Deformation theory of perfect complexes and traces

TL;DR

This paper explores the relationship between the deformation theory of perfect complexes and their determinants via trace maps, utilizing advanced $K$-theory and higher category theory techniques.

Contribution

It establishes a general connection between the deformation theories of perfect complexes and their determinants using trace maps, extending previous results to a broader sheaf setting.

Findings

Deformation theory of perfect complexes relates to their determinants via trace maps.

Uses $K$-theory and higher category theory to generalize from modules over rings to sheaves on a site.

Provides a framework for understanding deformations in a broad sheaf-theoretic context.

Abstract

We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.