The Atiyah class on algebraic stacks

TL;DR

This paper extends the concept of the Atiyah class to algebraic stacks, providing a global obstruction theory for moduli stacks of complexes without relying on derived algebraic geometry.

Contribution

It generalizes Illusie's Atiyah class to complexes on algebraic stacks, establishing new properties and compatibilities useful for algebraic geometry.

Findings

Defines Atiyah class on algebraic stacks

Provides a global obstruction theory for moduli stacks

Establishes compatibilities with tensor products and determinants

Abstract

We generalize Illusie's definition of the Atiyah class to complexes with quasi-coherent cohomology on arbitrary algebraic stacks. We show that this gives a global obstruction theory for moduli stacks of complexes in algebraic geometry without derived methods. We give a similar generalization of the reduced Atiyah class, and show various useful properties for working with Atiyah classes, such as compatibilities between the reduced and ordinary Atiyah class, and compatibility with tensor products and determinants.