Quadratic forms and Clifford algebras on derived stacks

TL;DR

This paper develops a framework for quadratic structures and Clifford algebras in derived algebraic geometry, introducing derived shifted quadratic forms, their Clifford algebras, and a derived Grothendieck-Witt group, with several existence and comparison results.

Contribution

It introduces derived n-shifted quadratic complexes and Clifford algebras in derived stacks, extending classical notions to the derived setting and establishing foundational existence results.

Findings

Defined derived n-shifted quadratic complexes on stacks.

Constructed derived Clifford algebras and compared with classical versions.

Proved existence theorems for derived shifted quadratic forms and related Grothendieck-Witt groups.

Abstract

In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We define the associated notion of derived Clifford algebra, in all these contexts, and compare it with its classical version, when they both apply. Finally, we prove three main existence results for derived shifted quadratic forms over derived stacks, define a derived version of the Grothendieck-Witt group of a derived stack, and compare it to the classical one.