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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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