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

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 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 Fg holds for $\Lambda$ if and only if 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.


Introduction
This paper deals with Gorenstein algebras/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 8.3, 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 Ext * kG (M, M ) is a finitely generated module over the group cohomology ring (see [29,31,61]). 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 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 Ext * Λ (M, M ) for a finitely generated Λ-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 Λ is a selfinjective algebra [26]. The above set of conditions is referred to as Fg (see [26,60]).
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 X, Orlov introduced the singularity category of X, as the Verdier quotient D sg (X) = D b (coh X)/perf(X), where D b (coh X) is the bounded derived category of coherent sheaves on X and perf(X) is the full subcategory consisting of perfect complexes on X. The singularity category D sg (X) captures many geometric properties of X. For instance, if the variety X is smooth, then the singularity category D sg (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 X but is also related to the Homological Mirror Symmetry Conjecture due to Kontsevich [42]. For more information we refer to [50,51,52].
Similarly, the singularity category over a noetherian ring R is defined [12] to be the Verdier quotient of the bounded derived category D b (mod R) of the finitely generated R-modules by the full subcategory perf(R) of perfect complexes and is denoted by D sg (R) = D b (mod R)/perf(R).
In this case the singularity category D sg (R) can be viewed as a categorical measure of the singularities of the spectrum 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 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 [15,17,19,20].
Next, from the perspective of support varieties, we describe some links between the above topics. Support varieties for D b (mod Λ) using the Hochschild cohomology ring of Λ were considered in [60] for a finite dimensional algebra Λ over a field k, where all the perfect complexes perf(Λ) 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 D b (mod Λ)/perf(Λ) -the singularity category.
To have an interesting theory of support, the finiteness condition Fg is pivotal. When Fg is satisfied for an algebra Λ, then Λ is Gorenstein [26,Proposition 1.2], or equivalently, mod Λ is a Gorenstein category.
As we pointed out above, when Λ is Gorenstein, then by Buchweitz-Happel the singularity category D b (mod Λ)/perf(Λ) is triangle equivalent to CM(Λ), the stable category of Cohen-Macaulay modules. When Λ is a selfinjective algebra, then Λ e is selfinjective and CM(Λ e ) = mod Λ e is a tensor triangulated category with Λ as a tensor identity. Let B be the full subcategory of CM(Λ e ) consisting of all bimodules which are projective as a left and as a right Λ-module. Then B is also a tensor triangulated category with tensor identity Λ. The strictly positive part of the graded endomorphism ring of the tensor identity Λ in CM(Λ e ) is isomorphic to the strictly positive part HH 1 (Λ) of the Hochschild cohomology ring of Λ. This is the relevant part for the theory of support varieties via the Hochschild cohomology ring. In addition B is a tensor triangulated category acting on the triangulated category CM(Λ), and we can consider a theory of support varieties for CM(Λ) using the framework described in the forthcoming paper [11]. Therefore the singularity category of the enveloping algebra Λ e encodes the geometric object for support varieties of modules and complexes over the algebra Λ.
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 0 / / A i / / B e / / C / / 0 where both the inclusion functor i : A −→ B and and the quotient functor e : B −→ 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 Pirashvilli 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 [25,53], 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 [1,2,3,4]. Also, Chen and Xi have investigated recollements in relation with tilting theory [22] and algebraic Ktheory [23,24]. 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/ e , 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 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 Λ and aΛa, where a is an idempotent of Λ. This is achieved using recollements of abelian categories. To summarize our main results we introduce the following notion. Given a functor f : B −→ 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 B, and every j > t, there is an isomorphism ). 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 recollements of abelian categories, are given in Corollary 3.6 and Proposition 3.7, Theorem 5.2 and Theorem 4.3.
Main Theorem. Let Λ be an artin algebra over a commutative ring k and let a be an idempotent element of Λ. Let e be the functor a− : mod Λ −→ mod aΛa given by multiplication by a. Consider the following conditions : 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 (A , B, C ). More precisely, we investigate when the exact functor e : B −→ 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. In the setting of the Main Theorem, we 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 : D −→ F between abelian categories, we investigate when Gorensteinness is transferred between D and F . Among other things, we prove that if f is an essentially surjective eventually homological isomorphism, then D is Gorenstein if and only if 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 (A , B, C ) of abelian categories. In fact, we give necessary and sufficient conditions for the quotient functor e : B −→ C to induce a triangle equivalence between the singularity categories of B and 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 Λ with an idempotent element a, we give a sufficient condition for the stable categories of Cohen-Macaulay modules of Λ and aΛa to be triangle equivalent, see Corollary 5.5.
In Section 6 and Section 7, which form a unit, we investigate the finite generation condition 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 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 Proposition 6.3 and Corollary 6.4. In Section 7, we use this result to show that Fg holds for a finite dimensional algebra Λ over a field if and only if Fg holds for the algebra aΛa, where a is an idempotent element of Λ which satisfies certain assumptions (see Theorem 7.10).
The final Section 8 is devoted to applications and examples of our main results. First we apply our results to triangular matrix algebras. For a triangular matrix algebra Λ = Σ 0 ΓMΣ Γ , we compare Λ to the algebras Σ and Γ with respect to the Fg condition, Gorensteinness and singularity categories. In particular, we recover a result by Chen [15] concerning the singularity categories of Λ and Σ. Then we consider some special cases where there are relations between the assumptions of our main results (see (α)-(δ) in Main Theorem) and provide an interpretation for quotients of path algebras. Finally, we compare our results to those of Nagase [47].
Conventions and Notation. For a ring R we work usually with left R-modules and the corresponding category is denoted by Mod R. The full subcategory of finitely presented R-modules is denoted by 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 rad R. By a module over an artin algebra Λ, we mean a finitely presented (generated) left Λ-module. Definition 2.1. A recollement situation between abelian categories A , B and C is a diagram 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 [32,54]. and (e, r) are isomorphisms : 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 Λ be an artin k-algebra, where k is a commutative artin ring, and let a be an idempotent element in Λ.
(i) We have the following recollement of abelian categories : The functor e : mod Λ −→ mod aΛa can be also described as follows : e = a(−) ∼ = Hom Λ (Λa, −) ∼ = aΛ ⊗ Λ −. We write a for the ideal of Λ generated by the idempotent element a. Then every left Λ/ a -module is annihilated by a and thus the category mod Λ/ a is the kernel of the functor a(−). (ii) Let Λ e = Λ ⊗ k Λ op be the enveloping algebra of Λ. The element ε = a ⊗ a op is an idempotent element of Λ e . Therefore as above we have the following recollement of abelian categories : Note that (aΛa) e ∼ = εΛ e ε 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/ e , 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 Fg, and recall some important results. For a more detailed exposition of these topics, see sections 2-5 of [60]. Let Λ be an artin algebra over a commutative ring k. We define the Hochschild cohomology ring HH * (Λ) of Λ by 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 Λ 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 Λ is projective over k. The following more general result was shown in [59, Theorem 1.1] (see also [62], which proves graded commutativity of several cohomology theories in a uniform way).
Theorem 2.5. Let Λ be an algebra over a commutative ring k such that Λ is flat as a module over k. Then the Hochschild cohomology ring HH * (Λ) is graded commutative.
To describe the finite generation condition Fg, we first need to define a HH * (Λ)-module structure on the direct sum of all extension groups of a Λ-module with itself (for more details about this module structure, see [59]). Assume that Λ is flat as k-module, and let M be a Λ-module. The direct sum of all extension groups of M with itself is a graded k-algebra with multiplication given by Yoneda product. We give it a graded HH * (Λ)-module structure by the graded ring homomorphism . which is defined in the following way. Any homogeneous element of positive degree in HH * (Λ) can be represented by an exact sequence of Λ e -modules, where every P i is projective. Tensoring this sequence throughout with M gives an exact sequence of Λ-modules (the exactness of this sequence follows from the facts that Λ is flat as k-module and that the modules P i are projective Λ e -modules). Using the isomorphism Λ ⊗ Λ M ∼ = M , we get an exact sequence of Λ-modules starting and ending in M ; we define ϕ M ([η]) to be the element of Ext * Λ (M, M ) represented by this sequence. For elements of degree zero in HH * (Λ), the map ϕ M is defined by tensoring with M and using the identification Λ ⊗ Λ M ∼ = M .
In [26], Erdmann-Holloway-Snashall-Solberg-Taillefer identified certain assumptions about an algebra Λ 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 Fg if it satisfies both Fg1 and Fg2. We use the following definition of Fg, which is equivalent (by [60,Proposition 5.7]) to the definition of Fg1 and Fg2 given in [26].
Definition 2.6. Let Λ be an algebra over a commutative ring k such that Λ is flat as a module over k. We say that Λ satisfies the Fg condition if the following is true: (i) The ring HH * (Λ) is noetherian. We end this section by describing a connection between the Fg condition and Gorensteinness.

Eventually homological isomorphisms in recollements
Given a functor f : D −→ F between abelian categories and an integer t, the functor f is called an t-homological isomorphism if there is an isomorphism ) for every pair of objects B and B ′ in B, and every j > t. 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 The functor e induces maps of extension groups for all objects X and Y in B and for every j ≥ 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 : B −→ 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 (mod Λ/ a , mod Λ, mod aΛa) of Example 2.3 (i). In particular, we characterize when the functor e : mod Λ −→ mod aΛa 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 Fg condition of the algebras Λ and aΛa, where a is an idempotent in Λ.
We start by fixing some notation. For an injective coresolution 0 −→ B −→ I 0 −→ I 1 −→ · · · of B in B, we say that the image of the morphism I n−1 −→ I n is an n-th cosyzygy of B, and we denote it by Σ n (B). Dually, if · · · −→ P 1 −→ P 0 −→ B −→ 0 is a projective resolution of B in B, then we say that the kernel of the morphism P n−1 −→ P n−2 is an n-th syzygy of B, and we denote it by Ω n (B). Also, if X is a class of objects in B, then we denote by X ⊥ = {B ∈ B | Hom B (X, B) = 0} the right orthogonal subcategory of X and by ⊥ X = {B ∈ B | Hom B (B, X) = 0} the left orthogonal subcategory of X.
We now describe precisely how the maps (3.1) induced by the functor e in a recollement are defined. Let D and F be abelian categories and f : D −→ F an exact functor which has a left and a right adjoint (for example, the functors i and e in a recollement have these properties). If is an exact sequence in D, then we denote by f (ξ) the exact sequence It is clear that this operation commutes with Yoneda product; that is, if ξ and ζ are composable exact sequences in D, then f (ξζ) = f (ξ) · f (ζ). For every pair of objects X and Y in D and every nonnegative integer j, we define a group homomorphism for a j-fold extension η of X by Y , where j > 0.
For an object X in D, the direct sum Ext * D (X, X) = ∞ j=0 Ext j 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 f * X,X : Ext * D (X, X) −→ Ext * F (f (X), f (X)).
Remark 3.1. We explain briefly why the maps f j X,Y and f * X,X defined above are homomorphisms.
(i) The functor f being a right and left adjoint implies that it preserves limits and colimits and therefore it preserves pullbacks and pushouts. Thus the map f j X,Y 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 induced by the functor e : B −→ C in a recollement, where we let one argument be fixed and the other vary over all objects of B. In [54], the first author studied when these maps are isomorphisms for all degrees up to some bound n, that is, for 0 ≤ j ≤ n. This immediately leads to a description of when these maps are isomorphisms in all degrees, which we state as the following theorem.
) is an isomorphism for every object B ′ in B and every nonnegative integer j.

(b) The object B has an injective coresolution of the form
The above theorem describes when the maps e j B,B ′ 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 A be an abelian category, n be an integer, and let
Then for every i > n and Z ∈ A , the map   We have the following relations between these statements :

Proof. (i) By dimension shift, statement (c) is equivalent to
be the beginning of the chosen projective resolution of B, where K = Ω n (B) is the n-th syzygy of B. Consider the following group homomorphisms : , e(B ′ )) (3.2) Here, the maps π * and (e(π)) * are isomorphisms by Lemma 3.3. Note that for (e(π)) * we use the fact that pd C e(P ) ≤ n for every projective object P in B. The map e j−n K,B ′ is an isomorphism by Theorem 3.2 (i). Thus, we have an isomorphism for every j ≥ m + n + 1 and B ′ ∈ B. We want to show that this is the same map as e j B,B ′ . We consider an element [η] ∈ Ext j−n B (K, B ′ ), and follow it through the maps (3.2). We then get the following elements : This shows that our isomorphism takes any element [ζ] ∈ Ext j B (B, B ′ ) to the element [e(ζ)] ∈ Ext j C (e(B), e(B ′ )). Thus, our isomorphism is e j B,B ′ . Dually to the above theorem, we have the following generalization of some of the implications in Theorem 3.2 (ii).  We have the following relations between these statements : In the above results, we fixed an object B of the category B, and considered the maps e j B,B ′ or e j B ′ ,B for all objects B ′ in 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 j B,B ′ are isomorphisms in almost all degrees j for all objects B and B ′ of 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 (A , B, C ) be a recollement and assume that B and C have enough projective and injective objects. Let m and n be two integers. Assume that one of the following conditions hold : (ǫ op ) Every object of B has an n-th cosyzygy which lies in i(Proj A ) ⊥ .
(δ) sup{id C e(I) | I ∈ Inj B} ≤ m. Then the functor e is an (m + n)-homological isomorphism, and in particular the map is an isomorphism for all objects B and B ′ of B and for every j > m + n.
We now show a partial converse of the above result.
Proposition 3.7. Let (A , B, C ) be a recollement and assume that B and C have enough projective and injective objects. Assume that the functor e is an eventually homological isomorphism. Then the following hold : In particular, if e is an t-homological isomorphism for a nonnegative integer t, then each of the above dimensions is bounded by t.
Proof. (α) Let A be an object of A . For every B in B and j > t, we get , 0) = 0, since ei = 0 by Proposition 2.2, and thus id B i(A) ≤ t. The proof of (γ) is similar.
(β) Let P be a projective object of B. For every C in C and j > t, we get , el(C)) ∼ = Ext j B (P, l(C)) = 0, since el ∼ = Id C by Proposition 2.2, and thus pd C e(P ) ≤ t. The proof of (δ) is similar.
We first need the following well-known observation. We also need the next easy result whose proof is left to the reader. Then the following inequalities hold : (i) pd aΛa e(P ) ≤ pd aΛa aΛ, for every P ∈ proj Λ.
The following is a consequence of Theorem 3.4 and Theorem 3.5 for artin algebras.
Corollary 3.11. Let Λ be an artin algebra and a an idempotent element in Λ, and let m and n be integers.
) is an isomorphism for every Λ-module N , and for every integer j > m + n.
) is an isomorphism for every Λ-module N , and for every integer j > m + n.
Proof. (i) Consider the recollement (mod Λ/ a , mod Λ, mod aΛa) of Example 2.3. Since every simple Λ/ a -module is also simple as a Λ-module it follows from Lemma 3.9 that This implies that Hom Λ (Ω m (M ), N ) = 0 for every Λ/ a -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 : mod Λ −→ mod aΛa is an eventually homological isomorphism. This constitutes the first part of the Main Theorem presented in the introduction. is an isomorphism. (ii) The functor e is an eventually homological isomorphism.
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).

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 : D −→ 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 : D −→ F , then D is Gorenstein if and only if 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 A be an abelian category with enough projective and injective objects. We consider the following invariants associated to A : Then we have the following notion of Gorensteinness for abelian categories. 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 A be an abelian category with enough projective and injective objects and let
X be an object of 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. (a) For every D in D: We have the following.
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 D is Gorenstein if and only if F is Gorenstein. (vi) If f is an eventually homological isomorphism and (b) holds, then F being Gorenstein
implies that D is Gorenstein.
Proof. We first assume that f is an essentially surjective t-homological isomorphism and show the inequality pd F f (D) ≤ sup{pd 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{pd D D, t} + 1. For any object X in F , there is an object X ′ in D with f (X ′ ) ∼ = X, since the functor f is essentially surjective. By using that f is a t-homological isomorphism, we get We now assume that (a) and (b) hold and f is essentially surjective, and show the inequality spli F ≤ sup{spli D, t}; the other inequalities in parts (iii) and (iv) are proved similarly. Let I be an injective object of F . Since f is essentially surjective, we can choose an object D in D such that f (D) ∼ = I. By (a), the object D has finite injective dimension, and then by Lemma 4.2, its projective dimension is at most spli D. Using (b), we get Since this holds for any injective object I in F , we have spli F ≤ sup{spli D, t}.
Parts (v) and (vi) follow by combining parts (i)-(iv).  (i) Assume that the categories B and C have enough projective and injective objects, and that the functor e is an eventually homological isomorphism. Then B is Gorenstein if and only if C is Gorenstein. (ii) Assume that the category B has enough projective and injective objects, and that we have either 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), A has enough projective and injective objects since B does (see [54,Remark 2.5]).
It follows from [54,Proposition 4.15] (or its dual) that the functor i : A −→ B is a homological embedding, i.e. the map i n X,Y is an isomorphism for all objects X and Y in A and every n ≥ 0. In particular, this means that i is a 0-homological isomorphism. By Theorem 4.3 (i), we have and for every object A in A . We show that spli A ≤ spli B. Let I be an injective object in A. By assumption, we have id B i(I) < ∞, and then by the first inequality in (4.1) and Lemma 4.2, we have Hence we have spli A ≤ spli B. By a similar argument, we have silp A ≤ silp B. The result follows.
In a recollement (A , B, C ) we have seen that the implications B Gorenstein if and only if C Gorenstein and B Gorenstein implies C Gorenstein hold under various additional assumptions. It is then natural to ask if the categories A and C being Gorenstein could imply that B is Gorenstein. The next example shows that this is not true in general.  Recall that an artin algebra Λ is called Gorenstein if id Λ Λ < ∞ and id Λ Λ < ∞ (see [5,6]). Note that mod Λ is a Gorenstein category if and only if Λ 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.
If the algebra Λ is Gorenstein, then the algebra Λ/ a is Gorenstein.

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 (A , B, C ). In particular we are interested in finding necessary and sufficient conditions such that the singularity categories of B and C are triangle equivalent. We start by recalling some well known facts about singularity categories.
Let B be an abelian category with enough projective objects. We denote by D b (B) the derived category of bounded complexes of objects of B and by K b (Proj B) the homotopy category of bounded complexes of projective objects of B. Then the singularity category of B ( [12,50]) is defined to be the Verdier quotient : It is well known that the singularity category D sg (B) carries a unique triangulated structure such that the quotient functor Q B : D b (B) −→ D sg (B) is triangulated, see [43,49,63]. Recall that the objects of the singularity category D sg (B) are the objects of the bounded derived category D b (B), the morphisms between two objects X • −→ Y • are equivalence classes of fractions (X • ← L • → Y • ) such that the cone of the morphism L • −→ X • belongs to K b (Proj B) and the exact triangles in D sg (B) are all the triangles which are isomorphic to images of exact triangles of D b (B) via the quotient functor Q B . Note that a complex X • is zero in D sg (B) if and only if X • ∈ K b (Proj B). Following Chen [19,20], we say that two abelian categories A and B are singularly equivalent if there is a triangle equivalence between the singularity categories D sg (A ) and D sg (B). This triangle equivalence is called a singular equivalence between A and B.
To proceed further we need the following well known result for exact triangles in derived categories. For a complex X • in an abelian category A we denote by σ >n (X • ) the truncation −→ X n+2 −→ · · · , and by H n (X • ) the n-th homology of X • .
Now we are ready to prove the main result of this section which gives necessary and sufficient conditions for the quotient functor e : B −→ C to induce a triangle equivalence between the singularity categories of B and C . This is a general version of the second part of the Main Theorem presented in the introduction. , see also [40], that 0 . Suppose first that P • is concentrated in degree zero, so we deal with a projective object P of B. Since the object e(P ) has finite projective dimension it follows that there is a quasi-isomorphism Q is a triangulated subcategory. Continuing inductively on the length of the complex P • we infer that the object D b (e)(P • ) lies in K b (Proj C ) and so our claim follows. Then since the triangulated functor • Q C factors uniquely through Q B via a triangulated functor D sg (e) : D sg (B) −→ D sg (C ), that is the following diagram is commutative : . Assume first that B • is concentrated in degree zero. Hence we deal with an object B ∈ B such that B ∼ = i(A) for some A ∈ A , and therefore our claim follows. Now consider a complex and . Then from the second triangle it follows that σ >0 (B • ) ∈ K b (Proj B) and therefore from the first triangle we get that σ >−1 ( . Using this we can form the quotient K b (Proj B)/D b A (B), and then we have the following exact commutative diagram : is an equivalence, where we denote it by K b (e). First from the above commutative diagram and since there is an equivalence , it follows that the functor K b (e) is fully faithful. Let P • : 0 −→ P n −→ · · · −→ P 1 −→ P 0 −→ 0 be an object of K b (Proj C ). Each P i is a projective object in C and from Proposition 2.2 we have el(P i ) ∼ = P i with l(P i ) ∈ Proj B. Then the complex l(P • ) : 0 −→ l(P n ) −→ · · · −→ l(P 1 ) −→ l(P 0 ) −→ 0 is such that K b (e)(l(P • )) = P • . This implies that the functor K b (e) is essentially surjective. Hence the functor K b (e) is an equivalence.
In conclusion, from the above exact commutative diagram we infer that the singularity categories of B and C are triangle equivalent. Applying Theorem 5.2 to the recollement of module categories (mod R/ e , mod R, 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 pd eRe eR < ∞ 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. We end this section with an application to stable categories of Cohen-Macaulay modules. Let Λ be a Gorenstein artin algebra. We denote by CM(Λ) the category of (maximal) Cohen-Macaulay modules defined as follows : Then it is known that the stable category CM(Λ) modulo projectives is a triangulated category, see [38], and moreover there is a triangle equivalence between the singularity category D sg (mod Λ) and the stable category CM(Λ), see [

Finite generation of cohomology rings
In this section, we describe a way to compare the Fg condition (see Definition 2.6) for two different algebras. This is used in the next section for the algebras Λ and aΛa, where Λ is a finite dimensional algebra over a field and a is an idempotent in Λ.
Let Λ and Γ be two artin algebras over a commutative ring k, and assume that they are flat as k-modules. Let M = Λ/(rad Λ) and N = Γ/(rad Γ). Assume that we have graded ring isomorphisms f and g making the diagram commute, where the maps ϕ M and ϕ N are defined in Subsection 2.2. Then it is clear that Fg for Λ is exactly the same as Fg for Γ, since all the relevant data for the Fg condition is exactly the same for the two algebras. However, we can come to the same conclusion even if the homology groups for Λ and Γ 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 Fg condition holds for Λ if and only if it holds for Γ. 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 Fg condition) in Proposition 6.2. Finally, we state the result for 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 graded over the nonnegative integers. We denote the set of nonnegative integers by N 0 . If R is a graded ring and n a nonnegative integer, we use the notation R ≥n for the graded ideal 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. 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 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 -module, and that every S i is finitely generated as left and as right S 0 -module. Let n be a nonnegative integer, and assume that there exists an isomorphism φ : R ≥n −→ S ≥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 φ −1 instead of φ.
Assume that R is not left noetherian. Let 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 For any degree d, we can make an ascending sequence I (0) d ⊆ · · · of R 0 -submodules of R d by taking the degree d part of each ideal in 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 For i ∈ N 0 , we define δ(i) as the smallest number such that These functions have the following interpretation. For a degree d, the number σ(d) is the index in the sequence I where the ideals in the sequence have stabilized up to degree d. For an index i, the number δ(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∈N0 of indices and a sequence (d j ) j∈N0 of degrees by We observe that for every positive integer j, we have We now construct a sequence J of graded left ideals in S. For every nonnegative integer j, we choose an element (this is possible because d j = δ(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 We let J be the sequence of these ideals : We want to show that each inclusion here is strict. This means that we must show, for every positive integer j, that φ(x j ) is not an element of J (j−1) . We show this by contradiction. Assume that there is a j such that φ(x j ) ∈ J (j−1) . Then we can write φ(x j ) as a sum where each s m is an element of S. Since φ(x j ) and every φ(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 Thus s m is either zero or in the image of φ. We use this to find corresponding elements in R. Let, for each m ∈ {1, . . . , j − 1}, Applying φ −1 gives Since we have x m ∈ I (im+1) ⊆ I (ij ) for every m, this means that x j ∈ I (ij ) . This is a contradiction, since x j is chosen so that it does not lie in I (ij ) .
We have shown that the sequence J is a strictly ascending sequence of graded left ideals in S. Thus S in 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 θ : R −→ M a graded ring homomorphism. View M as a graded left R-module with scalar multiplication given by θ. Assume that R 0 is noetherian, that every R i is finitely generated as left and as right R 0 -module, and that every M i is finitely generated as left R 0 -module.
Similarly, let R ′ and M ′ be graded rings, and θ ′ : R ′ −→ M ′ a graded ring homomorphism. View M ′ as a graded left R ′ -module with scalar multiplication given by θ ′ . Assume that R ′ 0 is noetherian, that every R ′ i is finitely generated as left and as right R ′ 0 -module, and that every M ′ i is finitely generated as left R ′ 0 -module.
Assume that there are graded rng isomorphisms φ : R ≥n −→ R ′ ≥n and ψ : M ≥n −→ M ′ ≥n (for some nonnegative integer n) such that the diagram commutes. Then the following two conditions are equivalent.
(i) R is noetherian and M is finitely generated as left R-module.
(ii) R ′ is noetherian and M ′ is finitely generated as left R ′ -module.
Proof. We prove that condition (i) implies condition (ii). The opposite implication is proved in exactly the same way by using φ −1 and ψ −1 instead of φ and ψ.
Assume that condition (i) holds. Then by Proposition 6.2, R ′ is noetherian. We need to show that M ′ is finitely generated as left R ′ -module.
We begin with choosing generating sets for things we know to be finitely generated. Note that the ideal R ≥n of R is finitely generated, since R is noetherian. Let A be a finite homogeneous generating set for R ≥n . Let G be a finite homogeneous generating set for M as left R-module. For every i, let B i be a finite generating set for M ′ i as left R ′ 0 -module. Let b R = max |a| a ∈ A and b M = max |g| g ∈ G be the maximal degrees of elements in our chosen generating sets for R and M , respectively. Let Define the set G ′ to be We want to show that G ′ generates M ′ as 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 ′ ∈ M ′ be a homogeneous element with |m ′ | > b. Let m = ψ −1 (m ′ ). We can write m as a sum 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 Thus r i lies in the ideal R ≥n , so we can write it as a sum 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 ≥n . For every u i,j , we have Now we can write the element m as If we have a i,j · g i = 0 for some terms in the sum, we ignore these terms. For every pair (i, j), we have |θ(u i,j )| = |u i,j | > n and |a i,j · g i | ≥ |a i,j | ≥ n.
This means that when applying ψ to a term in the above sum for m, we have Using this, we can write our element m ′ of M ′ in the following way : For every pair (i, j), we have so ψ(a i,j · 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 Fg condition, and obtain the main result of this section. Proposition 6.4. Let Λ and Γ be artin algebras over a commutative ring k, and assume that they are flat as k-modules. Let M and M ′ be Λ-modules, and let N and N ′ be Γ-modules, such that M ∼ = Λ/(rad Λ) and N ′ ∼ = Γ/(rad Γ). 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 : and HH ≥n (Λ) Then Λ satisfies Fg if and only if Γ satisfies 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 HH i (Λ), Ext i Λ (M, M ) and Ext i Λ (M ′ , M ′ ) are finitely generated as k-modules. Therefore, they are also finitely generated as HH 0 (Λ)-modules. The ring HH 0 (Λ) is noetherian since it is an artin algebra. Similarly, we see that HH i (Γ), Ext i Γ (N, N ) and Ext i Γ (N ′ , N ′ ) are finitely generated HH 0 (Γ)-modules, and that the ring HH 0 (Γ) is noetherian. Assume that Λ satisfies Fg. Then HH * (Λ) is noetherian, and by Theorem 2.7, Ext * Λ (M ′ , M ′ ) is a finitely generated HH * (Λ)-module. By applying Proposition 6.3 to the commutative diagram with f ′ and g ′ , we see that Γ satisfies Fg.
The opposite inclusion is proved in the same way by using the other commutative diagram.

Finite generation of cohomology rings in module recollements
We now investigate the relationship between the Fg condition (see Definition 2.6) for an algebra Λ and the algebra aΛa, where a is an idempotent of Λ. We show that, given some conditions on the idempotent a, the algebra Λ satisfies Fg if and only if the algebra aΛa satisfies 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, Λ a finite-dimensional k-algebra and a an idempotent in Λ. We denote by e and E the exact functors 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 id Λ Λ/ a rad Λ/ a and pd aΛa (aΛ) are finite, or, equivalently, that the two dimensions pd Λ Λ/ a rad Λ/ a and pd (aΛa) op (Λa) are finite. We show (given an additional technical assumption about the algebra Λ) that this is in fact also sufficient for the map E * Λ,Λ 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 * Λ,Λ 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 ϕ M and ϕ e(M) are defined by using tensor functors. It is convenient to have short names for these functors. For every Λ-module M , we define t M and T M to be the tensor functors  We are now able to show that the diagrams we consider are commutative.
where each P j is a projective Λ e -module. We apply the compositions of maps ϕ e(M) • E * Λ,Λ and e * M,M • ϕ M to [η], and show that we get the same result in both cases.
We first consider the map ϕ e(M) • E * Λ,Λ . If we apply the functor E to η, then we get the exact sequence 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 : pd εΛ e ε εΛ e < ∞ and Ext j Λ e Λ, Λ e / ε rad Λ e / ε = 0 for j ≫ 0.
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 Λ to be able to describe the simple modules over Λ e ), and finally tie it together in Proposition 7.9, where we show that E * Λ,Λ is an isomorphism in sufficiently high degrees.
First, we show how the projective dimension of the tensor product M ⊗ 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 Λ-module Λ M and N Λ both have finite projective dimension, then their tensor product M ⊗ k N has finite projective dimension as Λ emodule. Lemma 7.3. Let Σ and Γ be k-algebras, and let M be a Σ-module and N a Γ-module. If M has finite projective dimension as Σ-module and N has finite projective dimension as Γ-module, then M ⊗ k N has finite projective dimension as (Σ ⊗ k Γ)-module, and Proof. Assume that pd Σ M = m and pd Γ N = n. Then we have finite projective resolutions of M and N , respectively. Let P and Q denote the corresponding deleted resolutions. Consider the tensor product of the complexes P and Q. This is a bounded complex of projective (Σ ⊗ k Γ)-modules. We want to show that it is in fact a deleted projective resolution of the (Σ ⊗ k Γ)-module M ⊗ k N , which completes the proof. We need to show that the complex P ⊗ k Q is exact in all positive degrees and has homology M ⊗ k N in degree zero. Let us temporarily forget the Σ-and Γ-structures, and view P as a complex of right k-modules, Q as a complex of left k-modules, and P ⊗ k Q as a complex of abelian groups. Then by the Künneth formula for homology, see [57,Corollary 11.29], we have an isomorphism α : of abelian groups, given by α([p] ⊗ [q]) = [p ⊗ q], for p ∈ P i and q ∈ Q j . Observe that α preserves (Σ ⊗ k Γ)-module structure. Thus, α is a Σ ⊗ k Γ-module isomorphism, and we get This means that the complex P ⊗ k Q is a deleted projective resolution of the (Σ ⊗ k Γ)-module M ⊗ k N . Since the complex P ⊗ k Q is zero in all degrees above m + n, we get and the proof is complete.
Using the above result, we find that the assumptions we make about the left and right aΛamodules aΛ and Λa having finite projective dimension imply the first condition we need for applying Corollary 3.11 (i), namely that the εΛ e ε-module εΛ e has finite projective dimension. We state this as the following result.
Lemma 7.4. We have the following inequality : pd εΛ e ε εΛ e ≤ pd aΛa aΛ + pd (aΛa) op Λa Proof. Note that εΛ e is isomorphic to (aΛ ⊗ k Λa) as left (aΛa) e -modules and that the rings (aΛa) e and εΛ e ε are isomorphic. By using these isomorphisms and Lemma 7.3, we get that pd εΛ e ε εΛ e = pd (aΛa) e εΛ e = pd (aΛa) e (aΛ ⊗ k Λa) ≤ pd aΛa aΛ + pd (aΛa) op Λa. 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 Λ e to extension groups over Λ. Furthermore, we need to be able to describe the simple Λ e -modules in terms of simple Λmodules. It is reasonable to expect that taking the tensor product should produce all the simple Λ 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 of Λ e -modules if and only if the Λ e -module Proof. It is easy to show that as Λ e -modules, and that the ideal Λ ⊗ k (rad Λ op ) + (rad Λ) ⊗ k Λ op of Λ e is nilpotent. This means that if (Λ/ rad Λ) ⊗ k (Λ op / rad Λ op ) is a semisimple Λ e -module, then it is isomorphic to Λ e / rad Λ e . The opposite implication is obvious. Now we give an example showing that (Λ/ rad Λ) ⊗ k (Λ op / rad Λ op ) is not necessarily semisimple for a finite dimensional algebra Λ over a field k.
Example 7.7. Let k = Z 2 (x) be the field of rational functions in one indeterminant x over Z 2 , and let Λ be the 2-dimensional k-algebra k[y]/ y 2 − x . Then Λ is a field, so that rad Λ = (0). The element α = y ⊗ 1 + 1 ⊗ y satisfies α 2 = 0. Hence α is a nilpotent non-zero ideal in Λ e , and therefore Λ e is not semisimple.
We assume that (Λ/ rad Λ) ⊗ k (Λ op / rad Λ 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 Λ/ rad Λ is separable as k-algebra (by [21,Corollary 7.8 (i)]) if k is algebraically closed (this can be shown by using the Wedderburn-Artin Theorem), or if Λ 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 (Λ/ rad Λ) ⊗ k (Λ op / rad Λ op ) is a semisimple Λ e -module, and that we have Proof. By Lemma 7.6, every simple Λ e -module is a direct summand of a module of the form S ⊗ k D(T ) for some simple Λ-modules S and T , where D is the duality Hom k (−, k) : mod Λ −→ mod Λ op . If neither of the modules S or T is annihilated by the ideal a , then we have which means that no nonzero direct summand of the Λ e -module S ⊗ k D(T ) is a Λ e / ε -module. Let j be an integer such that In order to prove the result, it is sufficient to show that Ext j Λ e (Λ, U ) = 0 for every simple Λ e / εmodule U . By the above reasoning, every such U is a direct summand of a module S ⊗ k D(T ) for some simple Λ-modules S and T , where at least one of S and T is annihilated by a and is thus a simple Λ/ a -module. Using Lemma 7.5, we get Ext j Λ e (Λ, S ⊗ k D(T )) ∼ = Ext j Λ (T, S) = 0, since we have pd Λ T < j or id Λ S < j. It follows that Ext j Λ e (Λ, U ) = 0.
The following result summarizes the above work and shows that, with the assumptions we have indicated for the algebra Λ and the idempotent a, the functor E gives isomorphisms E j Λ,Λ : HH j (Λ) −→ HH j (aΛa) in almost all degrees j.
Proposition 7.9. Assume that (Λ/ rad Λ) ⊗ k (Λ op / rad Λ op ) is a semisimple Λ e -module, and that the functor e is an eventually homological isomorphism. Then the map ) is an isomorphism for every Λ e -module M and every integer j such that In particular, we have isomorphisms for almost all degrees j.
Proof. We use Corollary 3.11 (i) on the algebra Λ e , the idempotent ε = a⊗a op and the Λ e -module Λ. Let m and n be the integers m = max pd Λ Λ/ a rad Λ/ a , id Λ Λ/ a rad Λ/ a + 1 and n = pd aΛa aΛ + pd (aΛa) op Λa.
Note that m and n are finite by Corollary 3.12. By Lemma 7.8, we have Ext j Λ e Λ, Λ e / ε rad Λ e / ε = 0 for j ≥ m, and by Lemma 7.4, we have pd εΛ e ε εΛ e ≤ n. Now the result follows from Corollary 3.11 (i) by noting that (aΛa) e is the same algebra as εΛ e ε and that our functor E = a − a is the same as the functor ε− given by left multiplication with the idempotent ε. Since we have such diagrams for every Λ-module M and the functor e is essentially surjective (see Proposition 2.2), we can make one diagram with M = Λ/ rad Λ and another with e(M ) ∼ = aΛa/ rad aΛa. Then, by Proposition 6.4, it follows that Λ satisfies Fg if and only if aΛa satisfies Fg.

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 Λ be an artin algebra over a commutative ring k and let a be an idempotent element of Λ. Let e be the functor a− : mod Λ −→ mod aΛa given by multiplication by a. Consider the following conditions : Then the following hold. This section is divided into three subsections. In the first subsection, we apply Theorem 8.1 to the class of triangular matrix algebras. In the second subsection, we consider some cases where the conditions (α)-(δ) in Theorem 8.1 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 last subsection, we compare our work to that of Nagase in [47].
8.1. Triangular Matrix Algebras. Let Σ and Γ be two artin algebras over a commutative ring k, and let Γ M Σ be a Γ-Σ-bimodule such that M is finitely generated over k, and k acts centrally on M . Then we have the artin triangular matrix algebra where the addition and the multiplication are given by the ordinary operations on matrices.
The module category of Λ has a well known description, see [7,30]. In fact, a module over Λ is described as a triple (X, Y, f ), where X is a Σ-module, Y is a Γ-module and f : M ⊗ Σ X −→ Y is a Γ-homomorphism. A morphism between two triples (X, Y, f ) and (X ′ , Y ′ , f ′ ) is a pair of homomorphisms (a, b), where a ∈ Hom Σ (X, X ′ ) and b ∈ Hom Γ (Y, Y ′ ), such that the following diagram commutes : We define the following functors : (i) The functor T Σ : mod Σ −→ mod Λ is defined on Σ-modules X by T Σ (X) = (X, M ⊗ Σ X, Id M⊗X ) and given a Σ-homomorphism a : X −→ X ′ then T Σ (a) = (a, Id M ⊗a).
Similarly we define the functor U Γ : mod Λ −→ mod Γ. (iii) The functor Z Σ : mod Σ −→ mod Λ is defined on Σ-modules X by Z Σ (X) = (X, 0, 0) and given a Σ-homomorphism a : X −→ X ′ then Z Σ (a) = (a, 0). Similarly we define the functor Z Γ : mod Γ −→ mod Λ.  We want to use Theorem 8.1 to compare the triangular matrix algebra Λ with the algebras Σ and Γ. First consider the case where we compare Λ with Σ. We then take the idempotent a in the theorem to be e 1 , and we can reformulate the conditions (α), (β), (γ) and (δ) as follows: (α) The functor Z Γ sends every Γ-module to a Λ-module with finite injective dimension.
(β) The functor U Σ sends every projective Λ-module to a Σ-module with finite projective dimension. (γ) The functor Z Γ sends every Γ-module to a Λ-module with finite projective dimension. (δ) The functor U Σ sends every injective Λ-module to a Σ-module with finite injective dimension. By interchanging Σ and Γ, we get a similar reformulation of the conditions for the case where we compare Λ with Γ.
The next result clarifies when the above hold for the recollement (8.2) of a triangular matrix algebra Λ. (iii) Assume that gl. dim Σ < ∞. Then id Λ Z Σ (X) < ∞ for every Σ-module X.
Proof. (i) It is known, see [7], that the indecomposable projective Λ-modules are of the form T Σ (P ), where P is an indecomposable projective Σ-module, and Z Γ (Q), where Q is an indecomposable projective Γ-module. Hence it is enough to consider modules of these forms. We have is an adjoint pair and Z Γ is exact it follows that the functor U Γ preserves injectives.
(iv) This follows from [58, Lemma 2.4] since a Λ-module (X, Y, f ) has finite projective dimension if and only if the projective dimensions of X and Y are finite.
(iii) Assume that gl. dim Σ < ∞ and pd Γ M < ∞. If Γ is Gorenstein, then there is a triangle equivalence between the stable categories of Cohen-Macaulay modules of Λ and Γ : Algebras with ordered simples. In this subsection, we apply Theorem 8.1 to cases where there exists a total order of the simple Λ/ a -modules with the property that for every pair S and S ′ of simple Λ/ a -modules. With this assumption, we show that we have the implications (α) =⇒ (δ) and (γ) =⇒ (β) between the conditions in Theorem 8.1. We then consider some special cases where such orderings appear. We need the following preliminary results.
Lemma 8.8. Let Λ be an artin algebra, let M be a Λ-module with minimal projective resolution · · · −→ P 1 −→ P 0 −→ M −→ 0, and let S be a simple Λ-module. Then, for every nonnegative integer n, we have Ext n Λ (M, S) = 0 if and only if the projective cover of S is not a direct summand of P n . Lemma 8.9. Let Λ be an artin algebra, and let a be an idempotent in Λ. Let S be a simple Λ-module which is not annihilated by the ideal a , and let P be the projective cover of S. Then aP is a projective aΛa-module.

Proof. We have
Hom Λ (Λa, S) ∼ = aS = 0, so there exists a nonzero morphism f : Λa −→ S. Decomposing the idempotent a into a sum a = a 1 +· · ·+a t of orthogonal primitive idempotents gives a decomposition Λa ∼ = Λa 1 ⊕· · ·⊕Λa t of Λa into indecomposable projective modules. For some i, we must then have a nonzero morphism f i : Λa i −→ S. Since S is simple, this means that Λa i is its projective cover. Since a · a i = a i , we get aP ∼ = aΛa i = (aΛa)a i . Therefore aP is a projective aΛa-module. Proof. We show the second implication; the first can be showed in a similar way. Assume that (γ) holds, that is, every Λ/ a -module has finite projective dimension as a Λ-module. We want to show that (β) holds, that is, the aΛa-module aΛ has finite projective dimension. As in Section 7, we let e be the exact functor e = (a−) : mod Λ −→ mod aΛa given by multiplication with a. Then what we need to show is that e(Λ) has finite projective dimension as aΛa-module.
Let S 1 · · · S s be all the simple Λ/ a -modules (up to isomorphism), ordered by the total order . Let T 1 , . . . , T t be all the other simple Λ-modules (up to isomorphism). Let Q i be the projective cover of S i (considered as Λ-module) and Q ′ j the projective cover of T j , for every i and j. These are all the indecomposable projective Λ-modules up to isomorphism, so it is sufficient to show that e(Q i ) and e(Q ′ j ) have finite projective dimension as aΛa-modules for every i and j.
Lemma 8.13. Let Λ be an artin algebra, and let a be an idempotent in Λ. Assume that we have either Then there exists a total order on the simple Λ/ a -modules satisfying condition (8.3).
Proof. Assume that (γ 1 ) holds (the proof using (α 1 ) is similar). Let S 1 , . . . , S s be all the simple Λ/ a -modules (up to isomorphism), and let P 1 , . . . , P s be their projective covers as Λ-modules, such that P i /((rad Λ)P i ) ∼ = S i for every i. Assume that we have ordered these by increasing length of the projective covers, that is, length(P 1 ) ≤ length(P 2 ) ≤ · · · ≤ length(P s ).
For any i, the module S i has a projective resolution of the form Since the module Q has shorter length than the module P i , it can not have any of the modules P i , . . . , P s as direct summands. Then Lemma 8.8 implies that Ext >0 Λ (S i , S j ) = 0 for i ≤ j. By using Proposition 8.10, Lemma 8.13 and Theorem 8.1, we have the following.
Corollary 8.14. Let Λ be an artin algebra over a commutative ring k, and let a be an idempotent in Λ. Then the following hold, where (α), (β), (γ) and (δ) refer to the conditions in Theorem 8.1, and (α 1 ) and (γ 1 ) refer to the conditions in Lemma 8.13.
(i) If (γ 1 ) holds, then the singularity categories of Λ and aΛa are triangle equivalent.
For the following results, we let Λ = kQ/ ρ be a quotient of a path algebra, where k is a field, Q is a quiver, and ρ a minimal set of relations in kQ generating an admissible ideal ρ .
First we describe how the conditions (α 1 ) and (γ 1 ) can be interpreted for quotients of path algebras. The result follows directly from [10, Corollary, Section 1.1]. As a consequence of Lemma 8.15 and Corollary 8.14, we get the following results for path algebras. We apply the above result in the following example.
Example 8.18. Let Q be the quiver with relations ρ given by for some integers m ≥ 2 and n ≥ 2. Let Λ = kQ/ ρ , and let a = e 1 (the only vertex where a relation starts and ends). Then aΛa ∼ = k[x]/ x n , so aΛa satisfies Fg by [27,28]. By Corollary 8.17, the algebra Λ also satisfies Fg. By Corollary 8.16, the algebras Λ and k[x]/ x n are singularly equivalent. See [59] for a general discussion of the Hochschild cohomology ring of the path algebra kQ modulo one relation.
8.3. 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.
(2) Λ satisfies Fg if and only if so does aΛa. (

3) Λ is Gorenstein if and only if so is aΛa.
This work is based on the paper [41], where stratifying ideals a in a finite dimensional algebra Λ were used to show that the Hochschild cohomology groups of Λ and aΛa are isomorphic in almost all degrees.
We start by giving an example of a recollement (mod Λ/ a , mod Λ, mod aΛa), where the ideal a is not a stratifying ideal but it satisfies our conditions from Theorem 7.10. and ρ = {α 2 , γβ, βαδ}. Let Λ = kQ/ ρ for some field k, and let a = e 1 . We want to study the relationship between Λ and aΛa. Let S i denote the simple Λ-module associated to the vertex i for i = 1, 2, 3. Then pd Λ S 2 = 1, pd Λ S 3 = 3, id Λ S 2 = 2 and id Λ S 3 = 3. Furthermore, the left and right aΛa-module aΛ and Λa have finite projective dimension (they are projective) as aΛa-modules. Hence, according to Theorem 7.10 Λ satisfies Fg if and only if aΛa ∼ = k[x]/ x 2 does. We infer from this that Λ satisfies Fg. Moreover, the Hochschild cohomology groups of Λ and aΛa are isomorphic in almost all degrees by Proposition 7.9.
We claim that a is not a stratifying ideal. Recall that a is stratifying if (i) the multiplication map Λa ⊗ aΛa aΛ −→ ΛaΛ is an isomorphism and (ii) Tor aΛa i (Λa, aΛ) = (0) for i > 0. Using that (1−a)Λa ∼ = aΛa as a right aΛa-module, direct computations show that Λa⊗ aΛa aΛ has dimension 12, while a has dimension 10. Consequently a is not a stratifying ideal in Λ. However, the condition (ii) is satisfied since Λa is a projective aΛa-module.
Next we show that, when a is a stratifying ideal, then the property pd Λ e Λ/ a < ∞ is equivalent to the functor e : mod Λ −→ mod aΛa 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 8.21. Let Λ be a finite dimensional algebra over an algebraically closed field k.
(ii) Assume that a is a stratifying ideal in Λ. Then the following are equivalent.
(a) pd Λ e Λ/ a < ∞. The following result gives a characterization of the condition (γ) when a is a stratifying ideal. Lemma 8.22. Let Λ be an artin algebra and a an idempotent in Λ. Assume that a is a stratifying ideal in Λ. Then we have (γ) pd Λ Λ/ a rad Λ/ a < ∞ if and only if gl. dim Λ/ a < ∞ and pd Λ a < ∞. Moreover, if (γ) holds, then (β) holds.
Proof. Assume that (γ) pd Λ Λ/ a rad Λ/ a < ∞. It is clear that pd Λ a < ∞ if and only if pd Λ Λ/ a < ∞. Since Λ/ a as a Λ-module is filtered in simple modules occurring as direct summands in (Λ/ a )/(rad Λ/ a ), we infer that pd Λ Λ/ a < ∞ by the property (γ). Since a is a stratifying ideal in Λ, we have that Ext j Λ/ a (X, Y ) ∼ = Ext j Λ (X, Y ) for all j ≥ 0 and all modules X and Y in mod Λ/ a . Using the above isomorphism and the property (γ) again, we obtain that id Λ/ a Y ≤ pd Λ (Λ/ a )/(rad Λ/ a ) for all Y in mod Λ/ a . Hence gl. dim Λ/ a < ∞.