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Equivariant Witt groups. (English) Zbl 0674.10020

This paper studies for a number field K and a finite group \(\Gamma\) the cokernel of the residue homomorphism \(\partial: W(K\Gamma)\to \oplus_{{\mathfrak p}}W(k({\mathfrak p})\Gamma)\). It is shown for \(K={\mathbb{Q}}\) and \(\Gamma\) abelian that \(\partial\) is surjective. For a p-group \(\Gamma\) (not necessarily abelian) the cokernel of \(\partial\) turns out to be \(C(K)/C(K)^ 2\), where C(K) is the ideal class group of K, as in the non-equivariant case. It is also shown that for a semisimple algebra A over K endowed with an involution and an \(O_ K\)-order \(\Lambda\) \(\subset A\) preserved by the involution, there exists a natural homomorphism \(\phi\) : \(\oplus_{{\mathfrak p}}W(\Lambda /{\mathfrak p}\Lambda)\to H^ 1(C_ 2,G_ 0(\Lambda))\)- where \(G_ 0(\Lambda)\) is the Grothendieck group of the category of finitely generated \(\Lambda\)- modules - with the property \(\phi \partial =0\). In all examples considered the equality Ker \(\phi\) \(=Im \partial\) holds.
Reviewer: J.Morales

MSC:

11E16 General binary quadratic forms
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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