A transfer morphism for Witt groups

We construct a transfer homomorphism for coherent Witt groups of commutative rings with dualizing complexes and prove a projection formula. As a consequence we get a transfer morphism for the usual Witt groups. Namely, let p : R ! S be a finite morphism of regular rings of finite Krull dimension. If ht(ker pi) = 0 mod 4 and Spec S is connected then there exists a transfer morphism W(S, L) --> W(R), where L is an appropriate line bundle on Spec S. To calculate the line bundle L we show the following. If pi : R --> S is a finite morphism of Cohen-Macaulay rings of finite Krull dimension, l = ht(ker pi), and Omega is a canonical (or also called dualizing) module of R then Ext(R)(l)(S,Omega) is a canonical module of S. In the last section we construct a devissage map theta(n, S) : W*(S) --> W-(X1,W-...,W-Xn)*(+n) (S[X-1,..., X-n]) for any (commutative) ring S. If S is a finite dimensional regular ring we show that this morphism is equal to the transfer map constructed using coherent methods. It follows that for such a ring S any n-form on the Koszul complex K-.(X-1,..., X-n) over S[X-1,..., X-n] is a generator of the graded module <LF>W-(X1,W-...,W- Xn)* (S[X-1,..., X-n]) over the ring W*(S).