Articles
Canadian Journal of Mathematics (0008414X)77(3)pp. 975-1012
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 investigate a bijection between the collection of τ-cotorsion torsion triples in A and the collection of support 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 a unitary associative ring, we characterize all support τ-tilting (resp. all support τ−-tilting) subcategories of Mod-R in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support τ-tilting (resp. 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 . © The Author(s), 2024.
Journal of Pure and Applied Algebra (00224049)229(5)
Higher homological algebra, basically done in the framework of an n-cluster tilting subcategory M of an abelian category A, has been the topic of several recent researches. In this paper, we study a relative version, in the sense of Auslander-Solberg, of the higher homological algebra. To this end, we consider an additive sub-bifunctor F of ExtMn(−,−) as the basis of our relative theory. This, in turn, specifies a collection of n-exact sequences in M, which allows us to delve into the relative higher homological algebra. Our results include a proof of the relative n-Auslander-Reiten duality formula, as well as an exploration of relative Grothendieck groups, among other results. As an application, we provide necessary and sufficient conditions for M to be of finite type. © 2025 Elsevier B.V.
