The equivalence of two notions of discreteness of triangulated categories (Q2230603)

Derived-discreteness of a finite-dimensional algebra was introduced by \textit{D. Vossieck} [J. Algebra 243, No. 1, 168--176 (2001; Zbl 1038.16010)]. This notion has been generalized by \textit{N. Broomhead} et al. [Bull. Lond. Math. Soc. 50, No. 1, 174--188 (2018; Zbl 1391.18016)] to a notion of discreteness of a triangulated category with respect to (the heart of) a bounded \(t\)-structure. They also introduced a dual notion, namely, the notion of discreteness of a triangulated category with respect to a bounded co-\(t\)-structure. \textit{T. Adachi} et al. [Proc. Lond. Math. Soc. (3) 118, No. 1, 1--42 (2019; Zbl 1435.18007)] introduced the notion of ST-triples which provides a nice framework to study the interplay between \(t\)-structures and co-\(t\)-structures. Let \(\mathcal{C}\) and \(\mathcal{D}\) be triangulated categories and \(M\) a silting object of \(\mathcal{C}\) such that \((\mathcal{C}, \mathcal{D}, M)\) is an ST-triple. In this paper, the authors show the discreteness of \(\mathcal{C}\) with respect to the co-\(t\)-structure is equivalent to the discreteness of \(\mathcal{D}\) with respect to the \(t\)-structure. As a special case, the discreteness of \(D^b\pmod A\) in the sense of Vossieck is equivalent to the discreteness of \(K^b(\mathrm{proj} A)\) in a dual sense, where \(A\) is a finite-dimensional algebra.