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Quadratic forms with values in invertible modules
Authors:Stefaan Caenepeel and Freddy van Oystaeyen
Institution:(1) Faculty of Applied Sciences, Free University of Brussels (VUB), Pleinlaan 2, B-1050 Brussels, Belgium;(2) Department of Mathematics, University of Antwerp (UIA), Universiteitsplein 1, B-2610 Wilrijk, Belgium
Abstract:LetR be a commutative ring,I an invertibleR-module, and consider quadratic spaces with values inI. The Clifford algebra of such a quadratic space is an algebra over the generalized Rees ring associated toI. We discuss the relation between the Witt module of quadratic spaces with values inI and the graded Witt ring and the graded Brauer-Wall group of the generalized Rees ring. This leads to the introduction of three distinguished subgroups of the graded Brauer-Wall group of the generalized Rees ring. The image of the Clifford functor is a subgroup of one of these three subgroups (the lsquotype 1rsquo subgroup).
Keywords:Quadratic forms  Clifford functor  Witt ring  generalized Rees ring  Brauer-Wall group  graded Brauer group
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