### Clifford modules and invariants of quadratic forms, by Max Karoubi

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.

Max Karoubi <max.karoubi@gmail.com>