Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T10:30:23.017Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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.

Keywords

Azumaya algebrasWitt groups of hermitian forms
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 2 , October 2009 , pp. 347 - 377
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Balmer, P.. Triangular Witt groups I: The 12-term localization exact sequence, K-Theory 19 (2000), 311–363Google Scholar
2.Balmer, P.. Triangular Witt groups II: From usual to derived, Math. Z. 236 (2001), 351–382CrossRefGoogle Scholar
3.Balmer, P., Walter, C.. A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. (4) 35 (2002), 127–152Google 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), 1–16Google Scholar
9.Gille, S.. On Witt groups with support, Math. Ann. 322 (2002), 103–137CrossRefGoogle Scholar
10.Gille, S.. Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), 211–233Google Scholar
11.Gille, S.. On injective modules over Azumaya algebras over locally noetherian schemes, Manuscripta Math. 121 (2006), 437–450Google 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), 271–289Google 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), 181–198Google Scholar
20.Quebbemann, H., Scharlau, W., Schulte, M., Quadratic and Hermitian forms in additive and abelian categories, J. Algebra 59 (1979), 264–289Google Scholar
21.Quillen, D.. Higher algebraic K -theory, in: Proceedings of the Battele Conference, Seattle, Washington, 1972, Springer LNM 341 (1973), 85–147Google 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), 41–84Google Scholar