Tilting subcategories in extriangulated categories

Abstract

Let R be an artin ring and Θ = {Θ(1), Θ(2), · · · , Θ(n)} be a family of objects in an artin extriangulated R-category (C, E, s) such that E(Θ(j), Θ(i)) = 0 for all j ≥ i. In this paper, we show that the class P(Θ) of the Θ-projective objects is a precovering class and the class I(Θ) of the Θ-injective objects is a preenveloping one in C. Furthermore, if C has enough projectives and enough injectives, we show that the subcategory F(Θ) of Θ-filtered objects is functorially finite in C. As an appliacation, this generalizes the works by Ringel in a module category case and Mendoza-Santiago in a triangulated category case.

Tilting pairs in extriangulated categories

“…Hence and Using the above result, we have that is closed under direct summands. Similarly, one can see that is closed under extensions and direct summands. follows from [23, Lemma 8]. Since , there is an -triangle with and . Consider the following commutative diagram Since , by (3).…”

“…If is a silting complex (see [1, Definition 2.1] for the definition of silting objects in triangulated categories) in with and , then the pair is a -tilting pair by [10, Corollary 4.8]. Let be an Artin algebra, and . If is an -tilting pair in in the sense of [19], then is an -tilting pair in the bounded derived category . When is an extriangulated category with enough projectives and injectives, Zhu and Zhuang [23] defined -tilting subcategories in . If in addition , we call it an -tilting object.…”

“…If in addition , we call it an -tilting object. Now we set and be an -tilting subcategory in the sense of [23, Definition 7], then using [23, Remark 4], we have is an -tilting pair in .…”

“…When is an extriangulated category with enough projectives and injectives, Zhu and Zhuang [23] defined -tilting subcategories in . If in addition , we call it an -tilting object.…”

“…In § 4, we introduce the notion of tilting pairs in extriangulated categories. It unifies tilting pairs in the categories of finitely generated modules of Artin algebras [14], silting complexes in the bounded homotopy category of finite generated projective -modules over an associative ring [1], and meanwhile -tilting subcategories in extriangulated categories [23]. We give a Bazzoni characterization for -tilting pairs in this setting (see Theorem 4.9).…”