Clifford modules and invariants of quadratic forms

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.

💡 Research Summary

This paper introduces new characteristic classes for finitely generated projective modules ( V ) over a commutative ring ( A ), where ( 1/2 \in A ), equipped with non-degenerate quadratic forms. These classes belong to the usual K-theory of ( A ). When ( A ) is the ring of continuous functions on a compact space ( X ), these classes generalize the classical “cannibalistic” Bott classes in topological K-theory.

To define these characteristic classes, the authors replace the topological Thom isomorphism with a Morita equivalence between ( A )-modules and ( C(V) )-modules, where ( C(V) ) denotes the Clifford algebra of ( V ). This replacement assumes that the class of ( C(V) ) in the graded Brauer group of ( A ) is trivial. The authors then use ideas from Atiyah, Bott, and Shapiro, along with an alternative definition of 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. Additionally, a letter written by J.-P. Serre to the author provides insights into interpreting these classes on the Witt group ( W(A) ) of the ring ( A ). One key aspect of this letter is summarized in Lemma 3.5, where it is shown that under certain conditions, the Bott class has a canonical square root in the K-theory of ( A ).

The paper thus bridges algebraic and topological aspects by generalizing classical results to a broader algebraic setting while maintaining connections with geometric interpretations through the use of Clifford algebras and Morita equivalences.