×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Balmer P., Theory 19 pp 311– (2000)
[2] Balmer P., Math. Z. 236 pp 351– (2001)
[3] P. Balmer, C. Walter, A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. Eac. Norm. Sup. (4) 35 (2002), 127-152. · Zbl 1012.19003
[4] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Univ. Press, 1993. · Zbl 0788.13005
[5] Gille S., Math. Ann. 322 pp 103– (2002)
[6] Gille S., J. Algebra 261 pp 292– (2003)
[7] R. Hartshorne, Residues and duality, Lect. Notes Math. 20, Springer-Verlag, 1966. · Zbl 0212.26101
[8] Kawasaki T., Trans. Amer. Math. Soc. 354 pp 123– (2002)
[9] M. Knebusch, Symmetric bilinear forms over algebraic varieties, Conference on Quadratic Forms (Kingston 1976), Queen’s Papers Pure Appl. Math. 46, Queen’s Univ., Kingston (1977), 103-283.
[10] P. Roberts, Multiplicities and Chern classes in local algebra, Cambridge Univ. Press, 1998. · Zbl 0917.13007
[11] R. Y. Sharp, Necessary conditions for the existence of dualizing complexes in commutative algebra, Sem. Algebre P. Dubreil 1977/78, Lect. Notes Math. 740, Springer (1979), 213-229.
[12] J. St ckrad, W. Vogel, Buchsbaum rings and applications, Springer, 1986.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.