On $τ$-tilting subcategories

Abstract

The main theme of this paper is to study τ -tilting subcategories in an abelian category A with enough projective objects. We introduce the notion of τ -cotorsion torsion triples and show a bijection between the collection of τ -cotorsion torsion triples in A and the collection of τ -tilting subcategories of A , generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of A . General definitions and results are exemplified using persistent modules. If A = Mod-R, where R is an unitary associative ring, we characterize all support τ -tilting, resp. all support τ − -tilting, subcategories of Mod-R in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support τ -tilting, respectively support τ − -tilting, subcategory of Mod-R. We also study the theory in Rep(Q, A ), where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support τ -tilting subcategories in Rep(Q, A ) from certain support τ -tilting subcategories of A and present a systematic way to construct (n + 1)-tilting subcategories in Rep(Q, A ) from n-tilting subcategories in A .

Extending (τ-)tilting subcategories and (co)silting modules

“…In Section 6 the restriction and extension of τ -tilting subcategories will be studied. On the other hand, there is a bijection between the collection of all equivalence classes of τ -tilting subcategories of Mod-R and the collection of all equivalence classes of finendo quasi-tilting R-modules, when R is an arbitrary ring [6,Theorem 8.1.10]. In view of this result, we are able to show that restriction of a finendo quasi-tilting module T in Mod-A, i.e.…”

“…As an application of the results of the previous section, in view of [6], we are able to study the restriction and extension of τ -tilting subcategories. Let us first recall the definition of a τ -tilting subcategory of an abelian category with enough projective objects.…”

“…Recall that a Λ-module T , where T is an artin algebra, is called a tilting module if Ext 1 Λ (T, T ) = 0, the projective dimension of T , pd Λ T is at most 1, and there exists a short exact sequence 0 → Λ → T 0 → T 1 → 0, such that T 0 and T 1 are in add-T . It is easy to see that if T is a tilting B-module , then add-T is a tilting subcategory of mod-B, see [6]. Proof.…”

“…As a counterpart to a cotorsion torsion triple we have the notion of τ -cotorsion torsion triple. A pair of full subcategories (C , D) of A is called a τ -cotorsion pair [6] if C ∩D is a contravariantly finite subcategory of A , C = ⊥1 D and for every projective object P ∈ A , there exists an exact sequence…”

“…By Theorem 2.29 of [8], we know that there are bijections between tilting subcategories of A , where A is an abelian category with enough projective objects, and special triples of full subcategories of A , the so-called cotorsion torsion triples. Moreover, Theorem 5.7 of [6] implies that there is a bijection between the collection of all support τ -tilting subcategories of A and the collection of all τ -cotorsion torsion triples of A . Based on these bijections we study explicitly the behaviour of these triples under the restriction and extension functors.…”