}^\mathsf{b}(\mathscr{B}) / \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}}). \end{equation*}$$

See 18 for a discussion of more general quotients of $\mathsf{D}^\mathsf{b}(\mathscr{B})$ by $\mathsf{K}^\mathsf{b}(\mathcal{X})$, where $\mathcal{X}$ is a selforthogonal subcategory of $\mathscr{B}$.

It is well known that the singularity category $\mathsf{D}_\mathsf{sg}(\mathscr{B})$ carries a unique triangulated structure such that the quotient functor $Q_{\mathscr{B}}\colon \mathsf{D}^\mathsf{b}(\mathscr{B})\longrightarrow \mathsf{D}_\mathsf{sg}(\mathscr{B})$ is triangulated; see 434963. Recall that the objects of the singularity category $\mathsf{D}_\mathsf{sg}(\mathscr{B})$ are the objects of the bounded derived category $\mathsf{D}^\mathsf{b}(\mathscr{B})$, the morphisms between two objects $X^{\bullet }\longrightarrow Y^{\bullet }$ are equivalence classes of fractions $(X^{\bullet }\leftarrow L^{\bullet }\rightarrow Y^{\bullet })$ such that the cone of the morphism $L^{\bullet }\longrightarrow X^{\bullet }$ belongs to $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ and the exact triangles in $\mathsf{D}_\mathsf{sg}(\mathscr{B})$ are all the triangles which are isomorphic to images of exact triangles of $\mathsf{D}^\mathsf{b}(\mathscr{B})$ via the quotient functor $Q_{\mathscr{B}}$. Note that a complex $X^{\bullet }$ is zero in $\mathsf{D}_\mathsf{sg}(\mathscr{B})$ if and only if $X^{\bullet }\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$. Following Chen 1920, we say that two abelian categories $\mathscr{A}$ and $\mathscr{B}$ are singularly equivalent if there is a triangle equivalence between the singularity categories $\mathsf{D}_\mathsf{sg}(\mathscr{A})$ and $\mathsf{D}_\mathsf{sg}(\mathscr{B})$. This triangle equivalence is called a singular equivalence between $\mathscr{A}$ and $\mathscr{B}$.

To proceed further we need the following well known result for exact triangles in derived categories. For a complex $X^{\bullet }$ in an abelian category $\mathscr{A}$ we denote by $\sigma _{>n}(X^{\bullet })$ the truncation complex $\cdots \longrightarrow 0 \longrightarrow \operatorname {Im}{d^n}\longrightarrow X^{n+1}\stackrel{d^{n+1}}{\longrightarrow } X^{n+2}\longrightarrow \cdots$, and by $H^n(X^{\bullet })$ the $n$-th homology of $X^{\bullet }$.

Lemma 5.1.

Let $\mathscr{A}$ be an abelian category and let $X^{\bullet }$ be a complex in $\mathscr{A}$. Then we have the following triangle in $\mathsf{D}(\mathscr{A})\colon$

$$\begin{equation*} \vcenter{\img[][278pt][10pt][{$\xymatrix{ H^n(X^{\bullet})[-n] \ar[r]^{} & \sigma_{>n-1}(X^{\bullet}) \ar[r]^{} & \sigma_{>n}(X^{\bullet}) \ar[r]^{} & H^n(X^{\bullet})[1-n]. }$}]{Images/imgb21b245f77cb309fcf3fe1718209c518.svg}} \end{equation*}$$

Now we are ready to prove the main result of this section which gives necessary and sufficient conditions for the quotient functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ to induce a triangle equivalence between the singularity categories of $\mathscr{B}$ and $\mathscr{C}$. This is a general version of the second part of the Main Theorem presented in the Introduction.

Theorem 5.2.

Let $(\mathscr{A},\mathscr{B},\mathscr{C})$ be a recollement of abelian categories. Then the following statements are equivalent:

(i)

We have $\operatorname {\mathsf{pd}}_{\mathscr{B}}i(A)<\infty$ and $\operatorname {\mathsf{pd}}_{\mathscr{C}} e(P)< \infty$ for every $A\in \mathscr{A}$ and $P\in \operatorname {Proj}{\mathscr{B}}$.

(ii)

The functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ induces a singular equivalence between $\mathscr{B}$ and $\mathscr{C}\colon$$$\begin{equation*} \vcenter{\img[][136pt][13pt][{$\xymatrix@C=0.5cm{ \Dsg(e)\colon\Dsg(\B) \ \ar[rr]^{ \ \ \ \equivalence} && \ \Dsg(\C). }$}]{Images/imgc370aab17830fac56e29524b08ecb578.svg}} \end{equation*}$$

Proof.

(i) $\implies$ (ii) First note that we have a well defined derived functor $\mathsf{D}^\mathsf{b}(e)\colon \mathsf{D}^\mathsf{b}(\mathscr{B})\longrightarrow \mathsf{D}^\mathsf{b}(\mathscr{C})$ since the quotient functor $e\colon \mathscr{B}\longrightarrow \mathscr{C}$ is exact. Also the recollement situation $(\mathscr{A},\mathscr{B},\mathscr{C})$ implies that $0 \longrightarrow \mathscr{A}\longrightarrow \mathscr{B}\longrightarrow \mathscr{C}\longrightarrow 0$ is an exact sequence of abelian categories; see Proposition 2.2. Then it follows from 46, Theorem 3.2 (see also 40) that $0 \longrightarrow \mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B}) \longrightarrow \mathsf{D}^\mathsf{b}(\mathscr{B}) \longrightarrow \mathsf{D}^\mathsf{b}(\mathscr{C}) \longrightarrow 0$ is an exact sequence of triangulated categories, where $\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})$ is the full subcategory of $\mathsf{D}^\mathsf{b}(\mathscr{B})$ consisting of complexes whose homology lie in $\mathscr{A}$. Hence $\mathsf{D}^\mathsf{b}(e)$ is a quotient functor, i.e. $\mathsf{D}^\mathsf{b}(\mathscr{B})/\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})\simeq \mathsf{D}^\mathsf{b}(\mathscr{C})$. Next we claim that $\mathsf{D}^\mathsf{b}(e)(\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})) \subseteq \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$. Let $P^{\bullet }\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$. Suppose first that $P^{\bullet }$ is concentrated in degree zero, so we deal with a projective object $P$ of $\mathscr{B}$. Since the object $e(P)$ has finite projective dimension it follows that there is a quasi-isomorphism $Q^{\bullet }\longrightarrow e(P)[0]$, where $Q^{\bullet }\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$ is a projective resolution of $e(P)$. Then the object $e(P)$[0] is isomorphic with $Q^{\bullet }$ in $\mathsf{D}^\mathsf{b}(\mathscr{C})$ and therefore $e(P)\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$. Now let $P^{\bullet }=(0\longrightarrow P_0\longrightarrow P_1\longrightarrow 0)\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$. Then we have the triangle $P_0[0]\longrightarrow P_1[0]\longrightarrow P^{\bullet }\longrightarrow P_0[1]$ and if we apply the functor $\mathsf{D}^\mathsf{b}(e)$ we infer that $\mathsf{D}^\mathsf{b}(e)(P^{\bullet })\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$ since $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$ is a triangulated subcategory. Continuing inductively on the length of the complex $P^{\bullet }$ we infer that the object $\mathsf{D}^\mathsf{b}(e)(P^{\bullet })$ lies in $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$ and so our claim follows. Then since the triangulated functor $\mathsf{D}^\mathsf{b}(e)\circ Q_{\mathscr{C}}\colon \mathsf{D}^\mathsf{b}(\mathscr{B})\longrightarrow \mathsf{D}_\mathsf{sg}(\mathscr{C})$ annihilates $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ it follows that $\mathsf{D}^\mathsf{b}(e)\circ Q_{\mathscr{C}}$ factors uniquely through $Q_{\mathscr{B}}$ via a triangulated functor $\mathsf{D}_\mathsf{sg}(e)\colon \mathsf{D}_\mathsf{sg}(\mathscr{B})\longrightarrow \mathsf{D}_\mathsf{sg}(\mathscr{C})$, that is the following diagram is commutative:

$$\begin{equation*} \vcenter{\img[][129pt][58pt][{$\xymatrix{ \Db(\B) \ar[rr]^{Q_{\B}} \ar[d]_{\Db(e)} && \Dsg(\B) \ar[d]^{\Dsg(e)} \\ \Db(\C) \ar[rr]^{Q_{\C}} && \Dsg(\C) }$}]{Images/img40f7d10eafeb3f45c5afe54e1ffc6fe8.svg}} \end{equation*}$$

Next we show that $\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})\subseteq \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ in $\mathsf{D}^\mathsf{b}(\mathscr{B})$. Since the projective dimension of $i(A)$ is finite for all $A$ in $\mathscr{A}$, it follows that $i(\mathscr{A})\subseteq \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ in $\mathsf{D}^\mathsf{b}(\mathscr{B})$. Let $B^{\bullet }$ be an object of $\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})$. Assume first that $B^{\bullet }$ is concentrated in degree zero. Hence we deal with an object $B\in \mathscr{B}$ such that $B\cong i(A)$ for some $A\in \mathscr{A}$, and therefore our claim follows. Now consider a complex

$$\begin{equation*} B^{\bullet }\colon \vcenter{\img[][123pt][15pt][{$\xymatrix{ 0 \ar[r]^{} & B^0 \ar[r]^{d^0} & B^1 \ar[r]^{} & 0, }$}]{Images/img58799753058bd9d21825edd3b0fae9e8.svg}} \end{equation*}$$

where $H^0(B^{\bullet })$ and $H^1(B^{\bullet })$ lie in $\mathscr{A}$. From Lemma 5.1 we have the triangles

$$\begin{equation*} \vcenter{\img[][229pt][11pt][{$\xymatrix{ H^0(B^{\bullet}) \ar[r]^{} & \sigma_{>-1}(B^{\bullet}) \ar[r]^{} & \sigma_{>0}(B^{\bullet}) \ar[r]^{} & H^0(B^{\bullet})[1] }$}]{Images/imgf818433679d09668c93577dcefe5e06d.svg}} \end{equation*}$$

and

$$\begin{equation*} \vcenter{\img[][230pt][11pt][{$\xymatrix{ H^1(B^{\bullet})[-1] \ar[r]^{} & \sigma_{>0}(B^{\bullet}) \ar[r]^{} & \sigma_{>1}(B^{\bullet}) \ar[r]^{} & H^1(B^{\bullet}) }$}]{Images/img008a695c5964151323ffb10f86dacd2c.svg}} \end{equation*}$$

in $\mathsf{D}^\mathsf{b}(\mathscr{B})$. Then from the second triangle it follows that $\sigma _{>0}(B^{\bullet })\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ and therefore from the first triangle we get that $\sigma _{>-1}(B^{\bullet })=B^{\bullet }\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$. Continuing inductively on the length of the complex $B^{\bullet }$, we infer that $\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})\subseteq \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ in $\mathsf{D}^\mathsf{b}(\mathscr{B})$. Using this we can form the quotient $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})/\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})$, and then we have the following exact commutative diagram:

$$\begin{equation*} \vcenter{\img[][289pt][55pt][{$\xymatrix{ 0 \ar[r] & \Kb(\Proj{\B})/\Db_{\A}(\B) \ar[r]^{} \ar[d]_{} & \Db(\B)/\Db_{\A}(\B) \ar[d]^{\equivalence} \ar[r] & \Dsg(\B) \ar[r]^{} \ar[d]^{\Dsg(e)} & 0 \\ 0 \ar[r] & \Kb(\Proj{\C}) \ar[r]^{} & \Db(\C) \ar[r] & \Dsg(\C) \ar[r] & 0 }$}]{Images/img69caa9cbd69a8a2e7d17936493f4a2ec.svg}} \end{equation*}$$

We show that the functor $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})/\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})\longrightarrow \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$ is an equivalence, where we denote it by $\mathsf{K}^\mathsf{b}(e)$. First from the above commutative diagram and since there is an equivalence $\mathsf{D}^\mathsf{b}(\mathscr{B})/\mathsf{D}^\mathsf{b}_{\mathscr{A}}(\mathscr{B})\simeq \mathsf{D}^\mathsf{b}(\mathscr{C})$, it follows that the functor $\mathsf{K}^\mathsf{b}(e)$ is fully faithful. Let $P^{\bullet }\colon 0 \longrightarrow P_n \longrightarrow \cdots \longrightarrow P_1 \longrightarrow P_0 \longrightarrow 0$ be an object of $\mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$. Each $P_i$ is a projective object in $\mathscr{C}$ and from Proposition 2.2 we have $el(P_i)\cong P_i$ with $l(P_i)\in \operatorname {Proj}{\mathscr{B}}$. Then the complex $l(P^{\bullet })\colon 0 \longrightarrow l(P_n) \longrightarrow \cdots \longrightarrow l(P_1) \longrightarrow l(P_0) \longrightarrow 0$ is such that $\mathsf{K}^\mathsf{b}(e)(l(P^{\bullet }))=P^{\bullet }$. This implies that the functor $\mathsf{K}^\mathsf{b}(e)$ is essentially surjective. Hence the functor $\mathsf{K}^\mathsf{b}(e)$ is an equivalence.

In conclusion, from the above exact commutative diagram we infer that the singularity categories of $\mathscr{B}$ and $\mathscr{C}$ are triangle equivalent.

(ii) $\implies$ (i) Suppose that there is a triangle equivalence $\mathsf{D}_\mathsf{sg}(e)\colon \mathsf{D}_\mathsf{sg}(\mathscr{B})\stackrel{\simeq }{\longrightarrow } \mathsf{D}_\mathsf{sg}(\mathscr{C})$. Let $P$ be a projective object of $\mathscr{B}$. Then $P[0]\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$ and $\mathsf{D}^\mathsf{b}(e)(P[0])\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{C}})$. Thus the object $e(P)$ has finite projective dimension. Let $A\in \mathscr{A}$ and consider the object $i(A)$ of $\mathscr{B}$. Then from Proposition 2.2 we have $e(i(A))=0$. Since $\mathsf{D}_\mathsf{sg}(e)$ is an equivalence, the object $i(A)$ is zero in $\mathsf{D}_\mathsf{sg}(\mathscr{B})$, and therefore $i(A)\in \mathsf{K}^\mathsf{b}(\operatorname {Proj}{\mathscr{B}})$. We infer that $i(A)$ has finite projective dimension.

■

Remark 5.3.

If the functor $e \colon \mathscr{B}\longrightarrow \mathscr{C}$ is an eventually homological isomorphism, then statement (i) in Theorem 5.2 is true by Proposition 3.7. Thus Theorem 5.2 in particular says that if the functor $e \colon \mathscr{B}\longrightarrow \mathscr{C}$ in a recollement $(\mathscr{A},\mathscr{B},\mathscr{C})$ is an eventually homological isomorphism, then it induces a singular equivalence between $\mathscr{B}$ and $\mathscr{C}$. Example 5.5 below shows that the converse does not hold.

Note that statement (i) in Theorem 5.2 only states that each object of the form $i(A)$ or $e(P)$ has finite projective dimension, and not that there exists a finite bound for the projective dimensions of all such objects. In other words, the supremums

$$\begin{equation*} \sup \{ \operatorname {\mathsf{pd}}_\mathscr{B}i(A) \mid A \in \mathscr{A}\} \qquad \text{and}\qquad \sup \{ \operatorname {\mathsf{pd}}_\mathscr{C}e(P) \mid P \in \operatorname {Proj}\mathscr{B}\} \end{equation*}$$

(which are used in other parts of the paper) may be infinite even if statement (i) is true.

Applying Theorem 5.2 to the recollement of module categories $(\operatorname {mod}{R/\langle e \rangle },\operatorname {mod}{R},\operatorname {mod}{eRe})$ (see Example 2.3), we have the following consequence due to Chen; see 15, Theorem 2.1 and 16, Corollary 3.3. Note that our version is somewhat stronger; the difference is that Chen takes $\operatorname {\mathsf{pd}}_{eRe} eR <\infty$ as an assumption instead of including it in one of the equivalent statements. This result constitutes the second part of the Main Theorem presented in the Introduction.

Corollary 5.4.

Let $R$ be a left Noetherian ring and $e$ an idempotent element of $R$. Then the following statements are equivalent:

(i)

For every $R/\langle e \rangle$-module $X$ we have $\operatorname {\mathsf{pd}}_RX<\infty$, and $\operatorname {\mathsf{pd}}_{eRe} eR< \infty$.

(ii)

The functor $e(-)\colon \operatorname {mod}{R}\longrightarrow \operatorname {mod}{eRe}$ induces a singular equivalence between $\operatorname {mod}{R}$ and $\operatorname {mod}{eRe}\colon$$$\begin{equation*} \vcenter{\img[][198pt][13pt][{$\xymatrix@C=0.5cm{ \Dsg(e(-))\colon\Dsg(\fmod{R}) \ \ar[rr]^{ \ \ \ \ \equivalence} && \ \Dsg(\fmod{eRe}). }$}]{Images/img4fbfc008a8421bbfa8362fa835e463ed.svg}} \end{equation*}$$

Now we give an example where the functor $e=a(-)\colon \operatorname {mod}\Lambda \to \operatorname {mod}{a\Lambda a}$ induces a singular equivalence, but is not an eventually homological isomorphism.

Example 5.5.

We recycle the algebra from Example 4.5, that is, let $\Lambda$ be the triangular matrix algebra

$$\begin{equation*} \begin{pmatrix} k & k \\ 0 & k[x]/\langle x^2 \rangle \\ \end{pmatrix}, \end{equation*}$$

where $k$ is a field. We have the recollement of module categories $(\operatorname {mod}{k}, \operatorname {mod}{\Lambda }, \operatorname {mod}{k[x]/\langle x^2 \rangle })$, with $a=\left(\begin{smallmatrix} 0 & 0\\ 0 & 1\end{smallmatrix}\right)$ as the chosen idempotent. Then $(\Lambda /\langle a \rangle )/(\operatorname {rad}\Lambda /\langle a \rangle ) \cong \Lambda /\langle a \rangle \cong \Lambda \left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)$ as $\Lambda$-modules, which implies that all $\Lambda /\langle a \rangle$-modules are projective as $\Lambda$-modules. Furthermore, $a\Lambda \cong a{\Lambda }a$ as $a{\Lambda }a$-modules. Then it follows from Corollary 5.4 that the functor $e=a(-)$ induces a singular equivalence between $\operatorname {mod}\Lambda$ and $\operatorname {mod}{a\Lambda a}$. However, condition $(\alpha )$ in Corollary 3.12 is not satisfied, hence $e$ is not an eventually homological isomorphism.

The algebra $k[x]/\langle x^2 \rangle$ is a Gorenstein algebra, but the algebra $\Lambda$ is not Gorenstein. This example therefore shows that singular equivalences do not in general preserve Gorensteinness.

We end this section with an application to stable categories of Cohen–Macaulay modules. Let $\Lambda$ be a Gorenstein artin algebra. We denote by $\operatorname {CM}(\Lambda )$ the category of (maximal) Cohen–Macaulay modules defined as follows:

$$\begin{equation*} \operatorname {CM}(\Lambda ) = \{ X\in \operatorname {mod}{\Lambda } \mid \operatorname {Ext}_{\Lambda }^n(X,\Lambda )=0 \ \text{for all $n\geq 1$} \}. \end{equation*}$$

Then it is known that the stable category $\operatorname {\underline{CM}}(\Lambda )$ modulo projectives is a triangulated category (see 38), and moreover there is a triangle equivalence between the singularity category $\mathsf{D}_\mathsf{sg}(\operatorname {mod}{\Lambda })$ and the stable category $\operatorname {\underline{CM}}(\Lambda )$; see 12, Theorem 4.4.1 and 37, Theorem 4.6. As a consequence of Corollary 3.12, Corollary 4.7 and Corollary 5.4 we get the following.

Corollary 5.6.

Let $\Lambda$ be a Gorenstein artin algebra and $a$ an idempotent element of $\Lambda$. Assume that the functor $a-\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism. Then there is a triangle equivalence between the stable categories of Cohen–Macaulay modules of $\Lambda$ and $a\Lambda a$:

$$\begin{equation*} \vcenter{\img[][119pt][12pt][{$\xymatrix@C=0.5cm{ \uCM(\Lambda) \ \ar[rr]^{ \equivalence\ \ } && \ \uCM(a\Lambda a). }$}]{Images/imgb52073b21b25dc32f723e18067da366a.svg}} \end{equation*}$$

6. Finite generation of cohomology rings

In this section, we describe a way to compare the $\mathsf{Fg}$ condition (see Definition 2.6) for two different algebras. This is used in the next section for the algebras $\Lambda$ and $a{\Lambda }a$, where $\Lambda$ is a finite dimensional algebra over a field and $a$ is an idempotent in $\Lambda$.

Let $\Lambda$ and $\Gamma$ be two artin algebras over a commutative ring $k$, and assume that they are flat as $k$-modules. Let $M = \Lambda /(\operatorname {rad}\Lambda )$ and $N = \Gamma /(\operatorname {rad}\Gamma )$. Assume that we have graded ring isomorphisms $f$ and $g$ making the diagram

$$\begin{equation} \vcenter{\img[][113pt][57pt][{$\xymatrix{ {\HH*(\Lambda)} \ar[r]^-{\varphi_M} \ar[d]_{f}^{\iso} & {\Ext_\Lambda^*(M, M)} \ar[d]_g^{\iso} \\ {\HH*(\Gamma)} \ar[r]_-{\varphi_{N}} & {\Ext_{\Gamma}^*(N, N)} }$}]{Images/imgee3d94aa0aadf89f7dbf427896cc02fc.svg}} {\tag{6.1}} \end{equation}$$

commute, where the maps $\varphi _M$ and $\varphi _N$ are defined in Subsection 2.2. Then it is clear that $\mathsf{Fg}$ for $\Lambda$ is exactly the same as $\mathsf{Fg}$ for $\Gamma$, since all the relevant data for the $\mathsf{Fg}$ condition is exactly the same for the two algebras.

However, we can come to the same conclusion even if the homology groups for $\Lambda$ and $\Gamma$ are different in some degrees, as long as they are the same in all but finitely many degrees. In other words, if the maps $f$ and $g$ above are just graded ring homomorphisms such that $f_n$ and $g_n$ are group isomorphisms for almost all degrees $n$, then the $\mathsf{Fg}$ condition holds for $\Lambda$ if and only if it holds for $\Gamma$. The goal of this section is to show this.

We first prove the result in a more general setting, where we replace the rings in 6.1 by arbitrary graded rings satisfying appropriate assumptions. This is done in Proposition 6.3, after we have shown a part of the result (corresponding to part (i) of the $\mathsf{Fg}$ condition) in Proposition 6.2. Finally, we state the result for $\mathsf{Fg}$ in Proposition 6.4.

We now introduce some terminology and notation which is used in this section and the next. By graded ring we always mean a ring of the form

$$\begin{equation*} R = \bigoplus _{i=0}^\infty R_i \end{equation*}$$

graded over the nonnegative integers. We denote the set of nonnegative integers by $\mathbb{N}_0$. If $R$ is a graded ring and $n$ a nonnegative integer, we use the notation $R_{\ge n}$ for the graded ideal

$$\begin{equation*} R_{\ge n} = \bigoplus _{i=n}^\infty R_i \end{equation*}$$

in $R$. We use the term rng for a “ring without identity”, that is, an object which satisfies all the axioms for a ring except having a multiplicative identity element.

We use the following characterization of noetherianness for graded rings.

Theorem 6.1.

Let $R$ be a graded ring. Then $R$ is noetherian if and only if it satisfies the ascending chain condition on graded ideals.

Proof.

This follows directly from 48, Theorem 5.4.7.

■

We now begin the main work of this section by showing that an isomorphism in all but finitely many degrees between two sufficiently nice graded rings preserves noetherianness. This implies that such a map between Hochschild cohomology rings preserves part (i) of the $\mathsf{Fg}$ condition, and thus gives one half of the result we want.

Proposition 6.2.

Let $R$ and $S$ be graded rings. Assume that $R_0$ and $S_0$ are noetherian, that every $R_i$ is finitely generated as left and as right $R_0$-modules, and that every $S_i$ is finitely generated as left and as right $S_0$-modules. Let $n$ be a nonnegative integer, and assume that there exists an isomorphism $\phi \colon R_{\ge n} \longrightarrow S_{\ge n}$ of graded rngs. Then $R$ is noetherian if and only if $S$ is noetherian.

Proof.

We prove (by showing the contrapositive) that $R$ is left noetherian if $S$ is left noetherian. The corresponding result with right noetherian is proved in the same way. This gives one of the implications we need. The opposite implication is proved in the same way by interchanging $R$ and $S$ and using $\phi ^{-1}$ instead of $\phi$.

Assume that $R$ is not left noetherian. Let

$$\begin{equation*} \mathbb{I}\colon \quad I^{(0)} \subset I^{(1)} \subset \cdots \end{equation*}$$

be an infinite strictly ascending sequence of graded left ideals in $R$ (this is possible by Theorem 6.1). For every index $i$ in this sequence, we can write the ideal $I^{(i)}$ as a direct sum

$$\begin{equation*} I^{(i)} = \bigoplus _{d \in \mathbb{N}_0} I^{(i)}_d \end{equation*}$$

of abelian groups, where $I^{(i)}_d \subseteq R_d$ is the degree $d$ part of $I^{(i)}$. For any degree $d$, we can make an ascending sequence

$$\begin{equation*} I^{(0)}_d \subseteq I^{(1)}_d \subseteq \cdots \end{equation*}$$

of $R_0$-submodules of $R_d$ by taking the degree $d$ part of each ideal in $\mathbb{I}$. But $R_d$ is a noetherian $R_0$-module (since $R_0$ is noetherian and $R_d$ is a finitely generated $R_0$-module), and hence this sequence must stabilize at some point. Let $s(d)$ be the point where it stabilizes, that is, the smallest integer such that $I^{(s(d))}_d = I^{(i)}_d$ for every $i > s(d)$.

We now define two functions $\sigma \colon \mathbb{N}_0\longrightarrow \mathbb{N}_0$ and $\delta \colon \mathbb{N}_0\longrightarrow \mathbb{N}_0$. For $d \in \mathbb{N}_0$, we define

$$\begin{equation*} \sigma (d) = \max \{ s(0), s(1), \ldots , s(d) \}. \end{equation*}$$

For $i \in \mathbb{N}_0$, we define $\delta (i)$ as the smallest number such that

$$\begin{equation*} I^{(i)}_{\delta (i)} \ne I^{(i+1)}_{\delta (i)}. \end{equation*}$$

These functions have the following interpretation. For a degree $d$, the number $\sigma (d)$ is the index in the sequence $\mathbb{I}$ where the ideals in the sequence have stabilized up to degree $d$. For an index $i$, the number $\delta (i)$ is the lowest degree at which there is a difference from the ideal $I^{(i)}$ to the ideal $I^{(i+1)}$.

We now define a sequence $(i_j)_{j \in \mathbb{N}_0}$ of indices and a sequence $(d_j)_{j \in \mathbb{N}_0}$ of degrees by

$$\begin{align*} i_j &= \left\{ \begin{array}{ll} \sigma (n) &\text{if $j = 0$,} \\\sigma (d_{j-1} + n) &\text{otherwise,} \end{array} \right. \\ d_j &= \delta (i_j). \end{align*}$$

We observe that for every positive integer $j$, we have

$$\begin{equation*} i_j > i_{j-1} \qquad \text{and}\qquad d_j > d_{j-1} + n. \end{equation*}$$

We now construct a sequence $\mathbb{J}$ of graded left ideals in $S$. For every nonnegative integer $j$, we choose an element

$$\begin{equation*} x_j \in I^{(i_j+1)}_{d_j} - I^{(i_j)}_{d_j} \end{equation*}$$

(this is possible because $d_j = \delta (i_j)$). Note that the degree of $x_j$ is $d_j$, which is greater than $n$. We then define $J^{(j)}$ to be the left ideal of $S$ generated by the set

$$\begin{equation*} \{ \phi (x_0), \ldots , \phi (x_j) \}. \end{equation*}$$

We let $\mathbb{J}$ be the sequence of these ideals:

$$\begin{equation*} \mathbb{J}\colon \quad J^{(0)} \subseteq J^{(1)} \subseteq \cdots . \end{equation*}$$

We want to show that each inclusion here is strict. This means that we must show, for every positive integer $j$, that $\phi (x_j)$ is not an element of $J^{(j-1)}$.

We show this by contradiction. Assume that there is a $j$ such that $\phi (x_j) \in J^{(j-1)}$. Then we can write $\phi (x_j)$ as a sum

$$\begin{equation*} \phi (x_j) = \sum _{m=0}^{j-1} s_m \cdot \phi (x_m), \end{equation*}$$

where each $s_m$ is an element of $S$. Since $\phi (x_j)$ and every $\phi (x_m)$ are homogeneous elements, we can choose every $s_m$ to be homogeneous. For each $m$, we have that if $s_m$ is nonzero, then its degree is

$$\begin{equation*} \left|s_m\right| = \left|\phi (x_j)\right| - \left|\phi (x_m)\right| = \left|x_j\right| - \left|x_m\right| = d_j - d_m > n. \end{equation*}$$

Thus $s_m$ is either zero or in the image of $\phi$. We use this to find corresponding elements in $R$. Let, for each $m \in \{1,\ldots ,j-1\}$,

$$\begin{equation*} r_m = \left\{ \begin{array}{ll} 0 &\text{if $s_m = 0$,} \\\phi ^{-1}(s_m) &\text{otherwise.} \end{array} \right. \end{equation*}$$

Now we have

$$\begin{equation*} \phi (x_j) = \sum _{m=0}^{j-1} s_m \cdot \phi (x_m) = \phi \left( \sum _{m=0}^{j-1} r_m\cdot x_m \right). \end{equation*}$$

Applying $\phi ^{-1}$ gives

$$\begin{equation*} x_j = \sum _{m=0}^{j-1} r_m\cdot x_m. \end{equation*}$$

Since we have $x_m \in I^{(i_m+1)} \subseteq I^{(i_j)}$ for every $m$, this means that $x_j \in I^{(i_j)}$. This is a contradiction, since $x_j$ is chosen so that it does not lie in $I^{(i_j)}$.

We have shown that the sequence $\mathbb{J}$ is a strictly ascending sequence of graded left ideals in $S$. Thus $S$ is not left noetherian.

■

We now complete the picture by considering two graded rings and a graded module over each ring, and showing that isomorphisms in all but finitely many degrees preserve both noetherianness of the rings and finite generation of the modules (given that certain assumptions are satisfied).

Proposition 6.3.

Let $R$ and $M$ be graded rings, and $\theta \colon R \longrightarrow M$ a graded ring homomorphism. View $M$ as a graded left $R$-module with scalar multiplication given by $\theta$. Assume that $R_0$ is noetherian, that every $R_i$ is finitely generated as left and as right $R_0$-modules, and that every $M_i$ is finitely generated as a left $R_0$-module.

Similarly, let $R'$ and $M'$ be graded rings, and $\theta ' \colon R' \longrightarrow M'$ a graded ring homomorphism. View $M'$ as a graded left $R'$-module with scalar multiplication given by $\theta '$. Assume that $R'_0$ is noetherian, that every $R'_i$ is finitely generated as left and as right $R'_0$-modules, and that every $M'_i$ is finitely generated as a left $R'_0$-module.

Assume that there are graded rng isomorphisms $\phi \colon R_{\ge n} \longrightarrow R'_{\ge n}$ and $\psi \colon M_{\ge n} \longrightarrow M'_{\ge n}$ (for some nonnegative integer $n$) such that the diagram

$$\begin{equation*} \vcenter{\img[][91pt][62pt][{$\xymatrix@C=5em{ R_{\ge n} \ar[r]^{\theta_{\ge n}} \ar[d]_{\phi} & M_{\ge n} \ar[d]^{\psi} \\ R'_{\ge n} \ar[r]_{\theta'_{\ge n}} & M'_{\ge n} }$}]{Images/img3ccc49e59261473ec9cd5bac0d32a2b9.svg}} \end{equation*}$$

commutes. Then the following two conditions are equivalent.

(i)

$R$ is noetherian and $M$ is finitely generated as a left $R$-module.

(ii)

$R'$ is noetherian and $M'$ is finitely generated as a left $R'$-module.

Proof.

We prove that condition (i) implies condition (ii). The opposite implication is proved in exactly the same way by using $\phi ^{-1}$ and $\psi ^{-1}$ instead of $\phi$ and $\psi$.

Assume that condition (i) holds. Then by Proposition 6.2, $R'$ is noetherian. We need to show that $M'$ is finitely generated as a left $R'$-module.

We begin with choosing generating sets for things we know to be finitely generated. Note that the ideal $R_{\ge n}$ of $R$ is finitely generated, since $R$ is noetherian. Let $A$ be a finite homogeneous generating set for $R_{\ge n}$. Let $G$ be a finite homogeneous generating set for $M$ as a left $R$-module. For every $i$, let $B_i$ be a finite generating set for $M'_i$ as a left $R'_0$-module.

Let

$$\begin{equation*} b_R = \max \big \{ \left|a\right| \mathrel{\big |} a \in A \big \} \qquad \text{and}\qquad b_M = \max \big \{ \left|g\right| \mathrel{\big |} g \in G \big \} \end{equation*}$$

be the maximal degrees of elements in our chosen generating sets for $R$ and $M$, respectively. Let

$$\begin{equation*} b = b_R + b_M + n. \end{equation*}$$

Define the set $G'$ to be

$$\begin{equation*} G' = \bigcup _{i=0}^b B_i. \end{equation*}$$

We want to show that $G'$ generates $M'$ as a left $R'$-module.

Let $N'$ be the $R'$-submodule of $M'$ generated by $G'$. It is clear that $N'$ contains every homogeneous element of $M'$ with degree at most $b$. Let $m' \in M'$ be a homogeneous element with $\left|m'\right| > b$. Let $m = \psi ^{-1}(m')$. We can write $m$ as a sum

$$\begin{equation*} m = \sum _i \theta (r_i) \cdot g_i, \end{equation*}$$

where every $r_i$ is a homogeneous nonzero element of $R$ and every $g_i$ is an element of the generating set $G$ for $M$. For every $r_i$, we have

$$\begin{equation*} \left|r_i\right| = \left|m\right| - \left|g_i\right| = \left|m'\right| - \left|g_i\right| > b - b_M = b_R + n. \end{equation*}$$

Thus $r_i$ lies in the ideal $R_{\ge n}$, so we can write it as a sum

$$\begin{equation*} r_i = \sum _j u_{i,j} \cdot a_{i,j}, \end{equation*}$$

where every $u_{i,j}$ is a homogeneous nonzero element of $R$, and every $a_{i,j}$ is an element of the generating set $A$ for $R_{\ge n}$. For every $u_{i,j}$, we have

$$\begin{equation*} \left|u_{i,j}\right| = \left|r_i\right| - \left|a_{i,j}\right| > (b_R + n) - b_R = n. \end{equation*}$$

Now we can write the element $m$ as

$$\begin{equation*} m = \sum _{i,j} \theta (u_{i,j} \cdot a_{i,j}) \cdot g_i = \sum _{i,j} \theta (u_{i,j}) \cdot \theta (a_{i,j}) \cdot g_i = \sum _{i,j} \theta (u_{i,j}) \cdot (a_{i,j} \cdot g_i). \end{equation*}$$

If we have $a_{i,j} \cdot g_i = 0$ for some terms in the sum, we ignore these terms. For every pair $(i,j)$, we have

$$\begin{equation*} \left|\theta (u_{i,j})\right| = \left|u_{i,j}\right| > n \qquad \text{and}\qquad \left|a_{i,j} \cdot g_i\right| \ge \left|a_{i,j}\right| \ge n. \end{equation*}$$

This means that when applying $\psi$ to a term in the above sum for $m$, we have

$$\begin{equation*} \psi (\theta (u_{i,j}) \cdot (a_{i,j} \cdot g_i)) = \psi (\theta (u_{i,j})) \cdot \psi (a_{i,j} \cdot g_i). \end{equation*}$$

Using this, we can write our element $m'$ of $M'$ in the following way:

$$\begin{align} m' = \psi (m) & = \psi \Big ( \sum _{i,j} \theta (u_{i,j}) \cdot (a_{i,j} \cdot g_i) \Big )\\ & = \sum _{i,j} \psi (\theta (u_{i,j})) \cdot \psi (a_{i,j} \cdot g_i)\\ & = \sum _{i,j} \theta '(\phi (u_{i,j})) \cdot \psi (a_{i,j} \cdot g_i). \end{align}$$

For every pair $(i,j)$, we have

$$\begin{equation*} \left|\psi (a_{i,j} \cdot g_i)\right| = \left|a_{i,j} \cdot g_i\right| = \left|a_{i,j}\right| + \left|g_i\right| \le b_R + b_M \le b, \end{equation*}$$

so $\psi (a_{i,j} \cdot g_i)$ lies in the module $N'$ generated by $G'$. Thus $m'$ also lies in $N'$. Since every homogeneous element of $M'$ lies in $N'$, we have $M' = N'$, and hence $M'$ is finitely generated.

■

Finally, we apply the above result to the rings which are involved in the $\mathsf{Fg}$ condition, and obtain the main result of this section.

Proposition 6.4.

Let $\Lambda$ and $\Gamma$ be artin algebras over a commutative ring $k$, and assume that they are flat as $k$-modules. Let $M$ and $M'$ be $\Lambda$-modules, and let $N$ and $N'$ be $\Gamma$-modules, such that $M \cong \Lambda /(\operatorname {rad}\Lambda )$ and $N' \cong \Gamma /(\operatorname {rad}\Gamma )$. Let $n$ be some nonnegative integer, and assume that there are graded rng isomorphisms $f$, $g$, $f'$ and $g'$ making the following two diagrams commute:

$$\begin{equation*} \vcenter{ \vcenter{\img[][123pt][67pt][{$\xymatrix{ {\HH{\ge n}(\Lambda)} \ar[r]^-{\varphi_M^{\ge n}} \ar[d]_{f}^{\iso} & {\Ext_\Lambda^{\ge n}(M, M)} \ar[d]_{g}^{\iso} \\ {\HH{\ge n}(\Gamma)} \ar[r]_-{\varphi_{N}^{\ge n}} & {\Ext_{\Gamma}^{\ge n}(N, N)} }$}]{Images/img497d971109de1e371eddcf5887470f55.svg}}} \qquad \text{and}\qquad \vcenter{ \vcenter{\img[][129pt][69pt][{$\xymatrix{ {\HH{\ge n}(\Lambda)} \ar[r]^-{\varphi_{M'}^{\ge n}} \ar[d]_{f'}^{\iso} & {\Ext_\Lambda^{\ge n}(M', M')} \ar[d]_{g'}^{\iso} \\ {\HH{\ge n}(\Gamma)} \ar[r]_-{\varphi_{N'}^{\ge n}} & {\Ext_{\Gamma}^{\ge n}(N', N')} }$}]{Images/img8480a1f97a8a98ae6b2429b9aed2d563.svg}}} \end{equation*}$$

Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $\Gamma$ satisfies $\mathsf{Fg}$.

Proof.

We first check that the conditions on the graded rings in Proposition 6.3 are satisfied in this case. For every degree $i$, we have that $\operatorname {HH}^{i}(\Lambda )$, $\operatorname {Ext}_\Lambda ^i(M,M)$ and $\operatorname {Ext}_\Lambda ^i(M',M')$ are finitely generated as $k$-modules. Therefore, they are also finitely generated as $\operatorname {HH}^{0}(\Lambda )$-modules. The ring $\operatorname {HH}^{0}(\Lambda )$ is noetherian since it is an artin algebra. Similarly, we see that $\operatorname {HH}^{i}(\Gamma )$, $\operatorname {Ext}_\Gamma ^i(N,N)$ and $\operatorname {Ext}_\Gamma ^i(N',N')$ are finitely generated $\operatorname {HH}^{0}(\Gamma )$-modules, and that the ring $\operatorname {HH}^{0}(\Gamma )$ is noetherian.

Assume that $\Lambda$ satisfies $\mathsf{Fg}$. Then $\operatorname {HH}^{*}(\Lambda )$ is noetherian, and by Theorem 2.7, $\operatorname {Ext}_\Lambda ^*(M',M')$ is a finitely generated $\operatorname {HH}^{*}(\Lambda )$-module. By applying Proposition 6.3 to the commutative diagram with $f'$ and $g'$, we see that $\Gamma$ satisfies $\mathsf{Fg}$.

The opposite inclusion is proved in the same way by using the other commutative diagram.

■

7. Finite generation of cohomology rings in module recollements

We now investigate the relationship between the $\mathsf{Fg}$ condition (see Definition 2.6) for an algebra $\Lambda$ and the algebra $a{\Lambda }a$, where $a$ is an idempotent of $\Lambda$. We show that, given some conditions on the idempotent $a$, the algebra $\Lambda$ satisfies $\mathsf{Fg}$ if and only if the algebra $a{\Lambda }a$ satisfies $\mathsf{Fg}$. We prove this result only for finite dimensional algebras over a field, and not more general artin algebras.

Throughout this section, we let $k$ be a field, $\Lambda$ a finite dimensional $k$-algebra and $a$ an idempotent in $\Lambda$. We denote by $e$ and $E$ the exact functors

$$\begin{align*} e = (a-) &\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a, \\ E = (a-a) &\colon \operatorname {mod}\Lambda ^{\operatorname {e}} \longrightarrow \operatorname {mod}(a{\Lambda }a)^{\operatorname {e}}. \end{align*}$$

These functors fit into the recollements described in Example 2.3.

For a $\Lambda$-module $M$, we can construct the diagram

$$\begin{equation*} \vcenter{\img[][172pt][59pt][{$\xymatrix@C=4em{ {\HH*(\Lambda)} \ar[r]^-{\varphi_M} \ar[d]_{E_{\Lambda,\Lambda}^*} & {\Ext_\Lambda^*(M,M)} \ar[d]^{e_{M,M}^*} \\ {\HH*(a{\Lambda}a)} \ar[r]_-{\varphi_{e(M)}} & {\Ext_{a{\Lambda}a}^*(e(M), e(M))} }$}]{Images/img9a019c21f7ce209efe37bcdaa8314be2.svg}} \end{equation*}$$

where the maps $\varphi _M$ and $\varphi _{e(M)}$ are defined in Subsection 2.2, and the maps $E_{\Lambda ,\Lambda }^*$ and $e_{M,M}^*$ are defined in Section 3. We show that this diagram commutes, and that under certain conditions on $a$, the vertical maps are isomorphisms in almost all degrees. We then use Proposition 6.4 to show that $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

Let us consider what kind of conditions we need to put on the choice of the idempotent $a$. From Corollary 3.12, we know that the map $e_{M,M}^*$ in the above diagram is an isomorphism in all but finitely many degrees if the two dimensions

$$\begin{equation*} \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \qquad \text{and}\qquad \operatorname {\mathsf{pd}}_{a{\Lambda }a} (a\Lambda ) \end{equation*}$$

are finite, or, equivalently, that the two dimensions

$$\begin{equation*} \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \qquad \text{and}\qquad \operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}} ({\Lambda }a) \end{equation*}$$

are finite. We show (given an additional technical assumption about the algebra $\Lambda$) that this is in fact also sufficient for the map $E_{\Lambda ,\Lambda }^*$ to be an isomorphism in all but finitely many degrees.

This section is structured as follows. The first part considers the commutativity of the above diagram, concluding with Proposition 7.2. The second part considers when the map $E_{\Lambda ,\Lambda }^*$ is an isomorphism in high degrees, concluding with Proposition 7.9. Finally, the main result of this section is stated as Theorem 7.10.

We now show that the above diagram is commutative. The maps $\varphi _M$ and $\varphi _{e(M)}$ are defined by using tensor functors. It is convenient to have short names for these functors. For every $\Lambda$-module $M$, we define $t_M$ and $T_M$ to be the tensor functors

$$\begin{align*} t_M = (- \otimes _\Lambda M) &\colon \operatorname {mod}\Lambda ^{\operatorname {e}} \longrightarrow \operatorname {mod}\Lambda , \\ T_M = (- \otimes _{a{\Lambda }a} aM) &\colon \operatorname {mod}(a{\Lambda }a)^{\operatorname {e}} \longrightarrow \operatorname {mod}a{\Lambda }a. \end{align*}$$

Together with the functors $e$ and $E$ from above, these functors fit into the following diagram of categories and functors:

$$\begin{equation*} \vcenter{\img[][153pt][57pt][{$\xymatrix@C=6em{ {\fmod\envalg{\Lambda}} \ar[r]^{t_M} \ar[d]_{E} & {\fmod\Lambda} \ar[d]^{e} \\ {\fmod\envalg{(a \Lambda a)}} \ar[r]_{T_M } & {\fmod a \Lambda a} }$}]{Images/img42cd8c651875b552287c070183bc6ffe.svg}} \end{equation*}$$

We begin by showing that the two possible compositions of maps from upper left to lower right in this diagram are related by a natural transformation.

Lemma 7.1.

For every $\Lambda$-module $M$, there is a natural transformation $\tau ^M \colon T_M \circ E \longrightarrow e \circ t_M$.

Proof.

Note that we have

$$\begin{equation*} T_M E(N) = aNa \otimes _{a{\Lambda }a} aM \qquad \text{and}\qquad e t_M(N) = aN \otimes _\Lambda M \end{equation*}$$

for every $\Lambda ^{\operatorname {e}}$-module $N$. We define the maps $\tau ^M_N$ of the natural transformation $\tau ^M$ by

$$\begin{equation*} \tau ^M_N(n \otimes m) = n \otimes m \end{equation*}$$

for an element $n \otimes m$ of $T_M E(N)$. This gives well defined maps since $a\Lambda a \subseteq \Lambda$. It is easy to check that the compositions $et_M(f) \circ \tau ^M_N$ and $\tau ^M_{N'} \circ T_M E(f)$ are equal for a homomorphism $f\colon N \longrightarrow N'$ of $\Lambda ^{\operatorname {e}}$-modules, so $\tau ^M$ is a natural transformation.

■

We are now able to show that the diagrams we consider are commutative.

Proposition 7.2.

For any $\Lambda$-module $M$, the following diagram of graded rings commutes:

$$\begin{equation*} \vcenter{\img[][172pt][59pt][{$\xymatrix@C=4em{ {\HH*(\Lambda)} \ar[r]^-{\varphi_M} \ar[d]_{E_{\Lambda,\Lambda}^*} & {\Ext_\Lambda^*(M,M)} \ar[d]^{e_{M,M}^*} \\ {\HH*(a{\Lambda}a)} \ar[r]_-{\varphi_{e(M)}} & {\Ext_{a{\Lambda}a}^*(e(M), e(M))} }$}]{Images/img9a019c21f7ce209efe37bcdaa8314be2.svg}} \end{equation*}$$

Proof.

We show that the result holds in the positive degrees of the graded rings and graded ring homomorphisms in the diagram. Showing that it also holds in degree zero can be done in a similar way, by looking at elements given by homomorphisms instead of extensions.

Let $\mu$ and $\nu$ be the natural isomorphisms

$$\begin{equation*} \mu \colon \Lambda \otimes _\Lambda M \longrightarrow M \qquad \text{and}\qquad \nu \colon a{\Lambda }a \otimes _{a{\Lambda }a} e(M) \longrightarrow e(M) \end{equation*}$$

given by multiplication.

Consider, for some positive integer $i$, an element $[\eta ] \in \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^i(\Lambda ,\Lambda )$ which is represented by the exact sequence

$$\begin{equation*} \eta \colon \quad 0 \longrightarrow \Lambda \longrightarrow X \longrightarrow P_{i-2} \longrightarrow \cdots \longrightarrow P_0 \longrightarrow \Lambda \longrightarrow 0, \end{equation*}$$

where each $P_j$ is a projective $\Lambda ^{\operatorname {e}}$-module. We apply the compositions of maps $\varphi _{e(M)} \circ E_{\Lambda ,\Lambda }^*$ and $e_{M,M}^* \circ \varphi _M$ to $[\eta ]$, and show that we get the same result in both cases.

We first consider the map $\varphi _{e(M)} \circ E_{\Lambda ,\Lambda }^*$. If we apply the functor $E$ to $\eta$, then we get the exact sequence

$$\begin{equation*} E(\eta )\colon 0 \longrightarrow E(\Lambda ) \longrightarrow E(X) \longrightarrow E(P_{i-2}) \longrightarrow \cdots \longrightarrow E(P_0) \longrightarrow E(\Lambda ) \longrightarrow 0 \end{equation*}$$

of $(a \Lambda a)^{\operatorname {e}}$-modules, and we have that $E_{\Lambda ,\Lambda }^*([\eta ]) = [E(\eta )]$. Since the objects $E(P_j)$ are not necessarily projective, we may need to find a different representative of the element $[E(\eta )]$ in order to apply the map $\varphi _{e(M)}$. We construct the following commutative diagram with exact rows, where each $Q_j$ is a projective $(a \Lambda a)^{\operatorname {e}}$-module and the bottom row is $E(\eta )$:

$$\begin{equation*} \vcenter{\img[][279pt][49pt][{$\xymatrix@C=1.5em{ 0 \ar[r] & a{\Lambda}a \ar[r] \ar@{=}[d] & Y \ar[r] \ar[d]_{f_{i-1}} & Q_{i-2} \ar[r] \ar[d]_{f_{i-2}} & \cdots\ar[r] & Q_0 \ar[r] \ar[d]_{f_0} & a{\Lambda}a \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & E(\Lambda) \ar[r] & E(X) \ar[r] & E(P_{i-2}) \ar[r] & \cdots\ar[r] & E(P_0) \ar[r] & E(\Lambda) \ar[r] & 0 }$}]{Images/img1f597f718f301dfa104815e39d1c75ea.svg}} \end{equation*}$$

Note that both rows represent the same element in $\operatorname {Ext}_{(a \Lambda a)^{\operatorname {e}}}^i(a \Lambda a, a \Lambda a)$. Applying the functor $T_M$ to this diagram gives the two lower rows in the following commutative diagram of $a \Lambda a$-modules, where the two upper rows are exact:

$$\begin{equation*} \vcenter{\img[][333pt][89pt][{$\xymatrix@C=1.2em{ 0 \ar[r] & e(M) \ar[r] \ar[d]_{\nu^{-1}}^{\iso} & T_M(Y) \ar[r] \ar@{=}[d] & T_M(Q_{i-2}) \ar[r] \ar@{=}[d] & \cdots\ar[r] & T_M(Q_0) \ar[r] \ar@{=}[d] & e(M) \ar[r] \ar[d]_{\nu^{-1}}^{\iso} & 0 \\ 0 \ar[r] & T_M(a{\Lambda}a) \ar[r] \ar@{=}[d] & T_M(Y) \ar[r] \ar[d]_{T_M(f_{i-1})} & T_M(Q_{i-2}) \ar[r] \ar[d]_{T_M(f_{i-2})} & \cdots\ar[r] & T_M(Q_0) \ar[r] \ar[d]_{T_M(f_0)} & T_M(a{\Lambda}a) \ar[r] \ar@{=}[d] & 0 \\ & T_M E(\Lambda) \ar[r] & T_M E(X) \ar[r] & T_M E(P_{i-2}) \ar[r] & \cdots\ar[r] & T_M E(P_0) \ar[r] & T_M E(\Lambda) & }$}]{Images/img1d919545176bb7f7feb2f0543b3dad9c.svg}} \end{equation*}$$

The top row in this diagram is a representative for the element $(\varphi _{e(M)} \circ E_{\Lambda ,\Lambda }^*)([\eta ])$.

We now consider the map $e_{M,M}^* \circ \varphi _M$. Applying the functor $e \circ t_M$ to the exact sequence $\eta$ gives the top row in the following commutative diagram of $a \Lambda a$-modules with exact rows, where the bottom row is a representative of the element $(e_{M,M}^* \circ \varphi _M)([\eta ])$:

$$\begin{equation*} \vcenter{\img[][324pt][49pt][{$\xymatrix@C=1.5em{ 0 \ar[r] & e t_M(\Lambda) \ar[r] \ar[d]_{e(\mu)}^{\iso} & e t_M(X) \ar[r] \ar@{=}[d] & e t_M(P_{i-2}) \ar[r] \ar@{=}[d] & \cdots\ar[r] & e t_M(P_0) \ar[r] \ar@{=}[d] & e t_M(\Lambda) \ar[r] \ar[d]_{e(\mu)}^{\iso} & 0 \\ 0 \ar[r] & e(M) \ar[r] & e t_M(X) \ar[r] & e t_M(P_{i-2}) \ar[r] & \cdots\ar[r] & e t_M(P_0) \ar[r] & e(M) \ar[r] & 0 \\ }$}]{Images/img97f44b9847466bebbef300e9112c1d3b.svg}} \end{equation*}$$

Finally, we use the natural transformation $\tau ^M$ from Lemma 7.1 to combine the two above diagrams into the following commutative diagram of $a \Lambda a$-modules:

$$\begin{equation*} \vcenter{\img[][330pt][168pt][{$\xymatrix@C=1.2em{ 0 \ar[r] & e(M) \ar[r] \ar[d]_{\nu^{-1}}^{\iso} & T_M(Y) \ar[r] \ar@{=}[d] & T_M(Q_{i-2}) \ar[r] \ar@{=}[d] & \!\cdots\! \ar[r] & T_M(Q_0) \ar[r] \ar@{=}[d] & e(M) \ar[r] \ar[d]_{\nu^{-1}}^{\iso} & 0 \\ 0 \ar[r] & T_M(a{\Lambda}a) \ar[r] \ar@{=}[d] & T_M(Y) \ar[r] \ar[d]_{T_M(f_{i-1})} & T_M(Q_{i-2}) \ar[r] \ar[d]_{T_M(f_{i-2})} & \!\cdots\! \ar[r] & T_M(Q_0) \ar[r] \ar[d]_{T_M(f_0)} & T_M(a{\Lambda}a) \ar[r] \ar@{=}[d] & 0 \\ & T_M E(\Lambda) \ar[r] \ar[d]_{\tau^M_\Lambda} & T_M E(X) \ar[r] \ar[d]_{\tau^M_X} & T_M E(P_{i-2}) \ar[r] \ar[d]_{\tau^M_{P_{i-2}}} & \!\cdots\! \ar[r] & T_M E(P_0) \ar[r] \ar[d]_{\tau^M_{P_0}} & T_M E(\Lambda) \ar[d]_{\tau^M_\Lambda} & \\ 0 \ar[r] & e t_M(\Lambda) \ar[r] \ar[d]_{e(\mu)}^{\iso} & e t_M(X) \ar[r] \ar@{=}[d] & e t_M(P_{i-2}) \ar[r] \ar@{=}[d] & \!\cdots\! \ar[r] & e t_M(P_0) \ar[r] \ar@{=}[d] & e t_M(\Lambda) \ar[r] \ar[d]_{e(\mu)}^{\iso} & 0 \\ 0 \ar[r] & e(M) \ar[r] & e t_M(X) \ar[r] & e t_M(P_{i-2}) \ar[r] & \!\cdots\! \ar[r] & e t_M(P_0) \ar[r] & e(M) \ar[r] & 0 \\ }$}]{Images/imge29224b69403f2be0b656b3aa204d5e2.svg}} \end{equation*}$$

It is easy to check that the composition of maps along the leftmost column is the identity map on $e(M)$, and the same holds for the composition of maps along the rightmost column. Thus the top and bottom rows in this diagram represent the same element in $\operatorname {Ext}_{a \Lambda a}^i(e(M),e(M))$. Since the top row is a representative of the element $(\varphi _{e(M)} \circ E_{\Lambda ,\Lambda }^*)([\eta ])$ and the bottom row is a representative of the element $(e_{M,M}^* \circ \varphi _M)([\eta ])$, this means that $\varphi _{e(M)} \circ E_{\Lambda ,\Lambda }^* = e_{M,M}^* \circ \varphi _M$.

■

We thank the referee for pointing out an alternative proof of Proposition 7.2, which we sketch next.

Alternative proof of Proposition 7.2.

Consider the following diagram of functors between derived categories:

$$\begin{equation*} \vcenter{\img[][203pt][57pt][{$\xymatrix{ \Dminus(\fmod\envalg{\Lambda}) \ar[rr]^{-\otimes^{\mathbb{L}}_\Lambda M} \ar[d]_{(a\otimes a)\envalg{\Lambda}\otimes_{\envalg{\Lambda}}-} & & \Dminus(\fmod\Lambda)\ar[d]^{a\Lambda\otimes_\Lambda-}\\ \Dminus(\fmod\envalg{a\Lambda a}) \ar[rr]^{-\otimes^{\mathbb{L}}_{a\Lambda a} aM} & & \Dminus(\fmod a\Lambda a) }$}]{Images/imgbf1a8d5580a8ae58d69792983d98e753.svg}} \end{equation*}$$

Proposition 7.2 really amounts, starting with a morphism $\eta \colon \Lambda \to \Lambda [n]$ in

$$\begin{equation*} \operatorname {Hom}_{\mathsf{D}^{-}(\operatorname {mod}\Lambda ^{\operatorname {e}})}(\Lambda ,\Lambda [n])\cong \operatorname {HH}^{n}(\Lambda ) \end{equation*}$$

with $n\geqslant 0$, to showing that the images with the two different compositions in the above diagram coincide in

$$\begin{equation*} \operatorname {Hom}_{\mathsf{D}^{-}(\operatorname {mod}a\Lambda a)}(e(M),e(M)[n])\cong \operatorname {Ext}^n_{a\Lambda a}(e(M),e(M)). \end{equation*}$$

Let $\eta \colon \Lambda \to \Lambda [n]$ be a morphism in $\mathsf{D}^{-}(\operatorname {mod}\Lambda ^{\operatorname {e}})$, which we represent as

$$\begin{equation*} \vcenter{\img[][94pt][33pt][{$\xymatrix@R=10pt{ & \mathbb{P}\ar[dl]_s\ar[dr]^f & \\ \Lambda& & \Lambda[n] }$}]{Images/img783a64568dbb75f883566e684b940622.svg}} \end{equation*}$$

where $s\colon \mathbb{P}\to \Lambda$ is a projective resolution of $\Lambda$ over $\Lambda ^{\operatorname {e}}$.

The image of the composition via the upper right corner is

$$\begin{equation*} \vcenter{\img[][271pt][35pt][{$\xymatrix@R=10pt{ & a\Lambda\otimes_\Lambda\mathbb{P}\otimes_\Lambda M\ar[dl]_{1\otimes s\otimes1}\ar[dr]^{1\otimes f\otimes1} & \\ a\Lambda\otimes_\Lambda\Lambda\otimes_\Lambda M & & a\Lambda\otimes_\Lambda\Lambda[n]\otimes_\Lambda\Lambda}$}]{Images/imgce6555e3f9e09eaf9a77f328e94378a5.svg}} \end{equation*}$$

since $\mathbb{P}\to \Lambda \to 0$ as a sequence of right $\Lambda$-modules splits.

The image of the composition via the lower left corner we compute in two steps. The image of the first functor is:

$$\begin{equation*} \vcenter{\img[][127pt][32pt][{$\xymatrix@R=10pt{ a\Lambda a& & a\Lambda[n] a\\ & a\mathbb{P}a\ar[ul]^{asa}\ar[ur]_{afa} & }$}]{Images/img13e441d21577992d135bdcb380b9cbab.svg}} \end{equation*}$$

Let $\pi \colon \mathbb{P}_{a\Lambda a}\to a\Lambda a\to 0$ be a projective resolution of $a\Lambda a$ as a $a\Lambda a^{\operatorname {e}}$-module. Since $asa \colon a\mathbb{P}a\to a\Lambda a$ is a quasi-isomorphism, there exists a homomorphism of chain complexes $\varphi \colon \mathbb{P}_{a\Lambda a}\to a\mathbb{P}a$ such that we have the following commutative diagram:

$$\begin{equation*} \vcenter{\img[][142pt][54pt][{$\xymatrix@C=30pt{ a\Lambda a & a\mathbb{P}a\ar[l]_{asa}\ar[r]^{afa} & a\Lambda[n]a \\ & \mathbb{P}_{a\Lambda a}\ar[lu]^{\pi}\ar[ru]_{afa\varphi}\ar[u]^\varphi& }$}]{Images/imgc9e4e3c22d5f9793902f89cf67c46a9d.svg}} \end{equation*}$$

The image of the functor $-\otimes ^{\mathbb{L}}_{a\Lambda a} aM$ is computed applying $-\otimes _{a\Lambda a} aM$ to the lower roof in the above diagram, and we obtain the following diagram:

$$\begin{equation*} \vcenter{\img[][279pt][54pt][{$\xymatrix@C=35pt{ a\Lambda a\otimes_{a\Lambda a} aM & a\mathbb{P}a\otimes_{a\Lambda a} aM\ar[l]_{asa\otimes1}\ar[r]^{afa\otimes1} & a\Lambda[n]a\otimes_{a\Lambda a} aM\\ & \mathbb{P}_{a\Lambda a}\otimes_{a\Lambda a} aM\ar[lu]^{\pi\otimes1}\ar[ru]_{afa\varphi\otimes1}\ar[u]^{\varphi\otimes1} & }$}]{Images/imgc6d891fde9d7207b2d0af5cf69d3eca2.svg}} \end{equation*}$$

From this we can construct the following commutative diagram:

$$\begin{equation*} \vcenter{\img[][295pt][127pt][{$\xymatrix@C=35pt{ & a\Lambda\otimes_\Lambda\mathbb{P}\otimes_\Lambda M\ar[dl]_{1\otimes s\otimes1}\ar[dr]^{1\otimes f\otimes1} & \\ a\Lambda\otimes_\Lambda\Lambda\otimes_\Lambda M & & a\Lambda\otimes_\Lambda\Lambda[n]\otimes_\Lambda M\\ a\Lambda a\otimes_{a\Lambda a} aM \ar[u]^{\tau^M_\Lambda} & a\mathbb{P}a\otimes_{a\Lambda a} aM\ar[l]_{asa\otimes1}\ar[r]^{afa\otimes1}\ar[uu]^{\tau^M_{\mathbb{P}[n]}} & a\Lambda[n]a\otimes_{a\Lambda a} aM\ar[u]^{\tau^M_{\Lambda[n]}}\\ & \mathbb{P}_{a\Lambda a}\otimes_{a\Lambda a} aM\ar[lu]^{\pi\otimes1}\ar[ru]_{afa\varphi\otimes1}\ar[u]^{\varphi\otimes1} & }$}]{Images/img8fc1e55c4bdf61f9a630095f820c1a6e.svg}} \end{equation*}$$

where the extreme vertical morphisms are isomorphisms and $asa\otimes 1\colon a\mathbb{P}a\otimes _{a\Lambda a} aM\to a\Lambda a\otimes _{a\Lambda a} aM$ is a quasi-isomorphism (since $a\Lambda \otimes _\Lambda \mathbb{P}\to a\Lambda \otimes _\Lambda \Lambda \to 0$ as a sequence of right $\Lambda$-modules splits). This shows that the two images of $\eta \colon \Lambda \to \Lambda [n]$ in $\operatorname {Hom}_{\mathsf{D}^{-}(\operatorname {mod}\Lambda ^{\operatorname {e}})}(\Lambda ,\Lambda [n])$ are the same.

■

Having shown that our diagrams are commutative, we now move on to describing when the map $E_{\Lambda ,\Lambda }^*$ is an isomorphism in almost all degrees. For this, we use Corollary 3.11 (i) on the algebras $\Lambda ^{\operatorname {e}}$ and $(a\otimes a^{\operatorname {op}})\Lambda ^{\operatorname {e}}(a\otimes a^{\operatorname {op}})$ and the $\Lambda ^{\operatorname {e}}$-module $\Lambda$. We let $\varepsilon$ denote the element $a\otimes a^{\operatorname {op}}$ of $\Lambda ^{\operatorname {e}}$, so that we can write the algebra $(a\otimes a^{\operatorname {op}})\Lambda ^{\operatorname {e}}(a\otimes a^{\operatorname {op}})$ more simply as $\varepsilon \Lambda ^{\operatorname {e}}\varepsilon$. Note that Corollary 3.11 uses a recollement situation; in this case, the recollement is like the one in Example 2.3 (ii).

In order to use Corollary 3.11 (i) in this situation, we need to show the following:

$$\begin{equation*} \operatorname {\mathsf{pd}}_{\varepsilon \Lambda ^{\operatorname {e}}\varepsilon } \varepsilon \Lambda ^{\operatorname {e}} < \infty \qquad \text{and}\qquad \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j \Big ( \Lambda , \frac{\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle }{\operatorname {rad}\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle } \Big ) = 0 \quad \text{for $j \gg 0$.} \end{equation*}$$

We show the first of these conditions in Lemma 7.4, and the second one in Lemma 7.8 (here we need an additional technical assumption on $\Lambda$ to be able to describe the simple modules over $\Lambda ^{\operatorname {e}}$), and finally tie it together in Proposition 7.9, where we show that $E_{\Lambda ,\Lambda }^*$ is an isomorphism in sufficiently high degrees.

First, we show how the projective dimension of the tensor product $M \otimes _k N$ is related to the projective dimensions of $M$ and $N$, when $M$ and $N$ are modules over $k$-algebras. In particular, the following result implies that if a left and a right $\Lambda$-module ${}_\Lambda M$ and $N_\Lambda$ both have finite projective dimension, then their tensor product $M \otimes _k N$ has finite projective dimension as a $\Lambda ^{\operatorname {e}}$-module.

Lemma 7.3.

Let $\Sigma$ and $\Gamma$ be $k$-algebras, and let $M$ be a $\Sigma$-module and $N$ a $\Gamma$-module. If $M$ has finite projective dimension as a $\Sigma$-module and $N$ has finite projective dimension as a $\Gamma$-module, then $M \otimes _k N$ has finite projective dimension as a $(\Sigma \otimes _k \Gamma )$-module, and

$$\begin{equation*} \operatorname {\mathsf{pd}}_{\Sigma \otimes _k \Gamma }(M \otimes _k N) \le \operatorname {\mathsf{pd}}_\Sigma {M} + \operatorname {\mathsf{pd}}_\Gamma {N}. \end{equation*}$$

Proof.

Assume that $\operatorname {\mathsf{pd}}_{\Sigma }M=m$ and $\operatorname {\mathsf{pd}}_{\Gamma }N=n$. Then we have finite projective resolutions

$$\begin{equation*} 0 \to P_m \xrightarrow {} \cdots \xrightarrow {} P_0 \to M \to 0 \qquad \text{and}\qquad 0 \to Q_n \xrightarrow {} \cdots \xrightarrow {} Q_0 \to N \to 0 \end{equation*}$$

of $M$ and $N$, respectively. Let $P$ and $Q$ denote the corresponding deleted resolutions. Consider the tensor product

$$\begin{multline} P \otimes _k Q \colon \cdots \xrightarrow {} (P_0 \otimes _k Q_2) \oplus (P_1 \otimes _k Q_1) \oplus (P_2 \otimes _k Q_0) \\ \xrightarrow {} (P_0 \otimes _k Q_1) \oplus (P_1 \otimes _k Q_0) \xrightarrow {} P_0 \otimes _k Q_0 \to 0 \end{multline}$$

of the complexes $P$ and $Q$. This is a bounded complex of projective ($\Sigma \otimes _k \Gamma$)-modules. We want to show that it is in fact a deleted projective resolution of the $(\Sigma \otimes _k \Gamma )$-module $M \otimes _k N$, which completes the proof.

We need to show that the complex $P \otimes _k Q$ is exact in all positive degrees and has homology $M \otimes _k N$ in degree zero. Let us temporarily forget the $\Sigma$- and $\Gamma$-structures, and view $P$ as a complex of right $k$-modules, $Q$ as a complex of left $k$-modules, and $P \otimes _k Q$ as a complex of abelian groups. Then by the Künneth formula for homology (see 57, Corollary 11.29), we have an isomorphism

$$\begin{equation*} \alpha \colon \bigoplus _{i+j=n} H_i(P) \otimes _k H_j(Q) \xrightarrow {\cong } H_n(P \otimes _k Q) \end{equation*}$$

of abelian groups, given by $\alpha ([p] \otimes [q]) = [p \otimes q]$, for $p \in P_i$ and $q \in Q_j$. Observe that $\alpha$ preserves a $(\Sigma \otimes _k \Gamma )$-module structure. Thus, $\alpha$ is a $(\Sigma \otimes _k \Gamma )$-module isomorphism, and we get

$$\begin{equation*} H_n(P \otimes _k Q) \cong \bigoplus _{i+j=n} H_i(P) \otimes _k H_j(Q) \cong \left\{ \begin{array}{ll} M \otimes _k N & \text{if $n = 0$,} \\0 & \text{if $n > 0$.} \end{array} \right. \end{equation*}$$

This means that the complex $P \otimes _k Q$ is a deleted projective resolution of the $(\Sigma \otimes _k \Gamma )$-module $M \otimes _k N$. Since the complex $P \otimes _k Q$ is zero in all degrees above $m+n$, we get

$$\begin{equation*} \operatorname {\mathsf{pd}}_{\Sigma \otimes _k \Gamma } (M \otimes _k N) \le m + n = \operatorname {\mathsf{pd}}_\Sigma {M} + \operatorname {\mathsf{pd}}_\Gamma {N}, \end{equation*}$$

and the proof is complete.

■

Using the above result, we find that the assumptions we make about the left and right $a{\Lambda }a$-modules $a\Lambda$ and ${\Lambda }a$ having finite projective dimension imply the first condition we need for applying Corollary 3.11 (i), namely that the $\varepsilon \Lambda ^{\operatorname {e}}\varepsilon$-module $\varepsilon \Lambda ^{\operatorname {e}}$ has finite projective dimension. We state this as the following result.

Lemma 7.4.

We have the following inequality:

$$\begin{equation*} \operatorname {\mathsf{pd}}_{\varepsilon \Lambda ^{\operatorname {e}}\varepsilon }\varepsilon \Lambda ^{\operatorname {e}} \le \operatorname {\mathsf{pd}}_{a{\Lambda }a} a{\Lambda } + \operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}} {\Lambda }a. \end{equation*}$$

Proof.

Note that $\varepsilon \Lambda ^{\operatorname {e}}$ is isomorphic to $(a\Lambda \otimes _k\Lambda a)$ as left $(a\Lambda a)^{\operatorname {e}}$-modules and that the rings $(a{\Lambda }a)^{\operatorname {e}}$ and $\varepsilon \Lambda ^{\operatorname {e}}\varepsilon$ are isomorphic. By using these isomorphisms and Lemma 7.3, we get that

$$\begin{equation*} \operatorname {\mathsf{pd}}_{\varepsilon \Lambda ^{\operatorname {e}}\varepsilon }\varepsilon \Lambda ^{\operatorname {e}} = \operatorname {\mathsf{pd}}_{(a\Lambda a)^{\operatorname {e}}}\varepsilon \Lambda ^{\operatorname {e}} = \operatorname {\mathsf{pd}}_{(a\Lambda a)^{\operatorname {e}}}(a\Lambda \otimes _k \Lambda a) \leq \operatorname {\mathsf{pd}}_{a\Lambda a} a\Lambda +\operatorname {\mathsf{pd}}_{(a\Lambda a)^{\operatorname {op}}}\Lambda a. \end{equation*}$$■

Now we show how we get the second condition needed for applying Corollary 3.11 (i). We begin with a general result which relates extension groups over $\Lambda ^{\operatorname {e}}$ to extension groups over $\Lambda$.

Lemma 7.5.

Let $M$ and $N$ be $\Lambda$-modules. Let $D$ be the duality $\operatorname {Hom}_k(-,k)\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}\Lambda ^{\operatorname {op}}$. Then

$$\begin{equation*} \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j(\Lambda , M \otimes _k D(N)) \cong \operatorname {Ext}_\Lambda ^j(N, M) \end{equation*}$$

for every nonnegative integer $j$.

Proof.

This follows from 14, Corollary 4.4, Chapter IX by using the isomorphism $M \otimes _k D(N) \cong \operatorname {Hom}_k(N,M)$ of $\Lambda ^{\operatorname {e}}$-modules.

■

Furthermore, we need to be able to describe the simple $\Lambda ^{\operatorname {e}}$-modules in terms of simple $\Lambda$-modules. It is reasonable to expect that taking the tensor product

$$\begin{equation*} (\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}}) \end{equation*}$$

should produce all the simple $\Lambda ^{\operatorname {e}}$-modules. This is, however, not true for all finite dimensional algebras, as Example 7.7 shows. The following result describes when it is true.

Lemma 7.6.

We have an isomorphism

$$\begin{equation*} \Lambda ^{\operatorname {e}}/\operatorname {rad}\Lambda ^{\operatorname {e}} \cong (\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}}) \end{equation*}$$

of $\Lambda ^{\operatorname {e}}$-modules if and only if the $\Lambda ^{\operatorname {e}}$-module

$$\begin{equation*} (\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}}) \end{equation*}$$

is semisimple.

Proof.

It is easy to show that

$$\begin{equation*} (\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}}) \cong \frac{\Lambda ^{\operatorname {e}}}{\Lambda \otimes _k (\operatorname {rad}\Lambda ^{\operatorname {op}}) + (\operatorname {rad}\Lambda ) \otimes _k \Lambda ^{\operatorname {op}}} \end{equation*}$$

as $\Lambda ^{\operatorname {e}}$-modules, and that the ideal $\Lambda \otimes _k (\operatorname {rad}\Lambda ^{\operatorname {op}}) + (\operatorname {rad}\Lambda ) \otimes _k \Lambda ^{\operatorname {op}}$ of $\Lambda ^{\operatorname {e}}$ is nilpotent. This means that if $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module, then it is isomorphic to $\Lambda ^{\operatorname {e}}/\operatorname {rad}\Lambda ^{\operatorname {e}}$. The opposite implication is obvious.

■

Now we give an example showing that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is not necessarily semisimple for a finite dimensional algebra $\Lambda$ over a field $k$.

Example 7.7.

Let $k=\mathbb{Z}_2(x)$ be the field of rational functions in one indeterminant $x$ over $\mathbb{Z}_2$, and let $\Lambda$ be the $2$-dimensional $k$-algebra $k[y]/\langle y^2 - x \rangle$. Then $\Lambda$ is a field, so that $\operatorname {rad}\Lambda =(0)$. The element $\alpha = y\otimes 1 + 1\otimes y$ satisfies $\alpha ^2=0$. Hence $\langle \alpha \rangle$ is a nilpotent nonzero ideal in $\Lambda ^{\operatorname {e}}$, and therefore $\Lambda ^{\operatorname {e}}$ is not semisimple.

We assume that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is semisimple whenever we need it. In particular, this assumption is included in the main result at the end of this section. Note that this assumption is satisfied in many cases, for example if $\Lambda /\operatorname {rad}\Lambda$ is separable as a $k$-algebra (by 25, Corollary 7.8 (i)), if $k$ is algebraically closed (this can be shown by using the Wedderburn–Artin Theorem), or if $\Lambda$ is a quotient of a path algebra by an admissible ideal.

Now we can show how to get the second condition we need for applying Corollary 3.11 (i).

Lemma 7.8.

Assume that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module, and that we have

$$\begin{equation*} (\alpha ) \ \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty \qquad \text{and}\qquad (\gamma ) \ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty . \end{equation*}$$

Then

$$\begin{equation*} \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j \Big ( \Lambda , \frac{\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle }{\operatorname {rad}\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle } \Big ) = 0 \quad \text{for}\quad j > \max \Big \{ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ), \, \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \Big \}. \end{equation*}$$

Proof.

By Lemma 7.6, every simple $\Lambda ^{\operatorname {e}}$-module is a direct summand of a module of the form $S \otimes _k D(T)$ for some simple $\Lambda$-modules $S$ and $T$, where $D$ is the duality $\operatorname {Hom}_k(-,k)\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}\Lambda ^{\operatorname {op}}$. If neither of the modules $S$ or $T$ is annihilated by the ideal $\langle a \rangle$, then we have

$$\begin{equation*} \langle \varepsilon \rangle (S \otimes _k D(T)) = \langle a \otimes a^{\operatorname {op}} \rangle (S \otimes _k D(T)) = (\langle a \rangle S) \otimes _k D(\langle a \rangle T) = S \otimes _k D(T), \end{equation*}$$

which means that no nonzero direct summand of the $\Lambda ^{\operatorname {e}}$-module $S \otimes _k D(T)$ is a $\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle$-module.

Let $j$ be an integer such that

$$\begin{equation*} j > \max \Big \{ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ), \, \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \Big \}. \end{equation*}$$

In order to prove the result, it is sufficient to show that $\operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j(\Lambda , U) = 0$ for every simple $\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle$-module $U$. By the above reasoning, every such $U$ is a direct summand of a module $S \otimes _k D(T)$ for some simple $\Lambda$-modules $S$ and $T$, where at least one of $S$ and $T$ is annihilated by $\langle a \rangle$ and is thus a simple $\Lambda /\langle a \rangle$-module. Using Lemma 7.5, we get

$$\begin{equation*} \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j(\Lambda , S \otimes _k D(T)) \cong \operatorname {Ext}_\Lambda ^j(T, S) = 0, \end{equation*}$$

since we have $\operatorname {\mathsf{pd}}_\Lambda T < j$ or $\operatorname {\mathsf{id}}_\Lambda S < j$. It follows that $\operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j(\Lambda , U) = 0$.

■

The following result summarizes the above work and shows that, with the assumptions we have indicated for the algebra $\Lambda$ and the idempotent $a$, the functor $E$ gives isomorphisms $E_{\Lambda ,\Lambda }^j \colon \operatorname {HH}^{j}(\Lambda ) \longrightarrow \operatorname {HH}^{j}(a{\Lambda }a)$ in almost all degrees $j$.

Proposition 7.9.

Assume that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module, and that the functor $e$ is an eventually homological isomorphism. Then the map

$$\begin{equation*} E_{\Lambda ,M}^j \colon \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j(\Lambda , M) \longrightarrow \operatorname {Ext}_{(a{\Lambda }a)^{\operatorname {e}}}^j(E(\Lambda ), E(M)) \end{equation*}$$

is an isomorphism for every $\Lambda ^{\operatorname {e}}$-module $M$ and every integer $j$ such that

$$\begin{equation*} j > \max \Big \{ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ), \, \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \Big \} + \operatorname {\mathsf{pd}}_{a{\Lambda }a} a{\Lambda } + \operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}}{\Lambda }a + 1 < \infty . \end{equation*}$$

In particular, we have isomorphisms

$$\begin{equation*} \operatorname {HH}^{j}(\Lambda ) \cong \operatorname {HH}^{j}(a{\Lambda }a) \end{equation*}$$

for almost all degrees $j$.

Proof.

We use Corollary 3.11 (i) on the algebra $\Lambda ^{\operatorname {e}}$, the idempotent $\varepsilon = a \otimes a^{\operatorname {op}}$ and the $\Lambda ^{\operatorname {e}}$-module $\Lambda$. Let $m$ and $n$ be the integers

$$\begin{equation*} m = \max \Big \{ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ), \, \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \Big \} + 1 \end{equation*}$$

and

$$\begin{equation*} n = \operatorname {\mathsf{pd}}_{a{\Lambda }a} a{\Lambda } + \operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}}{\Lambda }a. \end{equation*}$$

Note that $m$ and $n$ are finite by Corollary 3.12. By Lemma 7.8, we have

$$\begin{equation*} \operatorname {Ext}_{\Lambda ^{\operatorname {e}}}^j \Big ( \Lambda , \frac{\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle }{\operatorname {rad}\Lambda ^{\operatorname {e}}/\langle \varepsilon \rangle } \Big ) = 0 \quad \text{for}\quad j \ge m, \end{equation*}$$

and by Lemma 7.4, we have

$$\begin{equation*} \operatorname {\mathsf{pd}}_{\varepsilon \Lambda ^{\operatorname {e}}\varepsilon }\varepsilon \Lambda ^{\operatorname {e}} \le n. \end{equation*}$$

Now the result follows from Corollary 3.11 (i) by noting that $(a{\Lambda }a)^{\operatorname {e}}$ is the same algebra as $\varepsilon \Lambda ^{\operatorname {e}}\varepsilon$ and that our functor $E = a-a$ is the same as the functor $\varepsilon -$ given by left multiplication with the idempotent $\varepsilon$.

■

Finally, we conclude this section by showing that the assumptions we have indicated imply that $\mathsf{Fg}$ holds for $\Lambda$ if and only if $\mathsf{Fg}$ holds for $a{\Lambda }a$. The following theorem is the main result of this section and constitutes the fourth part of the Main Theorem presented in the Introduction.

Theorem 7.10.

Let $\Lambda$ be a finite dimensional algebra over a field $k$, and let $a$ be an idempotent in $\Lambda$. Assume that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module, and that the functor $a-\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism. Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

Proof.

For every $\Lambda$-module $M$, we can make a diagram

$$\begin{equation*} \vcenter{\img[][172pt][59pt][{$\xymatrix@C=4em{ {\HH*(\Lambda)} \ar[r]^-{\varphi_M} \ar[d]_{E_{\Lambda,\Lambda}^*} & {\Ext_\Lambda^*(M,M)} \ar[d]^{e_{M,M}^*} \\ {\HH*(a{\Lambda}a)} \ar[r]_-{\varphi_{e(M)}} & {\Ext_{a{\Lambda}a}^*(e(M), e(M))} }$}]{Images/img9a019c21f7ce209efe37bcdaa8314be2.svg}} \end{equation*}$$

of graded rings and graded ring homomorphisms. This diagram commutes by Proposition 7.2, and the maps $E_{\Lambda ,\Lambda }^*$ and $e_{M,M}^*$ are isomorphisms in almost all degrees by Proposition 7.9 and Corollary 3.12, respectively.

Since we have such diagrams for every $\Lambda$-module $M$ and the functor $e$ is essentially surjective (see Proposition 2.2), we can make one diagram with $M = \Lambda /\operatorname {rad}\Lambda$ and another with $e(M) \cong a{\Lambda }a/\operatorname {rad}a{\Lambda }a$. Then, by Proposition 6.4, it follows that $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

■

8. Applications and examples

In this section we provide applications of our Main Theorem (stated in the Introduction), and examples illustrating its use. For ease of reference, we restate the Main Theorem here.

Theorem 8.1.

Let $\Lambda$ be an artin algebra over a commutative ring $k$ and let $a$ be an idempotent element of $\Lambda$. Let $e$ be the functor $a-\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{a\Lambda a}$ given by multiplication by $a$. Consider the following conditions:

$$\begin{align*} (\alpha ) \ \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) &< \infty & (\beta ) \ \operatorname {\mathsf{pd}}_{a{\Lambda }a} a{\Lambda } &< \infty \\ (\gamma ) \ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) &< \infty & (\delta ) \ \operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}}{\Lambda }a &< \infty \end{align*}$$

Then the following hold.

(i)

The following are equivalent:

(a)

$(\alpha )$ and $(\beta )$ hold.

(b)

$(\gamma )$ and $(\delta )$ hold.

(c)

The functor $e$ is an eventually homological isomorphism.

(ii)

The functor $a-\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{a\Lambda a}$ induces a singular equivalence between $\Lambda$ and $a\Lambda a$ if and only if conditions $(\beta )$ and $(\gamma )$ hold.

(iii)

Assume that $e$ is an eventually homological isomorphism. Then $\Lambda$ is Gorenstein if and only if $a{\Lambda }a$ is Gorenstein.

(iv)

Assume that $e$ is an eventually homological isomorphism, that $k$ is a field and that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module. Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

This section is divided into four subsections. In the first subsection, we discuss dependencies between conditions $(\alpha )$–$(\delta )$ in Theorem 8.1. In the second subsection, we consider some cases where these conditions are related. As a consequence, we find sufficient conditions, stated in terms of the quiver and relations, for applying Theorem 8.1 to a quotient of a path algebra. In the third subsection, we apply Theorem 8.1 to the class of triangular matrix algebras. In the last subsection, we compare our work to that of Nagase in 47.

8.1. Conditions in the Main Theorem

The first part of Theorem 8.1 says that if conditions $(\alpha )$ and $(\beta )$ hold, or conditions $(\gamma )$ and $(\delta )$ hold, then all four of these conditions hold, and the functor $e$ is an essentially homological isomorphism. We now give examples which show that assuming any other combination of two of the conditions is not sufficient to get the same conclusion.

Our first example shows that conditions $(\alpha )$ and $(\delta )$ do not imply conditions $(\beta )$ and $(\gamma )$.

Example 8.2.

Let $k$ be a field. Let the $k$-algebra $\Lambda = kQ/\langle \rho \rangle$ be given by the following quiver and relations:

$$\begin{equation*} Q\colon \vcenter{\img[][64pt][16pt][{$\xymatrix{ 1 \ar[r]^{\alpha} & 2 \ar@(ur,dr)^{\beta} }$}]{Images/img363896642e270b37fa80ba841dd6ad96.svg}} \qquad \qquad \rho = \{ \beta ^2, \beta \alpha \}. \end{equation*}$$

Let $a = e_2$. Then $(\Lambda /\langle a \rangle ) / (\operatorname {rad}\Lambda /\langle a \rangle ) \cong S_1$ as $\Lambda$-modules. We have $\operatorname {\mathsf{id}}_\Lambda S_1 = 0$ and $\operatorname {\mathsf{pd}}_\Lambda S_1 = \infty$, which means that $(\alpha )$ holds and $(\gamma )$ does not hold. Furthermore, we have $a{\Lambda }a \cong k[x]/\langle x^2 \rangle$ as algebras, $a\Lambda \cong a{\Lambda }a \oplus S$ as left $a{\Lambda }a$-modules, where $S$ is the simple $a{\Lambda }a$-module, and ${\Lambda }a \cong a{\Lambda }a$ as right $a{\Lambda }a$-modules. We therefore have $\operatorname {\mathsf{pd}}_{a{\Lambda }a} a\Lambda = \infty$ and $\operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}} {\Lambda }a = 0$, which means that $(\beta )$ does not hold, while $(\delta )$ holds.

By considering the opposite algebra of the algebra in the above example, we see that conditions $(\beta )$ and $(\gamma )$ do not imply conditions $(\alpha )$ and $(\delta )$.

The next example shows that conditions $(\alpha )$ and $(\gamma )$ do not imply conditions $(\beta )$ and $(\delta )$.

Example 8.3.

Let $k$ be a field. Let the $k$-algebra $\Lambda = kQ/\langle \rho \rangle$ be given by the following quiver and relations:

$$\begin{equation*} Q\colon \vcenter{\img[][40pt][29pt][{$\xymatrix{ {1} \ar@/^/[r]^\alpha& {2} \ar@/^/[l]^\beta}$}]{Images/img18a2c2934941f67362743c26c848ac75.svg}} \qquad \qquad \rho = \{ \alpha \beta \}. \end{equation*}$$

Let $a = e_1$. Let $S_2$ be the simple $\Lambda$-module associated to the vertex $2$. Conditions $(\alpha )$ and $(\gamma )$ are satisfied since $\operatorname {\mathsf{id}}_\Lambda S_2 = 2$ and $\operatorname {\mathsf{pd}}_\Lambda S_2 = 2$, respectively. Conditions $(\beta )$ and $(\delta )$ are not satisfied, since $\operatorname {\mathsf{pd}}_{a{\Lambda }a} a{\Lambda } = \infty$ and $\operatorname {\mathsf{pd}}_{(a{\Lambda }a)^{\operatorname {op}}} {\Lambda }a = \infty$.

Finally, the next example shows that conditions $(\beta )$ and $(\delta )$ do not imply conditions $(\alpha )$ and $(\gamma )$.

Example 8.4.

Let $\Lambda = k( \vcenter{\img[][64pt][28pt][{$\xymatrix{1\ar@(ul,dl)_\alpha\ar@<1ex>[r]^\beta& 2\ar@<1ex>[l]^\gamma}$}]{Images/imgffad615a7f99cba9e2920b35a131a939.svg}})/\langle \alpha ^2-\gamma \beta , \beta \gamma , \alpha \gamma , \beta \alpha \rangle$, which is Example 2.3 from 37. Let $a=e_2$. Then $a\Lambda a\cong k$, so both $(\beta )$ and $(\delta )$ are satisfied. However $\operatorname {\mathsf{id}}_\Lambda \Lambda /\langle a \rangle = \operatorname {\mathsf{id}}_\Lambda S_1 = \infty$ and $\operatorname {\mathsf{pd}}_\Lambda \Lambda /\langle a \rangle = \operatorname {\mathsf{pd}}_\Lambda S_1 = \infty$. Hence $(\alpha )$ and $(\gamma )$ are not satisfied.

We have now seen that in general, there are no more relations between conditions $(\alpha )$–$(\delta )$ than that described in Theorem 8.1. In the next subsection, we consider some special cases where there are more relations between the conditions.

8.2. Algebras with ordered simples

In this subsection, we apply Theorem 8.1 to cases where there exists a total order $\preceq$ of the simple $\Lambda /\langle a \rangle$-modules with the property that

$$\begin{equation} S \preceq S' \implies \operatorname {Ext}_\Lambda ^{>0}(S,S')=0 {\tag{8.1}} \end{equation}$$

for every pair $S$ and $S'$ of simple $\Lambda /\langle a \rangle$-modules. With this assumption, we show that we have the implications $(\alpha ) \implies (\delta )$ and $(\gamma ) \implies (\beta )$ between the conditions in Theorem 8.1. We then consider some special cases where such orderings appear.

We need the following preliminary results.

Lemma 8.5.

Let $\Lambda$ be an artin algebra, let $M$ be a $\Lambda$-module with minimal projective resolution $\cdots \longrightarrow P_1 \longrightarrow P_0 \longrightarrow M \longrightarrow 0$, and let $S$ be a simple $\Lambda$-module. Then, for every nonnegative integer $n$, we have $\operatorname {Ext}_\Lambda ^n(M, S) = 0$ if and only if the projective cover of $S$ is not a direct summand of $P_n$.

Lemma 8.6.

Let $\Lambda$ be an artin algebra, and let $a$ be an idempotent in $\Lambda$. Let $S$ be a simple $\Lambda$-module which is not annihilated by the ideal $\langle a \rangle$, and let $P$ be the projective cover of $S$. Then $aP$ is a projective $a{\Lambda }a$-module.

Proof.

We have

$$\begin{equation*} \operatorname {Hom}_\Lambda ({\Lambda }a, S) \cong aS \ne 0, \end{equation*}$$

so there exists a nonzero morphism $f \colon {\Lambda }a \longrightarrow S$. Decomposing the idempotent $a$ into a sum $a = a_1 + \cdots + a_t$ of orthogonal primitive idempotents gives a decomposition ${\Lambda }a \cong {\Lambda }a_1 \oplus \cdots \oplus {\Lambda }a_t$ of ${\Lambda }a$ into indecomposable projective modules. For some $i$, we must then have a nonzero morphism $f_i \colon {\Lambda }a_i \longrightarrow S$. Since $S$ is simple, this means that ${\Lambda }a_i$ is its projective cover. Since $a \cdot a_i = a_i$, we get

$$\begin{equation*} aP \cong a{\Lambda }a_i = (a{\Lambda }a) a_i. \end{equation*}$$

Therefore $aP$ is a projective $a{\Lambda }a$-module.

■

Now we show that the conditions of Theorem 8.1 are related when we have an ordering of the simple $\Lambda /\langle a \rangle$-modules.

Proposition 8.7.

Let $\Lambda$ be an artin algebra, and let $a$ be an idempotent in $\Lambda$. Assume that there is a total order $\preceq$ on the simple $\Lambda /\langle a \rangle$-modules satisfying condition 8.1. Then we have the following implications between the conditions of Theorem 8.1:

(i)

$(\alpha ) \implies (\delta )$.

(ii)

$(\gamma ) \implies (\beta )$.

In particular, we have that the functor $a-\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism if and only if conditions $(\alpha )$ and $(\gamma )$ hold.

Proof.

We show the second implication; the first can be shown in a similar way. The last claim follows directly from these two implications by Theorem 8.1 (i).

Assume that $(\gamma )$ holds, that is, every $\Lambda /\langle a \rangle$-module has finite projective dimension as a $\Lambda$-module. We want to show that $(\beta )$ holds, that is, the $a{\Lambda }a$-module $a\Lambda$ has finite projective dimension.

As in Section 7, we let $e$ be the exact functor $e=(a-)\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ given by multiplication with $a$. Then what we need to show is that $e(\Lambda )$ has finite projective dimension as a $a{\Lambda }a$-module.

Let $S_1 \preceq \cdots \preceq S_s$ be all the simple $\Lambda /\langle a \rangle$-modules (up to isomorphism), ordered by the total order $\preceq$. Let $T_1, \ldots , T_t$ be all the other simple $\Lambda$-modules (up to isomorphism). Let $Q_i$ be the projective cover of $S_i$ (considered as a $\Lambda$-module) and $Q'_j$ the projective cover of $T_j$, for every $i$ and $j$. These are all the indecomposable projective $\Lambda$-modules up to isomorphism, so it is sufficient to show that $e(Q_i)$ and $e(Q'_j)$ have finite projective dimension as $a{\Lambda }a$-modules for every $i$ and $j$.

For each of the modules $Q'_j$, we have that $e(Q'_j)$ is a projective $a{\Lambda }a$-module by Lemma 8.6. We need to check that $e(Q_i)$ has finite projective dimension for every $i$.

Consider the module $S_1$. By our assumptions, every simple $\Lambda /\langle a \rangle$-module has finite projective dimension over $\Lambda$. Let

$$\begin{equation*} \vcenter{\img[][297pt][14pt][{$\xymatrix{ 0 \ar[r]^{} & P^{(1)}_{n_1} \ar[r]^{} & \cdots\ar[r]^{} & P^{(1)}_2 \ar[r]^{} & P^{(1)}_1 \ar[r]^{} & Q_1 \ar[r]^{} & S_1 \ar[r] & 0 }$}]{Images/img86849199ce81b4518477a1f786642f5c.svg}} \end{equation*}$$

be a minimal projective resolution of $S_1$. Applying the functor $e$ to this sequence gives the exact sequence

$$\begin{equation} \vcenter{\img[][303pt][14pt][{$\xymatrix{ 0 \ar[r]^{} & e(P^{(1)}_{n_1}) \ar[r]^{} & \cdots\ar[r]^{} & e(P^{(1)}_2) \ar[r]^{} & e(P^{(1)}_1) \ar[r]^{} & e(Q_1) \ar[r]^{} & 0 }$}]{Images/imge4f1cdc824b4010d20adece8676e0030.svg}} {\tag{8.2}} \end{equation}$$

of $a{\Lambda }a$-modules, since $e(S_1) = 0$. Since we have $\operatorname {Ext}_\Lambda ^{>0}(S_1,S_i) = 0$ for every $i$, it follows from Lemma 8.5 that the only indecomposable projective $\Lambda$-modules which can occur as direct summands of the modules $P^{(1)}_1, \ldots , P^{(1)}_{n_1}$ are the modules $Q'_j$. Since we know that these are mapped to projective modules by $e$, the sequence 8.2 is a projective resolution of the $a{\Lambda }a$-module $e(Q_1)$.

We continue inductively. For every $i$, we apply the functor $e$ to a minimal projective resolution

$$\begin{equation*} \vcenter{\img[][291pt][14pt][{$\xymatrix{ 0 \ar[r]^{} & P^{(i)}_{n_i} \ar[r]^{} & \cdots\ar[r]^{} & P^{(i)}_2 \ar[r]^{} & P^{(i)}_1 \ar[r]^{} & Q_i \ar[r]^{} & S_i \ar[r] & 0 }$}]{Images/img6fef5f8927b50ea5b3ed97bf1e35a075.svg}} \end{equation*}$$

and obtain the sequence

$$\begin{equation*} \vcenter{\img[][298pt][14pt][{$\xymatrix{ 0 \ar[r]^{} & e(P^{(i)}_{n_i}) \ar[r]^{} & \cdots\ar[r]^{} & e(P^{(i)}_2) \ar[r]^{} & e(P^{(i)}_1) \ar[r]^{} & e(Q_i) \ar[r]^{} & 0 }$}]{Images/img06d5082be67dd417d0e4533efed5d5c9.svg}} \end{equation*}$$

of $a{\Lambda }a$-modules. Each of the modules $P^{(i)}_1, \ldots , P^{(i)}_{n_i}$ has only the indecomposable projective modules $Q'_1, \ldots , Q'_t, Q_1, \ldots , Q_{i-1}$ as direct summands. Therefore (by the induction assumption), all the modules $e(P^{(i)}_1), \ldots , e(P^{(i)}_{n_i})$ have finite projective dimension, and thus the module $e(Q_i)$ has finite projective dimension.

■

By combining Theorem 8.1 with Proposition 8.7, we get the following result.

Corollary 8.8.

Let $\Lambda$ be an artin algebra over a commutative ring $k$, and let $a$ be an idempotent in $\Lambda$. Assume that there is a total order $\preceq$ on the simple $\Lambda /\langle a \rangle$-modules satisfying condition 8.1. Then the following hold, where $(\alpha )$, $(\beta )$, $(\gamma )$ and $(\delta )$ refer to the conditions in Theorem 8.1.

(i)

The functor $a-\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{a\Lambda a}$ induces a singular equivalence between $\Lambda$ and $a\Lambda a$ if and only if $(\gamma )$ holds.

(ii)

Assume that $(\alpha )$ and $(\gamma )$ hold. Then $\Lambda$ is Gorenstein if and only if $a{\Lambda }a$ is Gorenstein.

(iii)

Assume that $(\alpha )$ and $(\gamma )$ hold, that $k$ is a field and $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module. Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

We now consider special cases of conditions $(\alpha )$ and $(\gamma )$ where the dimensions are not only finite, but at most one. We show that if one of these dimensions is at most one, then we have an ordering of the simple $\Lambda /\langle a \rangle$-modules as assumed in Proposition 8.7 and Corollary 8.8.

Lemma 8.9.

Let $\Lambda$ be an artin algebra, and let $a$ be an idempotent in $\Lambda$. Assume that we have either

$$\begin{equation*} (\alpha _1) \ \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \le 1 \qquad \text{or}\qquad (\gamma _1) \ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \le 1. \end{equation*}$$

Then there exists a total order $\preceq$ on the simple $\Lambda /\langle a \rangle$-modules satisfying condition 8.1.

Proof.

Assume that $(\gamma _1)$ holds (the proof using $(\alpha _1)$ is similar). Let $S_1, \ldots , S_s$ be all the simple $\Lambda /\langle a \rangle$-modules (up to isomorphism), and let $P_1, \ldots , P_s$ be their projective covers as $\Lambda$-modules, such that $P_i/((\operatorname {rad}\Lambda )P_i) \cong S_i$ for every $i$. Assume that we have ordered these by increasing length of the projective covers, that is,

$$\begin{equation*} \operatorname {length}(P_1) \le \operatorname {length}(P_2) \le \cdots \le \operatorname {length}(P_s). \end{equation*}$$

For any $i$, the module $S_i$ has a projective resolution of the form

$$\begin{equation*} \vcenter{\img[][153pt][10pt][{$\xymatrix{ 0 \ar[r]^{} & Q \ar[r]^{} & P_i \ar[r]^{} & S_i \ar[r]^{} & 0 . }$}]{Images/imgd83d404c6d96715fde9b1df81ae9c095.svg}} \end{equation*}$$

Since the module $Q$ has shorter length than the module $P_i$, it cannot have any of the modules $P_i, \ldots , P_s$ as direct summands. Then Lemma 8.5 implies that $\operatorname {Ext}_\Lambda ^{>0}(S_i, S_j) = 0$ for $i \le j$.

■

By using Proposition 8.7, Lemma 8.9 and Theorem 8.1, we have the following.

Corollary 8.10.

Let $\Lambda$ be an artin algebra over a commutative ring $k$, and let $a$ be an idempotent in $\Lambda$. Then the following hold, where $(\alpha )$, $(\beta )$, $(\gamma )$ and $(\delta )$ refer to the conditions in Theorem 8.1, and $(\alpha _1)$ and $(\gamma _1)$ refer to the conditions in Lemma 8.9.

(i)

If $(\gamma _1)$ holds, then the functor $a-\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{a\Lambda a}$ induces a singular equivalence between $\Lambda$ and $a\Lambda a$.

(ii)

Assume either that $(\alpha _1)$ and $(\gamma )$ hold, or that $(\alpha )$ and $(\gamma _1)$ hold. Then $\Lambda$ is Gorenstein if and only if $a{\Lambda }a$ is Gorenstein.

(iii)

Assume either that $(\alpha _1)$ and $(\gamma )$ hold, or that $(\alpha )$ and $(\gamma _1)$ hold. Furthermore, assume that $k$ is a field and $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module. Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

For the following results, we let $\Lambda = kQ/\langle \rho \rangle$ be a quotient of a path algebra, where $k$ is a field, $Q$ is a quiver, and $\rho$ is a minimal set of relations in $kQ$ generating an admissible ideal $\langle \rho \rangle$.

First we describe how conditions $(\alpha _1)$ and $(\gamma _1)$ can be interpreted for quotients of path algebras. The result follows directly from 10, Corollary, Section 1.1.

Lemma 8.11.

Let $S$ be the simple $\Lambda$-module corresponding to a vertex $v$ in the quiver $Q$.

(i)

We have $\operatorname {\mathsf{pd}}_\Lambda S \le 1$ if and only if no relation starts in the vertex $v$.

(ii)

We have $\operatorname {\mathsf{id}}_\Lambda S \le 1$ if and only if no relation ends in the vertex $v$.

As a consequence of Lemma 8.11 and Corollary 8.10, we get the following results for path algebras.

Corollary 8.12.

Let $\Lambda = kQ/\langle \rho \rangle$ be a quotient of a path algebra as above. Choose some vertices in $Q$ where no relations start, and let $a$ be the sum of all vertices except these. Then the functor $a-\colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ induces a singular equivalence between $\Lambda$ and $a{\Lambda }a\colon$

$$\begin{equation*} \mathsf{D}_\mathsf{sg}(a-)\colon \mathsf{D}_\mathsf{sg}(\operatorname {mod}\Lambda ) \stackrel{\simeq }{\longrightarrow } \mathsf{D}_\mathsf{sg}(\operatorname {mod}a{\Lambda }a). \end{equation*}$$

Corollary 8.13.

Let $\Lambda = kQ/\langle \rho \rangle$ be a quotient of a path algebra as above. Choose some vertices in $Q$ where no relations start and no relations end, and let $a$ be the sum of all vertices except these. Then the following hold:

(i)

$\Lambda$ is Gorenstein if and only if $a{\Lambda }a$ is Gorenstein.

(ii)

$\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a{\Lambda }a$ satisfies $\mathsf{Fg}$.

We apply the above result in the following example.

Example 8.14.

Let $Q$ be the quiver with relations $\rho$ given by

$$\begin{equation*} Q\colon \vcenter{\img[][136pt][36pt][{$\xymatrix@C=3em{ 1 \ar[r]^{\alpha_1} & 2 \ar[r]^-{\alpha_2} & {\cdots} \ar[r]^{\alpha_{m-1}} & m \ar@(dl,dr)[lll]^{\alpha_m} }$}]{Images/img4da865ec0f7530b7475334c53b768bba.svg}} \qquad \text{and}\qquad \rho =\{(\alpha _m \cdots \alpha _1)^n\}, \end{equation*}$$

for some integers $m \ge 2$ and $n \ge 2$. Let $\Lambda = kQ/\langle \rho \rangle$, and let $a = e_1$ (the only vertex where a relation starts and ends). Then $a{\Lambda }a \cong k[x]/\langle x^n \rangle$, so $a{\Lambda }a$ satisfies $\mathsf{Fg}$ by 2728. By Corollary 8.13, the algebra $\Lambda$ also satisfies $\mathsf{Fg}$. By Corollary 8.12, the algebras $\Lambda$ and $k[x]/\langle x^n \rangle$ are singularly equivalent. See 59 for a general discussion of the Hochschild cohomology ring of the path algebra $kQ$ modulo one relation.

8.3. Triangular matrix algebras

Let $\Sigma$ and $\Gamma$ be two artin algebras over a commutative ring $k$, and let ${_{\Gamma }M_{\Sigma }}$ be a $\Gamma$-$\Sigma$-bimodule such that $M$ is finitely generated over $k$, and $k$ acts centrally on $M$. Then we have the artin triangular matrix algebra

$$\begin{equation*} \Lambda = \begin{pmatrix} \Sigma & 0 \\ {_\Gamma M_\Sigma } & \Gamma \\ \end{pmatrix}, \end{equation*}$$

where the addition and the multiplication are given by the ordinary operations on matrices.

The module category of $\Lambda$ has a well known description; see 730. In fact, a module over $\Lambda$ is described as a triple $(X,Y,f)$, where $X$ is a $\Sigma$-module, $Y$ is a $\Gamma$-module and $f\colon M\otimes _{\Sigma }X\longrightarrow Y$ is a $\Gamma$-homomorphism. A morphism between two triples $(X,Y,f)$ and $(X',Y',f')$ is a pair of homomorphisms $(a,b)$, where $a\in \operatorname {Hom}_{\Sigma }(X,X')$ and $b\in \operatorname {Hom}_{\Gamma }(Y,Y')$, such that the following diagram commutes:

$$\begin{equation*} \vcenter{\img[][90pt][54pt][{$\xymatrix{ M\tensor_{\Sigma}X \ar[r]^{ \ \ \ f} \ar[d]_{\Id_M\tensor a} & Y \ar[d]^{b} \\ M\tensor_{\Sigma}X'\ar[r]^{ \ \ \ f'} & Y' }$}]{Images/img37b7e088984b47a09a4628f2ed875572.svg}} \end{equation*}$$

We define the following functors:

(i)

The functor $T_{\Sigma }\colon \operatorname {mod}{\Sigma }\longrightarrow \operatorname {mod}{\Lambda }$ is defined on $\Sigma$-modules $X$ by $T_{\Sigma }(X)=(X,M\otimes _{\Sigma }X,\operatorname {Id}_{M\otimes X})$ and given a $\Sigma$-homomorphism $a\colon X\longrightarrow X'$ then $T_{\Sigma }(a)=(a,\operatorname {Id}_{M}\otimes a)$.

(ii)

The functor $U_{\Sigma }\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{\Sigma }$ is defined on $\Lambda$-modules $(X,Y,f)$ by $U_{\Sigma }(X,Y,f)=X$ and given a $\Lambda$-homomorphism $(a,b)\colon (X,Y,f)\longrightarrow (X',Y',f')$ then $U_{\Sigma }(a,b)=a$. Similarly we define the functor $U_{\Gamma }\colon \operatorname {mod}{\Lambda }\longrightarrow \operatorname {mod}{\Gamma }$.

(iii)

The functor $Z_{\Sigma }\colon \operatorname {mod}{\Sigma }\longrightarrow \operatorname {mod}{\Lambda }$ is defined on $\Sigma$-modules $X$ by $Z_{\Sigma }(X)=(X,0,0)$ and given a $\Sigma$-homomorphism $a\colon X\longrightarrow X'$ then $Z_{\Sigma }(a)=(a,0)$. Similarly we define the functor $Z_{\Gamma }\colon \operatorname {mod}{\Gamma }\longrightarrow \operatorname {mod}{\Lambda }$.

(iv)

The functor $H_{\Gamma }\colon \operatorname {mod}{\Gamma }\longrightarrow \operatorname {mod}{\Lambda }$ is defined by $H_{\Gamma }(Y)=(\operatorname {Hom}_{\Gamma }(M,Y),Y,\epsilon _X)$ on $\Gamma$-modules $Y$ and given a $\Gamma$-homomorphism $b\colon Y\longrightarrow Y'$ then $H_{\Gamma }(b)=(\operatorname {Hom}_{\Gamma }(M,b),b)$.

Then from Example 2.3 (see also 54, Example 2.12), using the idempotent elements $e_1=\bigl (\begin{smallmatrix} 1_{\Sigma } & 0 \\ 0 & 0 \end{smallmatrix}\bigr )$ and $e_2=\bigl (\begin{smallmatrix} 0 & 0 \\ 0 & 1_{\Gamma } \end{smallmatrix}\bigr )$, we have the following recollements of abelian categories:

$$\begin{equation} \vcenter{\img[][202pt][55pt][{$\xymatrix@C=0.5cm{ \fmod{\Gamma} \ar[rrr]^{Z_{\Gamma}} &&& \fmod{\Lambda} \ar[rrr]^{U_{\Sigma}} \ar@/_1.5pc/[lll]_{q} \ar@/^1.5pc/[lll]^{U_{\Gamma}} &&& \fmod{\Sigma} \ar@/_1.5pc/[lll]_{T_{\Sigma}} \ar@/^1.5pc/[lll]^{Z_{\Sigma}} }$}]{Images/img323b97ab7d75377a7a380d30c84ae70b.svg}} {\tag{8.3}} \end{equation}$$

and

$$\begin{equation} \vcenter{\img[][202pt][55pt][{$\xymatrix@C=0.5cm{ \fmod{\Sigma} \ar[rrr]^{Z_{\Sigma}} &&& \fmod{\Lambda} \ar[rrr]^{U_{\Gamma}} \ar@/_1.5pc/[lll]_{U_{\Sigma}} \ar@/^1.5pc/[lll]^{p} &&& \fmod{\Gamma} \ar@/_1.5pc/[lll]_{Z_{\Gamma}} \ar@/^1.5pc/[lll]^{H_{\Gamma}} }$}]{Images/img8d19207ed898f0262a4082c1447687cc.svg}} {\tag{8.4}} \end{equation}$$

The functors $q$ and $p$ are induced from the adjoint pairs $(T_{\Sigma },U_{\Sigma })$ and $(U_{\Gamma },H_{\Gamma })$, respectively; see 54, Remark 2.3 for more details.

We want to use Theorem 8.1 to compare the triangular matrix algebra $\Lambda$ with the algebras $\Sigma$ and $\Gamma$. First consider the case where we compare $\Lambda$ with $\Sigma$. We then take the idempotent $a$ in the theorem to be $e_1$, and we can reformulate conditions $(\alpha )$, $(\beta )$, $(\gamma )$ and $(\delta )$ as follows:

$(\alpha )$

The functor $Z_\Gamma$ sends every $\Gamma$-module to a $\Lambda$-module with finite injective dimension.

$(\beta )$

The functor $U_\Sigma$ sends every projective $\Lambda$-module to a $\Sigma$-module with finite projective dimension.

$(\gamma )$

The functor $Z_\Gamma$ sends every $\Gamma$-module to a $\Lambda$-module with finite projective dimension.

$(\delta )$

The functor $U_\Sigma$ sends every injective $\Lambda$-module to a $\Sigma$-module with finite injective dimension.

By interchanging $\Sigma$ and $\Gamma$, we get a similar reformulation of the conditions for the case where we compare $\Lambda$ with $\Gamma$.

The next result clarifies when the above hold for the recollement 8.4 of a triangular matrix algebra $\Lambda$.

Lemma 8.15.

Let $\Lambda =\bigl (\begin{smallmatrix} \Sigma & 0 \\ {_\Gamma M_\Sigma } & \Gamma \end{smallmatrix}\bigr )$ be a triangular matrix algebra. The following hold.

(i)

If $\operatorname {\mathsf{pd}}_{\Gamma }M<\infty$, then the functor $U_{\Gamma }$ sends projective $\Lambda$-modules to $\Gamma$-modules of finite projective dimension.

(ii)

The functor $U_{\Gamma }$ preserves injectives.

(iii)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Sigma }<\infty$. Then $\operatorname {\mathsf{id}}_{\Lambda }Z_{\Sigma }(X)<\infty$ for every $\Sigma$-module $X$.

(iv)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Sigma }<\infty$ and $\operatorname {\mathsf{pd}}_{\Gamma }M<\infty$. Then we have $\operatorname {\mathsf{pd}}_{\Lambda }Z_{\Sigma }(X)<\infty$ for all $\Sigma$-modules $X$.

Proof.

(i) It is known (see 7) that the indecomposable projective $\Lambda$-modules are of the forms $T_{\Sigma }(P)$, where $P$ is an indecomposable projective $\Sigma$-module, and $Z_{\Gamma }(Q)$, where $Q$ is an indecomposable projective $\Gamma$-module. Hence it is enough to consider modules of these forms. We have $U_{\Gamma }Z_{\Gamma }(Q)=Q$, and since $\operatorname {\mathsf{pd}}_{\Gamma }M<\infty$ it follows that $\operatorname {\mathsf{pd}}_{\Gamma }U_{\Gamma }T_{\Sigma }(P)=\operatorname {\mathsf{pd}}_{\Gamma }(M\otimes _{\Sigma }P)<\infty$.

(ii) Since $(Z_{\Gamma }, U_{\Gamma })$ is an adjoint pair and $Z_{\Gamma }$ is exact it follows that the functor $U_{\Gamma }$ preserves injectives.

(iii) Let $0\longrightarrow X\longrightarrow I^0\longrightarrow \cdots \longrightarrow I^n\longrightarrow 0$ be a finite injective resolution of a $\Sigma$-module $X$. Then applying the functor $Z_{\Sigma }$ we get the exact sequence $0\longrightarrow Z_{\Sigma }(X)\longrightarrow Z_{\Sigma }(I^0)\longrightarrow \cdots \longrightarrow Z_{\Sigma }(I^n)\longrightarrow 0$, where every $Z_{\Sigma }(I^i)$ is an injective $\Lambda$-module since we have the adjoint pair $(U_{\Sigma },Z_{\Sigma })$ and $U_{\Sigma }$ is exact. Hence the injective dimension of $Z_{\Sigma }(X)$ is finite.

(iv) This follows from 58, Lemma 2.4, which says that, if $\operatorname {\mathsf{pd}}_\Gamma M < \infty$, then a $\Lambda$-module $(X,Y,f)$ has finite projective dimension if and only if the projective dimensions of $X$ and $Y$ are finite.

■

Using now the recollement 8.3 we have the following dual result of Lemma 8.15. The proof is left to the reader.

Lemma 8.16.

Let $\Lambda =\bigl (\begin{smallmatrix} \Sigma & 0 \\ {_\Gamma M_\Sigma } & \Gamma \end{smallmatrix}\bigr )$ be a triangular matrix algebra. The following hold.

(i)

The functor $U_{\Sigma }$ preserves projectives.

(ii)

If $\operatorname {\mathsf{pd}}_{\Sigma } M_{\Sigma }<\infty$, then the functor $U_{\Sigma }$ sends injective $\Lambda$-modules to $\Sigma$-modules of finite injective dimension.

(iii)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Gamma }<\infty$. Then $\operatorname {\mathsf{pd}}_{\Lambda }Z_{\Gamma }(Y)<\infty$ for every $\Gamma$-module $Y$.

(iv)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Gamma }<\infty$ and $\operatorname {\mathsf{pd}}_{\Sigma } M_{\Sigma }<\infty$. Then for every $\Gamma$-module $Y$ we have $\operatorname {\mathsf{id}}_{\Lambda }Z_{\Gamma }(Y)<\infty$.

As a consequence of Lemma 8.15 and Theorem 8.1 we have the following result. For similar characterizations with (ii) see 64.

Corollary 8.17.

Let $\Lambda =\bigl (\begin{smallmatrix} \Sigma & 0 \\ {_{\Gamma }M_{\Sigma }} & \Gamma \end{smallmatrix}\bigr )$ be an artin triangular matrix algebra over a commutative ring $k$ such that $\operatorname *{\mathsf{gl.dim}}{\Sigma }<\infty$ and $\operatorname {\mathsf{pd}}_{\Gamma }M<\infty$. Then the following hold.

(i)

The singularity categories of $\Lambda$ and $\Gamma$ are triangle equivalent:$$\begin{equation*} \vcenter{\img[][183pt][13pt][{$\xymatrix@C=0.5cm{ \Dsg(U_{\Gamma})\colon\Dsg(\fmod{\Lambda}) \ \ar[rr]^{ \ \ \ \ \equivalence} && \ \Dsg(\fmod{\Gamma}). }$}]{Images/img41319b6d3f5fa156535436336d367fc0.svg}} \end{equation*}$$

(ii)

$\Lambda$ is Gorenstein if and only if $\Gamma$ is Gorenstein.

(iii)

Assume that $k$ is a field and that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module. Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $\Gamma$ satisfies $\mathsf{Fg}$.

Remark 8.18.

The algebra $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ being semisimple (as required in part (iii) above) can be shown to be equivalent to the following three algebras being semisimple: $(\Sigma /\operatorname {rad}\Sigma ) \otimes _k (\Sigma ^{\operatorname {op}}/\operatorname {rad}\Sigma ^{\operatorname {op}})$, $(\Sigma /\operatorname {rad}\Sigma ) \otimes _k (\Gamma ^{\operatorname {op}}/\operatorname {rad}\Gamma ^{\operatorname {op}})$ and $(\Gamma /\operatorname {rad}\Gamma ) \otimes _k (\Gamma ^{\operatorname {op}}/\operatorname {rad}\Gamma ^{\operatorname {op}})$.

We also have the following consequence, obtained now from Lemma 8.16 and Theorem 8.1. Note that in the first statement we recover a theorem by Chen 15.

Corollary 8.19.

Let $\Lambda =\bigl (\begin{smallmatrix} \Sigma & 0 \\ {_{\Gamma }M_{\Sigma }} & \Gamma \end{smallmatrix}\bigr )$ be an artin triangular matrix algebra over a commutative ring $k$.

(i)

15, Theorem 4.1 Assume that $\operatorname *{\mathsf{gl.dim}}{\Gamma }<\infty$. Then there is a triangle equivalence:$$\begin{equation*} \vcenter{\img[][142pt][18pt][{$\xymatrix@C=0.5cm{ \Dsg(\fmod{\Lambda}) \ \ar[rr]^{ \equivalence\ }_{\Dsg(U_{\Sigma}) \ } && \ \Dsg(\fmod{\Sigma}). }$}]{Images/img9e3451dbf15774995c64ac70e1a1112a.svg}} \end{equation*}$$

(ii)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Gamma }<\infty$ and $\operatorname {\mathsf{pd}}_{\Sigma } M_{\Sigma }<\infty$. Then the following hold.

(a)

$\Lambda$ is Gorenstein if and only if $\Sigma$ is Gorenstein.

(b)

Assume that $k$ is a field and that $(\Lambda /\operatorname {rad}\Lambda ) \otimes _k (\Lambda ^{\operatorname {op}}/\operatorname {rad}\Lambda ^{\operatorname {op}})$ is a semisimple $\Lambda ^{\operatorname {e}}$-module. Then $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $\Sigma$ satisfies $\mathsf{Fg}$.

From the above corollaries and the classical result of Buchweitz–Happel (see the text before Corollary 5.6) we have the following result for stable categories of Cohen–Macaulay modules.

Corollary 8.20.

Let $\Lambda =\bigl (\begin{smallmatrix} \Sigma & 0 \\ {_{\Gamma }M_{\Sigma }} & \Gamma \end{smallmatrix}\bigr )$ be an artin triangular matrix algebra.

(i)

15, Corollary 4.2 Assume that $\operatorname *{\mathsf{gl.dim}}{\Gamma }<\infty$ and $\Sigma$ is Gorenstein. Then there is a triangle equivalence:$$\begin{equation*} \vcenter{\img[][124pt][13pt][{$\xymatrix@C=0.5cm{ \Dsg(\fmod{\Lambda}) \ \ar[rr]^{ \ \equivalence} && \ \uCM(\Sigma). }$}]{Images/img3fc1e1b37f3e721f288ec39d15ee7a62.svg}} \end{equation*}$$

(ii)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Gamma }<\infty$ and $\operatorname {\mathsf{pd}}_{\Sigma } M_{\Sigma }<\infty$. If $\Sigma$ is Gorenstein, then there is a triangle equivalence between the stable categories of Cohen–Macaulay modules of $\Lambda$ and $\Sigma$:$$\begin{equation*} \vcenter{\img[][107pt][12pt][{$\xymatrix@C=0.5cm{ \uCM(\Lambda) \ \ar[rr]^{ \equivalence} && \ \uCM(\Sigma). }$}]{Images/imgb302ff061f0a17caae685dfce6ef09ad.svg}} \end{equation*}$$

(iii)

Assume that $\operatorname *{\mathsf{gl.dim}}{\Sigma }<\infty$ and $\operatorname {\mathsf{pd}}_{\Gamma }M<\infty$. If $\Gamma$ is Gorenstein, then there is a triangle equivalence between the stable categories of Cohen–Macaulay modules of $\Lambda$ and $\Gamma$:$$\begin{equation*} \vcenter{\img[][106pt][12pt][{$\xymatrix@C=0.5cm{ \uCM(\Lambda) \ \ar[rr]^{ \equivalence} && \ \uCM(\Gamma). }$}]{Images/imgf941645411eb44ff11c52f0e6896f1bf.svg}} \end{equation*}$$

8.4. Comparison to work by Nagase

In this subsection we recall a result of Hiroshi Nagase 47 and relate his set of assumptions to ours.

In 47 Hiroshi Nagase proves the following result.

Proposition 8.21.

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field with a stratifying ideal $\langle a \rangle$ for an idempotent $a$ in $\Lambda$. Suppose $\operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}} \Lambda /\langle a \rangle < \infty$. Then we have

(1)

$\operatorname {HH}^{\ge n}(\Lambda )\cong \operatorname {HH}^{\ge n}(a\Lambda a)$ as graded algebras, where $n = \operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}} \Lambda /\langle a \rangle + 1$.

(2)

$\Lambda$ satisfies $\mathsf{Fg}$ if and only if so does $a\Lambda a$.

(3)

$\Lambda$ is Gorenstein if and only if so is $a\Lambda a$.

This work is based on the paper 41, where stratifying ideals $\langle a \rangle$ in a finite dimensional algebra $\Lambda$ were used to show that the Hochschild cohomology groups of $\Lambda$ and $a\Lambda a$ are isomorphic in almost all degrees.

We start by giving an example of a recollement $(\operatorname {mod}\Lambda /\langle a \rangle , \operatorname {mod}\Lambda , \operatorname {mod}a\Lambda a)$, where the ideal $\langle a \rangle$ is not a stratifying ideal but it satisfies our conditions from Theorem 7.10.

Example 8.22.

Let $Q$ be the quiver with relations $\rho$ given by

$$\begin{equation*} \vcenter{\img[][69pt][42pt][{$\xymatrix@R=5pt{ & 2\ar[dd]^\gamma\\ 1\ar@(ul,dl)_{\alpha}\ar[ur]^\beta& \\ & 3\ar[ul]^\delta}$}]{Images/img12477851d2bedd7888216613c3851f79.svg}} \end{equation*}$$

and $\rho =\{\alpha ^2, \gamma \beta , \beta \alpha \delta \}$. Let $\Lambda = kQ/\langle \rho \rangle$ for some field $k$, and let $a = e_1$. We want to study the relationship between $\Lambda$ and $a\Lambda a$. Let $S_i$ denote the simple $\Lambda$-module associated to the vertex $i$ for $i=1,2,3$. Then $\operatorname {\mathsf{pd}}_\Lambda S_2 = 1$, $\operatorname {\mathsf{pd}}_\Lambda S_3 = 3$, $\operatorname {\mathsf{id}}_\Lambda S_2 = 2$ and $\operatorname {\mathsf{id}}_\Lambda S_3 = 3$. Furthermore, the left and right $a\Lambda a$-modules $a\Lambda$ and $\Lambda a$ have finite projective dimension (they are projective) as $a\Lambda a$-modules. Hence, according to Theorem 7.10 $\Lambda$ satisfies $\mathsf{Fg}$ if and only if $a\Lambda a\cong k[x]/\langle x^2 \rangle$ does. We infer from this that $\Lambda$ satisfies $\mathsf{Fg}$. Moreover, the Hochschild cohomology groups of $\Lambda$ and $a\Lambda a$ are isomorphic in almost all degrees by Proposition 7.9.

We claim that $\langle a \rangle$ is not a stratifying ideal. Recall that $\langle a \rangle$ is stratifying if (i) the multiplication map $\Lambda a\otimes _{a\Lambda a} a\Lambda \longrightarrow \Lambda a \Lambda$ is an isomorphism and (ii) $\operatorname {Tor}^{a\Lambda a}_i(\Lambda a,a\Lambda ) = (0)$ for $i>0$. Using that $(1-a)\Lambda a\cong a\Lambda a$ as a right $a\Lambda a$-module, direct computations show that $\Lambda a\otimes _{a\Lambda a} a\Lambda$ has dimension $12$, while $\langle a \rangle$ has dimension $10$. Consequently $\langle a \rangle$ is not a stratifying ideal in $\Lambda$. However, the condition (ii) is satisfied since $\Lambda a$ is a projective $a\Lambda a$-module.

Next we show that, when $\langle a \rangle$ is a stratifying ideal, then the property $\operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}} \Lambda /\langle a \rangle < \infty$ is equivalent to the functor $e \colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ being an eventually homological isomorphism. We thank Hiroshi Nagase for pointing out that (a) implies (b) in the second part of the following result. This led to a much better understanding of the conditions occurring in the main results.

Lemma 8.23.

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field $k$.

(i)

Assume that $(\alpha )\ \operatorname {\mathsf{id}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty$ and $(\gamma )\ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty$. Then $\operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}} \Lambda /\langle a \rangle < \infty$.

(ii)

Assume that $\langle a \rangle$ is a stratifying ideal in $\Lambda$. Then the following are equivalent.

(a)

$\operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}} \Lambda /\langle a \rangle < \infty$.

(b)

The functor $e \colon \operatorname {mod}\Lambda \longrightarrow \operatorname {mod}a{\Lambda }a$ is an eventually homological isomorphism.

Proof.

(i) For two primitive idempotents $u$ and $v$ in $\Lambda$, we have that

$$\begin{equation*} \operatorname {Hom}_{\Lambda ^{\operatorname {e}}}(\Lambda ^{\operatorname {e}}(u\otimes v), \Lambda /\langle a \rangle ) \cong u(\Lambda /\langle a \rangle )v. \end{equation*}$$

Then, if $u$ or $v$ occurs in $a$, then this homomorphism set is zero. Consequently we infer that the composition factors of $\Lambda /\langle a \rangle$ are direct summands of the semisimple module $\Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \otimes _k \Big ( \frac{\Lambda ^{\operatorname {op}}/\langle a \rangle }{\operatorname {rad}\Lambda ^{\operatorname {op}}/\langle a \rangle } \Big )$. By Lemma 7.3 $\operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}} \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) \otimes _k \Big ( \frac{\Lambda ^{\operatorname {op}}/\langle a \rangle }{\operatorname {rad}\Lambda ^{\operatorname {op}}/\langle a \rangle } \Big )$ is finite, hence the claim follows.

(ii) By Corollary 3.12 and part (i), statement (b) implies (a).

Conversely, assume (a). For $j> \operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}}\Lambda /\langle a \rangle$ and any $\Lambda$-modules $M$ and $N$ we have that

$$\begin{equation*} \operatorname {Ext}^j_{\Lambda ^{\operatorname {e}}}(\Lambda ,\operatorname {Hom}_k(M,N)) \cong \operatorname {Ext}^j_{\Lambda ^{\operatorname {e}}}(\langle a \rangle , \operatorname {Hom}_k(M,N)). \end{equation*}$$

Using the isomorphism in the proof of Proposition 3.3 in 41,

$$\begin{equation*} \operatorname {Ext}^i_{\Lambda ^{\operatorname {e}}}(\langle a \rangle ,X) \cong \operatorname {Ext}^i_{a\Lambda a^{\operatorname {e}}}(a\Lambda a, aXa), \end{equation*}$$

we obtain that

$$\begin{align*} \operatorname {Ext}^i_{\Lambda ^{\operatorname {e}}}(\langle a \rangle ,\operatorname {Hom}_k(M,N)) &\cong \operatorname {Ext}^i_{a\Lambda a^{\operatorname {e}}}(a\Lambda a, a\operatorname {Hom}_k(M,N)a)\\ &\cong \operatorname {Ext}^i_{a\Lambda a^{\operatorname {e}}}(a\Lambda a, \operatorname {Hom}_k(aM,aN))\\ &\cong \operatorname {Ext}^i_{a\Lambda a}(aM,aN)) \end{align*}$$

for all $\Lambda$-modules $M$ and $N$. Since $\operatorname {Ext}^i_{\Lambda ^{\operatorname {e}}}(\Lambda ,\operatorname {Hom}_k(M,N)) \cong \operatorname {Ext}^i_{\Lambda }(M,N)$, we obtain the isomorphism

$$\begin{equation*} \operatorname {Ext}_\Lambda ^j (M, N) \cong \operatorname {Ext}_{a\Lambda a}^j (aM, aN) \end{equation*}$$

for all $j > \operatorname {\mathsf{pd}}_{\Lambda ^{\operatorname {e}}}\Lambda /\langle a \rangle$ and all $\Lambda$-modules $M$ and $N$. Hence $e$ is an eventually homological isomorphism.

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The following result gives a characterization of condition $(\gamma )$ when $\langle a \rangle$ is a stratifying ideal.

Lemma 8.24.

Let $\Lambda$ be an artin algebra and $a$ an idempotent in $\Lambda$. Assume that $\langle a \rangle$ is a stratifying ideal in $\Lambda$. Then we have $(\gamma )\ \operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty$ if and only if $\operatorname *{\mathsf{gl.dim}}{\Lambda /\langle a \rangle }<\infty$ and $\operatorname {\mathsf{pd}}_{\Lambda } \langle a \rangle <\infty$. Moreover, if $(\gamma )$ holds, then $(\beta )$ holds.

Proof.

Assume that $(\gamma )$ $\operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty$. It is clear that $\operatorname {\mathsf{pd}}_\Lambda \langle a \rangle < \infty$ if and only if $\operatorname {\mathsf{pd}}_\Lambda \Lambda /\langle a \rangle < \infty$. Since $\Lambda /\langle a \rangle$ as a $\Lambda$-module is filtered in simple modules occurring as direct summands in $(\Lambda /\langle a \rangle )/(\operatorname {rad}\Lambda /\langle a \rangle )$, we infer that $\operatorname {\mathsf{pd}}_\Lambda \Lambda /\langle a \rangle < \infty$ by the property $(\gamma )$. Since $\langle a \rangle$ is a stratifying ideal in $\Lambda$, we have that

$$\begin{equation*} \operatorname {Ext}^j_{\Lambda /\langle a \rangle }(X,Y) \cong \operatorname {Ext}^j_\Lambda (X,Y) \end{equation*}$$

for all $j\ge 0$ and all modules $X$ and $Y$ in $\operatorname {mod}\Lambda /\langle a \rangle$. Using the above isomorphism and property $(\gamma )$ again, we obtain that $\operatorname {\mathsf{id}}_{\Lambda /\langle a \rangle } Y \leq \operatorname {\mathsf{pd}}_\Lambda (\Lambda /\langle a \rangle )/(\operatorname {rad}\Lambda /\langle a \rangle )$ for all $Y$ in $\operatorname {mod}\Lambda /\langle a \rangle$. Hence $\operatorname *{\mathsf{gl.dim}}\Lambda /\langle a \rangle < \infty$.

Assume conversely that $\operatorname *{\mathsf{gl.dim}}{\Lambda /\langle a \rangle }<\infty$ and $\operatorname {\mathsf{pd}}_{\Lambda } \langle a \rangle <\infty$. From 54, Theorem $3.9$ we have a finite projective resolution $0\longrightarrow \Lambda a\otimes _{a\Lambda a}Q_n\longrightarrow \cdots \longrightarrow \Lambda a\otimes _{a\Lambda a}Q_0\longrightarrow \langle a \rangle \longrightarrow 0$, where $Q_i$ are projective $a\Lambda a$-modules. Then applying the exact functor $e=a-$, it follows from Proposition 2.2 that the sequence $0\longrightarrow Q_n\longrightarrow \cdots \longrightarrow Q_0\longrightarrow a(\langle a \rangle )\longrightarrow 0$ is exact. We infer that $(\beta )\ \operatorname {\mathsf{pd}}_{a\Lambda a} a\Lambda < \infty$, since $a\langle a \rangle \cong a\Lambda$. Since $\operatorname *{\mathsf{gl.dim}}\Lambda /\langle a \rangle < \infty$ and $\operatorname {\mathsf{pd}}_{\Lambda } \Lambda /\langle a \rangle <\infty$, we have that $\operatorname {\mathsf{pd}}_{\Lambda }X\leq \operatorname {\mathsf{pd}}_{\Lambda /\langle a \rangle }X + \operatorname {\mathsf{pd}}_{\Lambda }\Lambda /\langle a \rangle$. We infer that $(\gamma )$ $\operatorname {\mathsf{pd}}_\Lambda \Big ( \frac{\Lambda /\langle a \rangle }{\operatorname {rad}\Lambda /\langle a \rangle } \Big ) < \infty$ holds.

The last claim follows immediately from the above.

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