A transfer morphism for Witt groups. (English) Zbl 1050.11046

The trace map or Scharlau transfer is an important tool in the classical theory of quadratic forms. The present paper contains a vast generalization of this construction. Both coherent and derived Witt groups are considered. Let \(R\) be a ring. The family \((\widetilde W^i(R,I_.))_{i\in\mathbb{Z}}\) of coherent Witt groups is constructed using a dualizing complex of injective \(R\)-modules. For Cohen-Macaulay rings with finite Krull dimension dualizing complexes are closely related to canonical modules. The construction of the derived Witt groups \((W^i(R,{\mathcal E}))_{i\in\mathbb{Z}}\) uses a locally free \(R\)-module \({\mathcal E}\) of rank 1. Any finite injective resolution \(0\to{\mathcal E}\to I_0\to I_1\to\cdots\) leads to a (non-canonical) homomorphism \(W^i(R,{\mathcal E})\to\widetilde W^i(R,I_.)\), which sometimes is an isomorphism. The derived Witt groups with respect to the module \({\mathcal E}= R\) form a graded ring \(W^*(R)\), the coherent Witt groups form a graded module over this graded ring. Given a finite ring homomorphism \(\pi: R\to S\), every dualizing complex \(I_.\) for \(R\) yields a natural dualizing complex \(\pi^\#(I_.)\) for \(S\). These complexes are used to define the transfer map \(\text{Tr}_{S/R}:\widetilde W^i(S,\pi^\#(I_.))\to\widetilde W^i(R,I_.)\). A projection formula shows that the transfer is a homogeneous \(W^*(R)\)-homomorphism. The transfer map is considered more closely for the case that \(\pi: R\to R/{\mathfrak a}\) is the factor map belonging to some ideal \({\mathfrak a}\subseteq R\).


11E81 Algebraic theory of quadratic forms; Witt groups and rings
13D02 Syzygies, resolutions, complexes and commutative rings
19G12 Witt groups of rings
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