Twisted forms of perfect complexes and Hilbert 90

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.