Pairings in triangular Witt theory

Given a product between triangulated categories with duality, we show that under some conditions there exist naturally two different pairings , where W* denotes the triangulated Witt functor of Balmer [P. Balmer, K-theory 19 (2000) 311–363]. Our main example of such a situation is the case that is the bounded derived category of vector bundles over a scheme X and is the (derived) tensor product. The derived Witt groups of this scheme 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.