Clifford modules and invariants of quadratic forms

TL;DR

This paper introduces new characteristic classes for projective modules with quadratic forms over rings, generalizing classical topological classes using Clifford algebras and Morita equivalences, with applications to algebraic K-theory and Witt groups.

Contribution

It defines novel characteristic classes for quadratic modules over rings, extending classical topological invariants to algebraic settings via Clifford algebras and Morita equivalences.

Findings

Classes generalize Bott classes in topological K-theory

Characteristic classes take values in twisted K-theory when Clifford algebra is non-trivial

Bott class has a canonical square root in K-theory

Abstract

Let A be a commutative ring with 1/2 in A. In this paper, we define new characteristic classes for finitely generated projective A-modules V provided with a non degenerate quadratic form. These classes belong to the usual K-theory of A. They generalize in some sense the classical "cannibalistic" Bott classes in topological K-theory, when A is the ring of continuous functions on a compact space X. To define these classes, we replace the topological Thom isomorphism by a Morita equivalence between A-modules and C(V)-modules, where C(V) denotes the Clifford algebra of V, assuming that the class of C(V) in the graded Brauer group of A is trivial. We then essentially use ideas going back to Atiyah, Bott and Shapiro together with an alternative definition of the Adams operations due to Atiyah. When C(V) is not trivial in the graded Brauer group, the characteristic classes take their values in an algebraic analog of twisted K-theory. Finally, we also make use of a letter written by J.-P. Serre to the author, in order to interpret these classes as defined on the Witt group W(A) of the ring A. One aspect of this letter is summarized in our Lemma 3.5 where it is shown that in our situation the Bott class has a canonical square root in the K-theory of A.