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A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution of the second kind

Published online by Cambridge University Press:  04 September 2008

Stefan Gille
Affiliation:
[email protected] InstitutUniversität MünchenTheresienstrasse 3980333 MünchenGermany
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Abstract

Let X be a regular noetherian scheme of finite Krull dimension with involution σ and an Azumaya algebra over X with involution τ of the second kind with respect to σ. We construct a hermitian and a skew-hermitian Gersten-Witt complex for (, τ) and show that these complexes are exact if X = Spec R is the spectrum of a regular semilocal ring R of geometric type, such that R is a quadratic étale extension of the fixed ring of σ.

Type
Research Article
Copyright
Copyright © ISOPP 2009

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References

1.Balmer, P.. Triangular Witt groups I: The 12-term localization exact sequence, K-Theory 19 (2000), 311363Google Scholar
2.Balmer, P.. Triangular Witt groups II: From usual to derived, Math. Z. 236 (2001), 351382CrossRefGoogle Scholar
3.Balmer, P., Walter, C.. A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. (4) 35 (2002), 127152Google Scholar
4.Berthelot, P., Grothendieck, A., Illusie, L., Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Berthelot, Dirigé par P., Grothendieck, A. et Illusie, L., Springer LNM 225 (1971)Google Scholar
5.Bruns, W., Herzog, J., Cohen-Macaulay rings, Cambridge Univ. Press 1993Google Scholar
6.Curtis, C., Reiner, I.. Methods of representation theory. Vol. I. With applications to finite groups and orders, Wiley-Interscience 1981Google Scholar
7.DeMeyer, F., Ingraham, E., Separable algebras over commutative rings, Springer LNM 181 (1971)CrossRefGoogle Scholar
8.Eilenberg, S., Nakayama, T.. On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. 9 (1955), 116Google Scholar
9.Gille, S.. On Witt groups with support, Math. Ann. 322 (2002), 103137CrossRefGoogle Scholar
10.Gille, S.. Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), 211233Google Scholar
11.Gille, S.. On injective modules over Azumaya algebras over locally noetherian schemes, Manuscripta Math. 121 (2006), 437450Google Scholar
12.Gille, S.. A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes I: Involution of the first kind, Compos. Math. 143 (2007), 271289Google Scholar
13.Grothendieck, A., Dieudonné, J., Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967)Google Scholar
14.Hahn, A.. Quadratic algebras, Clifford algebras, and arithmetic Witt groups, Springer 1994Google Scholar
15.Hartshorne, R.. Residues and duality, Springer LNM 20 (1966)Google Scholar
16.Knus, M.. Quadratic and Hermitian forms over rings, Springer 1991Google Scholar
17.Knus, M., Ojanguren, M.. Théorie de la descente et algèbres d'Azumaya, Springer LNM 389 (1974)Google Scholar
18.Milne, J.. Étale cohomology, Princeton Univ. Press 1980Google Scholar
19.Ojanguren, M., Panin, I.. Rationally trivial Hermitian spaces are locally trivial, Math. Z. 237 (2001), 181198Google Scholar
20.Quebbemann, H., Scharlau, W., Schulte, M., Quadratic and Hermitian forms in additive and abelian categories, J. Algebra 59 (1979), 264289Google Scholar
21.Quillen, D.. Higher algebraic K -theory, in: Proceedings of the Battele Conference, Seattle, Washington, 1972, Springer LNM 341 (1973), 85147Google Scholar
22.Raynaud, M.. Anneaux Locaux Henséliens, Springer LNM 169 (1970)CrossRefGoogle Scholar
23.Reiner, I.. Maximal orders, Academic Press 1975Google Scholar
24.Yekutieli, A.. Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), 4184Google Scholar