## 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 $${\mathfrak a}\subseteq R$$.

### MSC:

 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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### References:

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