K-theory of semi-local rings with finite coefficients and etale cohomology

Let A be a commutative semi-local ring containing 1/2. We construct natural isomor-phisms [GRAPHICS] if A is 'nonexceptional'. We deduce that, for a nonexceptional scheme X quasi-projective or regular over Z[1/2], the groups K-n(X,Z/2(nu)) and K'(n)(X,Z/2(nu)) are finite for ngreater than or equal to dim(X)-1. When X is a variety over Fp or Qp with p odd, we also obtain finiteness results for K-*(X) and K'(*)(X). Finally, using higher Chern classes with values in truncated etale cohomology, we show that, for X over Z[1/2], of Krull dimension d, quasi-projective over an affine base (resp. smooth over a field or a discrete valua-tion ring), K-n(X,Z/2(nu)) is isomorphic for ngreater than or equal to 3 (resp. for ngreater than or equal to 2) to coproduct (igreater than or equal to1) H-Zar(2i-n)(X,tauless than or equal to(i) Ralpha(*)mu(2nu)(xi)), up to controlled torsion depending only on n and d (not on nu). Here, alpha is the projection from the etale site of X to its Zariski site and tau denotes truncation in the derived category.