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 <>
Alexander Nenashev <>