The azumaya complex of a commutative ring

Abstract

This paper has as its aim the geometric description of three important groups classically associated with a commutative ring R as the first (and only) three non-trivial homotopy groups of a reduced (and hence connected) simplicial Kan-complex AZ(R) which we wi l l call the Azumaya complex of the ring. This description, which places the Brauer group Br(R) of equivalence classes of Azumaya algebras over the ring as T~, of the complex, the Picard group Pic(R) of isomorphism classes of invertible modules over it as rr a, and the multiplicative group Gm(R) [= R x, the group of units] of the ring as W3, is stable under arbitrary change of the commutative base ring and (among other things) allows the recovery of several wel l-known exact sequences linking these groups as instances of exact sequences of groups obtained in simplicial homotopy theory. This conference paper wi l l be devoted to a direct description of the simplicial complex itself wi th the applications which involve the computation of a particular spectral sequence to appear elsewhere. The complex we construct here is, in fact, one which is associated with a particular 3-category AZI~ which has a single object (i.e.,0-ceil) the commutative ring R. Its 1-cells are Azumaya algebras over R, A R )R 5