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On Karoubi’s theorem: \(W(A)=W(A[t])\). (English) Zbl 0552.18003
Let A be an associative ring with involution in which 2 is invertible and fix \(\epsilon =\pm 1\). An \(\epsilon\)-Hermitian space (M,q) over A consists of a finitely generated projective right A-module and an A- linear isomorphism \(q: M\to^{\simeq}M^*=Hom_ A(M,A)\) such that \(q=\epsilon q^*\), where \(M^*\) is viewed as a right A-module via the involution of A. The Witt group \(W_{\epsilon}(A)\) is the quotient of the free abelian group on the set of (suitably defined) isometry classes of \(\epsilon\)-Hermitian spaces by the subgroup generated by the hyperbolic spaces. In this paper, the author gives a short and simple proof of Karoubi’s theorem: the inclusion of A into the polynomial ring A[t] induces an isomorphism \(W_{\epsilon}(A)\to W_{\epsilon}(A[t])\).
Reviewer: M.R.Stein

MSC:
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
16E20 Grothendieck groups, \(K\)-theory, etc.
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
11E16 General binary quadratic forms
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References:
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[2] M. Karoubi, P?riodicit? de laK-th?orie hermitienne. LNM343, 301-411 Berlin-Heidelberg-New York 1973.
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