8 October 2025 — Padova

Generation time for perfect complexes and weak global dimension

Jan Šťovíček, Charles University Prague

Abstract

We prove that for any ring \(A\) (commutative or not), the number of extensions needed to generate the category of perfect complexes from the ring itself equals the weak global dimensions of \(A\). Using a recent technique of Neeman, the result also globalizes to all quasi-compact separated schemes, characterizing locally finite weak global dimension in terms of strong generation of the perfect derived category.

This result has an interesting history and can be seen as a culmination of a longer effort to relate homological dimensions of a ring or scheme to structural properties of the corresponding derived categories. Namely, an analogous problem in a much greater generality of ring spectra was studied in a series of papers by Hovey and Lockridge, and a (not entirely correct) version of the result was claimed by Neeman and later fixed to the present for coherent rings and schemes by Stevenson. Here we remove the coherence assumption and show that this comes at no cost of making the argument more complicated.