Etale K-theory of Azumaya algebras

Abstract

This paper extends our earlier study of the algebraic K-theory of commutative rings to certain non-commutative rings. Our method is to use etale descent to represent linear automorphisms of free modules over an Azumaya algebra by group schemes over its center. This passage from the context of non-commutative rings to algebraic geometry enables us to apply our etale-theoretic methods. In particular, we obtain a natural surjective map from algebraic to etale K-theory (with coefficients) of an Azumaya algebra over a ring of S-integers in a global field, and we prove that the etale K-theory of such an Azumaya algebra is isomorphic to that of its center. In brief, the paper is organized as follows. Section 1 presents constructions of etale K-theory for Azumaya algebras which closely parallel those of [2) for commutative rings. In particular, there is an Atiyah-Hirzebruch type spectral sequence converging to the etale K-theory of an Azumaya algebra whose &-term is isomorphic to the corresponding &term of its center. In Section 2, we establish various structures on etale K-theory of Azumaya algebras module structure, transfer, secondary transfer which arise exactly as in the commutative case. We construct a homotopy theoretic analogue of the reduced norm map in Section 3, as well as prove certain surjectivity theorems relating algebraic to etale K-theory. The second author gratefully acknowledges the hospitality of the LJniversity of Paris VII and the Max Planck Institut during the developent of this paper.