### Pairings in triangular Witt theory, by Stefan Gille and Alexander Nenashev

This paper has been accepted for publication, and an updated version posted,
June 29, 2002.

Given a product ${\cal K}\times {\cal L}\rightarrow {\cal M}$ between
triangulated categories with duality, we show that under some conditions
there exists naturally two different pairings $W^{i}({\cal K})\times
W^{j}({\cal L})\rightarrow W^{i+j}({\cal M})$, where $W^*$ denotes the
triangulated Witt functor of Balmer. Our main example of such a situation is
the case that ${\cal K}={\cal L}={\cal M}$ is the bounded derived category of
vector bundles over a scheme $X$ and the product is the (derived) tensor
product. The derived Witt groups of this scheme $W^*(X):=\oplus_{i\in\Z}
W^{i}(X)$ become a graded skew-commutative ring with two different product
structures which are both equally natural. In the last section we prove then
a projection formula for our product and show as an application that a
Springer-type theorem is true for the derived Witt groups, too.

Stefan Gille <gilles@math.uni-muenster.de>

Alexander Nenashev <nenashev@snoopy.usask.ca>