# Twisted forms of perfect complexes and Hilbert 90

## TL;DR

This paper explores the structure and deformation theory of automorphism stacks of perfect complexes, establishing conditions for smoothness and discussing a Hilbert 90 analogue in this context.

## Contribution

It introduces a framework for understanding automorphism stacks of perfect complexes as $
abla$-groups and provides a criterion for their formal smoothness, extending classical results.

## Key findings

- Automorphism stacks form $
abla$-groups with rich structure.
- Criteria for formal smoothness of these stacks are established.
- A version of Hilbert 90 for perfect complexes is discussed.

## Abstract

Automorphisms of a perfect complex naturally have the structure of an $\infty$-group: the 1-morphisms are quasi-isomorphisms, the 2-morphisms are homotopies, etc. This article starts by proving some basic properties of this $\infty$-group. We go on to study the deformation theory of this stack of $\infty$-groups and give a criterion for this stack to be formally smooth. The classifying stack of this $\infty$-group classifies forms of a complex. We discuss a version of Hilbert 90 for perfect complexes.