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The Equivalence of Two Notions of Discreteness of Triangulated Categories Lingling Yao1 · Dong Yang2 Received: 28 November 2018 / Accepted: 8 September 2020 / © Springer Nature B.V. 2020

Abstract Given an ST-triple (C , D, M) one can associate a co-t-structure on C and a t-structure on D. It is shown that the discreteness of C with respect to the co-t-structure is equivalent to the discreteness of D with respect to the t-structure. As a special case, the discreteness of Db (mod A) in the sense of Vossieck is equivalent to the discreteness of K b (proj A) in a dual sense, where A is a finite-dimensional algebra. Keywords Derived discrete · Discreteness of triangulated category · ST-triple · T-structure · Co-t-structure Mathematics Subject Classification (2010) 18E30 · 16E35

1 Introduction Derived-discreteness of a finite-dimensional algebra was introduced by Vossieck in [20]. It is defined by counting the number of indecomposable objects in the bounded derived category. Recently this notion has been generalised by Broomhead, Pauksztello and Ploog in [10] to a notion of discreteness of a triangulated category with respect to (the heart of) a

Presented by: Vyjayanthi Chari The authors thank Bernhard Keller for correcting a grammatical mistake in the title. They thank the referee for his/her helpful comments.  Lingling Yao

[email protected] Dong Yang [email protected] 1

School of Mathematics, Southeast University, Nanjing, 210096, China

2

Department of Mathematics, Nanjing University, Nanjing, 210093, China

L. Yao, D. Yang

bounded t-structure. In [10] they also introduced a dual notion, namely, the notion of discreteness of a triangulated category with respect to a bounded co-t-structure (equivalently, a silting subcategory). It turns out that ST-triples introduced in [1] provide a nice framework to study the interplay between t-structures and co-t-structures. Let C and D be triangulated categories and M a silting object of C such that (C , D, M) is an ST-triple. Then on D there is a natural bounded t-structure, say, with heart H. Our main result is Theorem (4.1) The category C is M-discrete if and only if the category D is H-discrete. In the literature there are another two notions of discreteness of triangulated categories, namely, silting-discreteness [2] and t-discreteness [1], defined by counting the number of silting objects and bounded t-structures, respectively. In [1] it is shown that C is siltingdiscrete if and only if D is t-discrete, and that if D is H-discrete then D is t-discrete. Together with these results Theorem 4.1 implies the following corollary, which completes the picture. Corollary (4.2) If C is M-discrete, then C is silting-discrete. The paper is structured as follows. In Section 2 we fix the notation and briefly recall the definitions of t-structure, silting object and co-t-structure. In Section 3 we recall the definitions of ST-triple and discreteness of triangulated categories. In Section 4 we prove Theorem 4.1. In the final section we apply Theorem 4.1 to finite-dimensional algebr