Equivalences Induced by Infinitely Generated Silting Modules
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Equivalences Induced by Infinitely Generated Silting Modules Simion Breaz1 · George Ciprian Modoi1 Received: 17 April 2019 / Accepted: 9 October 2019 / © Springer Nature B.V. 2019
Abstract We study equivalences induced by a complex P, consisting of projectives and concentrated in degrees −1 and 0, which is silting in the derived category D(R) of a ring R. Keywords Silting module · Silting complex · Endomorphism ring · Endomorphism dg-algebra · dg-module · Derived functors Mathematics Subject Classification (2010) 16E30 · 18E30 · 16D90
1 Introduction A torsion theory in an abelian category A (e.g. A = Mod(R) is the category of right Rmodules) is a pair τ = (T , F ), such that the classes T and F satisfy that HomA (T , F ) = 0, and for every X ∈ A there is a short exact sequence 0 → T → X → F → 0, such that T ∈ T and F ∈ F . Then T and F are called the torsion class, respectively the torsion free class of τ . In the context of a triangulated category D endowed with the shift functor −[1] : D → D (e.g. D = D(R) the derived category of the category of R-modules), a t-structure is a pair (A, B ) of full subcategories if D such that (1) (2) (3)
HomD (A, B [−1]) = 0. A ⊆ A[−1] (or equivalently B [−1] ⊆ B ). + For every X ∈ D there is a triangle X → X → X →, where X ∈ A and X ∈ B [−1].
Presented by: Michela Varagnolo. George Ciprian Modoi
[email protected] Simion Breaz [email protected] 1
Faculty of Mathematics and Computer Science 1, Babes¸–Bolyai University, Mihail Kog˘alniceanu, 400084 Cluj–Napoca, Romania
Simion Breaz, George Ciprian Modoi
The heart of a t-structure (A, B ) is defined to be the subcategory H = A ∩ B . We recall that the heart H is an abelian category. Note that the definition of a t-structure implies immediately that the inclusion functors A → D and B → D have a right, respectively a left adjoint. For more informations about torsion pairs and t-structures one can consult [19, Chapter I, Section 2]. One of the central results in Tilting Theory is the Tilting Theorem, [13, Theorem 3.5.1], which states that if (T , F ) the torsion theory generated by a finitely presented (i.e. classical) tilting right R-module T then there exists a torsion theory (X , Y ) in the category of right E-modules (E is the endomorphism ring of T ) and a pair of equivalences HomR (T , −) : T Y : − ⊗E T and Ext1R (T , −) : T X : TorE 1 (−, T ). Such a pair of equivalences is called a counter-equivalence. It was proven in [12] and [15] that the existence of a counter equivalence is strongly related to the existence of a classical tilting module which generates T . In the case of infinitely generated tilting modules, versions of Tilting Theorem was formulated at the level of for derived category in [6] and [7]. The main idea is that every tilting module S is equivalent to a good tilting module T ∈ Mod(R) which induces an equivalence between the derived category D(R) and a subcategory of the derived category D(End(T )). This equivalence also induces a counter equivalence at the level of module categories
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