The Merkuriev-suslin Theorem for Any Semi-local Ring

We introduce here a method which uses étale neighborhoods to extend results from smooth semi-local rings to arbitrary semi-local rings A by passing to the henselization of a smooth presentation of A. The technique is used to show that étale cohomology of A agrees with Galois cohomology, the Merkuriev-Suslin theorem holds for A, and to describe torsion in K2(A). We introduce here a method which uses étale neighborhoods to extend results from smooth semi-local rings to arbitrary semi-local rings. Three applications are given. In the first and last, A is a connected, semi-local ring containing a field k while the second application holds for any connected, semi-local ring A. 1) Let X = Spec (A), and let n an integer with (n, char(k)) = 1. If F is a finite, locally constant sheaf of Z/n-modules for the étale site on X, we show H(G(As/A), F (As)) → ≈ H(X,F ) where As is the separable closure of A, the left hand side is the Galois cohomology of A with coefficients in the G(As/A)-module F (As) and the right hand side is the étale cohomology group of the semi-local scheme X with coefficients in F . 2) We extend the Merkuriev-Suslin theorem to a connected, semilocal ring A; that is, for n relatively prime to the residue characteristics of A, the Galois symbol map K2(A)/n → ≈ H(A,Z/n(2)) is an isomorphism where, as usual, Z/n(i) = μ n . Since this implies the cup product map is surjective, we conclude that any Azumaya algebra of order n in Br(A) is similar to a tensor product of symbol algebras if A contains a primitive n root of unity. 3) We extend Suslin’s computation of the l-primary component of the torsion in K2(k), k a field, to A. Fix notation as follows. H(X,F ) (or, if X = Spec (A), H(A,F )) denotes the étale cohomology group of X with coefficients in the étale