8 July 2026 — Padova

\(t\)-proxy smallness

Sergio Pavon, University of Verona

Abstract

In this talk I will introduce the notion of \(t\)-proxy small objects in a compactly generated triangulated category, which is a refinement of the more classical notion of proxy small objects. I will highlight the difference between the two by mentioning a few interesting examples in the derived category of various commutative noetherian rings. Despite this, I will then show that one of the main theorems on proxy small objects (the characterisation of locally complete intersection rings \(R\) as those for which every object of \(D^b(\mathsf{mod} R)\) is proxy small) works verbatim if one uses \(t\)-proxy small objects instead. I will then present our main application for \(t\)-proxy small objects, which comes from the homotopy category of injectives \(K(\mathsf{Inj} R)\), for \(R\) a commutative noetherian ring. I will explain the relation between the \(t\)-proxy small objects of \(D^b(\mathsf{mod} R)\) and the compactly generated \(t\)-structures of \(K(\mathsf{Inj} R)\) which glue along Krause’s recollement. As a corollary, we obtain a classification for compactly generated \(t\)-structures of \(K(\mathsf{Inj} R)\) where \(R\) is locally complete intersection.

This talk is based on joint work with Michal Hrbek, Pat Lank and Giovanna Le Gros, arxiv:2605.26057 .