November 11, 2025, 16:00 +0100
Verona, Ca’ Vignal 2 Sala Riunioni

Detecting derived equivalences via the CHZ criterion

Sergio Pavon, University of Verona

Abstract

Let \(\mathcal{A}\) be an abelian category, and \((\mathcal T,\mathcal F)\) a torsion pair in \(\mathcal A\). Following Happel, Reiten and Smalø, one can use \((\mathcal T,\mathcal F)\) to deform \(\mathcal A\) into a new abelian category \(\mathcal B=\mathcal F\ast \mathcal T[-1]\). In most concrete situations, there is a functor \(D^b(\mathcal B)\to D^b(\mathcal A)\) between the derived categories of \(\mathcal A\) and \(\mathcal B\), and one can ask if it is an equivalence, for example because this means that \(\mathcal A\) and \(\mathcal B\), despite being possibly very different, share the same derived invariants. In 2019, Chen, Han and Zhou obtained a criterion to detect when a torsion pair induces an equivalence between derived categories. This criterion is internal to the category \(\mathcal A\) (that is, it does not appeal to \(D^b(\mathcal A)\)). In this talk we aim to showcase a few applications of this criterion in various settings (often, over finite-dimensional algebras), to convince the audience that it is a practical tool with useful applications. This talk is based on the preprint arxiv:2509.12983.