Equivalences induced by infinitely generated tilting modules

We generalize Brenner and Butler's Theorem as well as Happel's Theorem on the equivalences induced by a finitely generated tilting module over artin algebras, to the case of an infinitely generated tilting module over an arbitrary associative ring establishing the equivalences induced between subcategories of module categories and also at the level of derived categories.


Introduction
Tilting theory started in the context of finitely generated modules over artin algebras and was further generalized over arbitrary associative rings with unit and to infinitely generated modules (see [6], [8], [9], [1]).
One of the most important features in classical tilting theory is the famous Brenner and Butler's Theorem [5] establishing two equivalences between suitable categories of finitely generated modules.
A finitely generated tilting module T over an artin algebra Λ gives rise to a torsion pair (T , F), where T is the class of modules generated by T . If D denotes the standard duality and Γ is the endomorphism ring of T , then D(T ) is a cotilting Γ-modules with an associated torsion pair (X , Y) where Y is the class modules cogenerated by D(T ). The Brenner and Butler's Theorem states that the functor Hom Λ (T, −) induces an equivalence between the categories T and Y with inverse the functor − ⊗ Γ T , and the functor Ext 1 Λ (T, −) induces an equivalence between F and X with inverse the functor Tor Γ 1 (−, T ). (See [16] and [17]). Moreover, T is the kernel of the functor Ext 1 Λ (T, −), Y is the kernel of Tor Γ 1 (−, T ), F is the kernel of Hom Λ (T, −) and X is the kernel of − ⊗ Γ T . Later on, Happel [15] observed that the natural context in which to interpret the above equivalences is that of derived categories. He proved that the total right derived functor of the functor Hom Λ (T, −) induces a derived equivalence between the bounded derived categories of finitely generated Λmodules and the bounded derived categories of finitely generated Γ-modules.
Colby and Fuller [6] proved a "Tilting Theorem" for finitely presented tilting modules over an arbitrary associative ring, generalizing Brenner and Butler's Theorem, and Colpi [7] extended the Tilting Theorem to the wider context of Grothedieck categories.
The first instance of a generalization of Brenner and Butler's Theorem to infinitely generated tilting modules, appears in two papers by Facchini [11], [12] where he studied the equivalences induced by the tilting module ∂, a divisible module introduced by Fuchs [13] over commutative domains. The theorems proved by Facchini provide a link between the Brenner and Butler tilting equivalences and the equivalences established by Harrison and Matlis between subcategories of modules over a commutative domain R.
If Q is the quotient filed of a commutative domain R and K is the module Q/R, then Harrison and Matlis' Theorem states that the functor Hom R (K, −) induces an equivalence between the category of h-divisible torsion modules and the category of torsion free cotorsion modules. Moreover, the functor Ext 1 (K, −) gives an equivalence between the category of h-reduced torsion R-modules and the category of special cotorsion modules. Thus the similarity with tilting equivalences was evident and the papers by Facchini showed the advantage to work with a tilting module, namely the module ∂ rather than the module K, even though the formal definition of an infinitely generated tilting module was not yet available.
In this paper we generalize both Brenner and Butler Theorem's and Facchini results, to the case of an arbitrary (infinitely generated) tilting module over an associative ring R. If Mod-R is the category of all right R-modules and T ∈ Mod-R is a tilting module, T induces a torsion pair (T , F) in Mod-R, where T is the class of modules genrated by T . If S is the endomorphism ring of T , we prove that the dual T d of T with respect to an injective cogenerator of Mod-R, is a partial cotilting right S-module inducing a torsion pair (T T d , F T d ) in Mod-S. By Theorem 4.5, we prove that the functor Hom R (T, −) induces an equivalence between the category T and the intersection of F T d with a suitable subcategory M of Mod-S, namely the double perpendicular category of the module T d (see definition in Section 4). Secondly, the functor Ext 1 R (T, −) induces an equivalence between F and the intersection of T T d with the subcategory M. Moreover, the inverses of these equivalences are given by the functors − ⊗ S T and Tor S 1 (−, T ). The subcategories of Mod-S equivalent to T and F in the above equivalences cannot be interpreted as Gabriel quotients of Mod-S, since there are no Serre subcategories arising in the process. Thus again, as in the case of finitely generated tilting modules, the situation can be better illustrated in the context of derived categories, where the equivalences involved can be formulated in a concise and more expressive way. In fact, if D(R) and D(S) are the (unbounded) derived categories of the categories Mod-R and Mod-S respectively, we prove that the total right derived functor of the functor Hom R (T, −), that is the functor RHom R (T, −), induces an equivalence between D(R) and the quotient category of D(S) modulo the full triangulated subcategory Ker(− L ⊗ S T ), namely the kernel of the total left derived functor of the functor − ⊗ S T .
Acknowledgement I wish to thank Bernhard Keller for his help in proving a crucial step in the proof of Lemma 5.2 and Pedro Nicolás for suggesting the use of the powerful Proposition 1.3 in Gabriel Zisman's book [14].

Preliminaries
In what follows all rings are associative with unit. We recall some definitions and results.
For a ring R, Mod-R (R-Mod) will denote the category of all right (left) R-modules.
For an R-module M we denote by p.d. M and i.d. M the projective and injective dimension of M , respectively.
If λ is a cardinal, M (λ) and M λ will denote the direct sum and the direct product of λ copies of M , respectively.
Let C ⊆ Mod-R. Define Here, Add T denotes the class of all direct summands of arbitrary direct sums of copies of T .
If T is an 1-tilting module, T ⊥ is called 1-tilting class.
and T ⊥ is closed under direct sums.
We have also dual definitions.
= 0 for each i ≥ 1 and every cardinal λ, and (C3) there exists an exact sequence Here, Prod C denotes the class of all direct summands of arbitrary direct products of copies of C.
If C is an 1-cotilting module, ⊥ C is called 1-cotilting class.
We recall some results on infinitely generated 1-tilting and 1-cotilting modules which give a better understanding of their properties. • GenT is the class of modules generated by T . • By [3] If T is a 1-tilting module, then the tilting class T ⊥ is of finite type, that is there is a set S of finitely presented modules of projective dimension at most 1, such that S ⊥ = T ⊥ . • By [2] 1-cotilting modules are pure injective.
• As a consequence of the above results, we have that every 1-tilting right R-module T induces a torsion pair (T , F) in Mod-R where T = GenT = T ⊥ and F = Ker(Hom R (T, −)).
Moreover, the cotilting torsion free class F is closed under epimorphic images.
In this section we adapt the results proved by Facchini in [11] and [12] for the case of the tilting module ∂ defined over a commutative domain, to the case of a tilting module over an arbitrary associative ring.
First of all we have to make a suitable choice of a representative in the equivalence class of a 1-tilting module.
Proposition 3.1. Let R be a ring and let T R be a 1-tilting module. Up to equivalence we can assume that T fits in an exact sequence of the form: Proof. From condition (T3) in the definition of tilting modules, we have an Then we have an exact sequence: In fact, T ′ satisfies conditions (T1) and (T2) since T ′ ∈ Add T ; it satisfies also (T3), by the above sequence. Moreover, T ′ and T are equivalent, since T ⊥ ⊂ T ′⊥ and T ′⊥ = GenT ′ ⊆ GenT = T ⊥ . Notation 3.2. From now on we assume that T is a 1-tilting right R-module such that the short exact sequence of condition (T3) has the form where T 1 is a direct summand of T . Moreover, we denote by S the endomorphism ring of T . As in [11] we fix the following notations: Proof. (1) and (2) follow by applying the functor Hom R (−, T ) to the exact sequence (a). So S T is cyclically presented; (3) and (4)   (2) The proof is the same as in [11,Proposition 4.2], but we repeat the argument because our context is different. Let N be a right S-module; applying the functor N ⊗ S − to the exact sequence (b), we get that Tor  ( (2).
Recall that if M is an R-module over a ring R, the preradical Rej M is the subfunctor of the identity functor defined by Rej M (X) = ∩Ker{f | f ∈ Hom R (X, M )}, for every R-module X. Rej M is always a radical and if it is also idempotent, then it is a torsion radical (see from [21]). In this case the associated torsion class consists of the modules X such that Hom R (X, M ) = 0 and the torsion free class is Cogen M . (1) T d S is a direct summand of a 1-cotilting right S-module C such that ⊥ C = ⊥ T d . We define now the subcategories of Mod-S which will play a crucial role in establishing the equivalences which will be proved by Theorem 4.5.

(2) The preradical Rej T d is an idempotent radical inducing a torsion pair
First, we recall the notion of perpendicular categories. If C is a category of R-modules the right perpendicular category C ⊥ is    ( (2) E is closed under direct sums, direct summands and has the 2 out of 3 property. Proof.
(1) The equality follows by the usual homological formulas and by the fact that i.d.T d ≤ 1.
(2) Follows by a direct check.
Consider now the right perpendicular category M of E, that is The next theorem, inspired by Facchini's Theorems, is the generalization of the equivalences proved by Brenner and Bluter [5] in the case of a classical 1-tilting module (that is finitely generated) over artin algebras.
Theorem 4.5. ( [11], [12]) Let R be a ring, T R a 1-tilting module as in Notation 3.2 and let (T , F) be the tilting torsion pair associated to T . Let S = End R (T ). The following hold.
(1) There is an equivalence The proof of the two equivalences is essentially the same as [11] and [12] with the suitable translation of the terminology.
In those papers the modules named I-divisible are the modules in T T d , that is the right S-modules N such that Hom S (N, T d ) = or equivalently, N ⊗ S T = 0. The modules called I-reduced are the modules in F T d . Moreover, the modules in the class E are called I-divisible and I-torsion free.
(1) Is proved by the same arguments as in [11], once it is observed that the modules named I-cotorsion in that paper are the modules in the class First one shows as in [11,Theorem 7.1], that for every M ∈ Mod-R, Hom R (T, M ) ∈ F T d ∩ M. Then one uses that, by Proposition 3.4 (1), φ : Hom R (T, M ) ⊗ S T → M is an isomorphism if and only if M in the tilting torsion class T . Finally one verifies that η : N → Hom R (T, N ⊗ S T ) is an isomorphism if and only if N is a right S-module in the class F T d ∩ M. This is obtained following the proofs of [11,Theorem 7.2,7.3].
(2) This is proved as in [12] noticing that there, slightly differently from the definitions in [11], the modules named I-cotorsion are the modules in the class M.
First, as in [12, Lemma 1], one proves that Ext 1 R (T, M ) is in the class T T d ∩ M, for every right R-module M . Secondly one shows that, if M in the torsion free class F, then the natural homomorphism ξ : Tor S 1 (Ext 1 R (T, M ), T ) → M is an isomorphism (see [12,Lemma 1]).
Then, one proves that Tor S 1 (N, T ) ∈ F, for every N ∈ T T d ∩ M and that the natural homomorphism θ : N → Ext 1 R (T, Tor S 1 (N, T )) is an isomorphism if and only if N ∈ T T d ∩ M (see [12,Lemma 2]). Remark 1. If T is a finitely presented 1-tilting module, then the dual module T d is a 1-cotilting module over the endomorphism ring of T . Hence, in this case, the category E = ⊥ T d is zero, so M coincides with Mod-S and (X , Y) is the cotilting torsion pair associated to the 1-cotilting module T d . Thus, we recover both Brenner and Butler's Theroem for the case of artin algebras and Colby-Fuller Tilting theorem over an arbitrary ring. So, Theorem 4.5 can be viewed as the generalization to the case of infinitely generated 1tilting modules of Brenner and Butler's and Colby-Fuller's Theorems.
The categories Ker(− ⊗ S T ) and Ker(Tor S 1 (−, T ) are not Serre subcategories of Mod-S in general. Thus, we cannot perform the corresponding quotient categories in Gabriel sense. However, we can localize the category Mod-S at a suitable multiplicative system as we are going to explain.
In the next proposition we use the terminology as in the Gabriel and Zisman's book [14]. Proposition 4.6. Let T be a 1 tilting right R-module as in Notation 3.2 and let (T , F) be the associated torsion pair in Mod-R. Let Σ be the system of morphisms u ∈ Mod-S such that u ⊗ S 1 T is invertible in Mod-R. Then the following hold: (1) Σ admits a calculus of left fractions.
(2) There is an equivalence ρ : is the canonical localization functor. Remark 2. We couldn't get an analogous result for the pair of functors Ext 1 R (T, −) and Tor S 1 (−, T ) because they are not an adjoint pair in general. Moreover, we don't know wether the category of fractions Mod-S[Σ −1 ], considered in Proposition 4.6, is the quotient of Mod-S modulo a suitable subcategory.
The above remark indicate that a better understanding of the whole situation can be obtained in the setting of derived categories.

Derived equivalence
Before stating the main result of this section we recall some notions and facts about derived categories which will be used later on.
Let D(R) and D(S) be the unbounded derived categories of Mod-R and Mod-S respectively. The following hold.
• (Bökstedt and Neeman [4] or Spaltenstein [20]) For every complex M · ∈ D(R) there is a quasi isomorphism M · → I · where I · is a complex with injective terms. I · is also denoted by iM · and called a K-injective or fibrant resolution of M · . Symmetrically, for every complex M · ∈ D(R) there is a quasi isomorphism P · → M · where P · is a complex with projective terms. P · is also denoted by pM · and called a K-projective or cofibrant resolution of M · . Proof. Let M · be a complex in D(R) and consider a K-injective resolution iM · of M · . We have: Let C · = H(iM · ). C · is a complex of right S-modules and LG where pC · is a K-projective resolution of C · as a complex in D(S).

Consider the complex T
From the quasi-isomorphism pT · → T · and pC · → C · we get the chain of quasi-isomorphisms: Thus, LG(C · ) = C · ⊗ S pT · and this gives LG(C · ) = C · L ⊗ S T = Cone(1 ⊗ δ).
Letting I · = iM · , we have C · = Hom R (T, I · ) and we have also the commutative diagram:  1), since I · is a complex of injective right R-modules, hence belonging to the tilting class T ⊥ .
Hence, Hom R (T, I n ) ⊗ S T and I · are canonically isomorphic as complexes of R-modules, so we have: LG(RH(M · )) = H(iM · ) Proof. of Theorem 5.1 Condition (1) is proved by Lemma 5.2 and the equivalence of (1) with the other conditions follows essentially by applying [14,Proposition 1.3].
To complete the proof we add only a few comments. The functor LG = − L ⊗ S T is a triangle functor, hence Ker(LG) is a full triangulated subcategory of D(S). It is well known that the quotient category D(S)/Ker(LG) is the localization of D(S) at the multiplicative system Σ given by the morphisms u ∈ D(S) such that there exists a trinagle: where K · ∈ Ker(LG) and M · , N · ∈ D(S). Thus Σ coincides with the systems of morphisms u ∈ D(S) such that LG(u) is invertible in D(R).