Derived equivalences induced by big cotilting modules

Abstract

Abstract. We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique way. We also prove that the triangle equivalence is at the base of an equivalence of derivators, which in turn is induced by a Quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes. Contents Introduction IntroductionIf G is a Grothendieck category with a tilting object T ∈ G (i.e. T is a rigid compact generator for the unbounded derived category of G) and R is the endomorphism ring of T , then we have a triangle equivalence from D(G) to D(R) which sends T to R. A motivating question for this paper, answered in [HRS96, CGM07] for tilting objects of projective dimension 1, is when we can go back. That is, if we start with a ring R, under what circumstances can we find a Grothendieck category G with a tilting object T and a triangle equivalenceOur approach is inspired by classical representation theory where, in the context of module categories over artin algebras, a finitely generated injective cogenerator of G is always sent to a finitely generated cotilting module. The general situation is analogous, but an injective cogenerator of an arbitrary Grothendieck category G is far from being finitely generated in any reasonable sense. The solution is to pass to so-called big cotilting modules, introduced in [AHC01]. Now we can explain our first result (Theorem 1.7): If G is a Grothendieck category, T is a tilting object of finite projective dimension (in the sense that Ext i G (T, −) ≡ 0 for i ≫ 0) and W ∈ G is an injective cogenerator, then the derived equivalence D(G) → D(R) sends W to a big cotilting R-module C.

The Tilting–Cotilting Correspondence

“…In the last decade, there have been several attempts to recover a part of this picture outside of the realm of finite-dimensional algebras. Based on an existing theory of infinitely generated (co)tilting modules, similar results were obtained in [81] in the case Figure 1. The tilting-cotilting correspondence: T ∈ A is a tilting object and W ∈ A an injective cogenerator, while T ∈ B is a projective generator and W ∈ B is a cotilting object.…”

“…We exhibit an equally symmetric and easy-to-state correspondence (which we call the tilting-cotilting correspondence) in the context of very general abelian categories. This puts several recent generalizations of the Brenner-Butler correspondence (see for instance [13,15,56,74,81]) into a unified framework.…”

Purity in compactly generated derivators and t-structures with Grothendieck hearts

“…The t-structure induced by a large tilting module T has a Grothendieck heart exactly when T is pure-projective [4,Thm. 7.5] and the t-structure induced by a large cotilting module always has a Grothendieck heart [35]. A more general version of these results can be found in [3,Thm.…”

“…Strategy 1: Consider T as a subcategory of a Grothendieck category A(T ) and understand direct limits in G in terms of direct limits in A(T ) ( [3], [4], [25] and [35]).…”

∞-tilting theory

“…The main innovation in [49], which allows to obtain very naturally derived equivalences from big n-tilting objects and which is based on the ideas previously developed in [52], [50], [39], and [47], is that to a big tilting object T in an abelian category A one can assign a richer structure than its ring of endomorphisms Hom A (T, T ). For any set X, consider the set of all morphisms T −→ T (X) in A, where T (X) denotes the coproduct of X copies of T .…”