Topological models for arithmetic

Abstract

THE etale topological type A,, of a commutative ring A is an interesting topological object (technically a pro-space) associated to A. In [4] we introduced the “etale K-groups” of A as local coefficient I-adic topological K-theory groups of Act, where 1 is a prime invertible in A. Building upon work of C. Soule [19], we described a natural map from the algebraic K-groups of A (completed at I) to the etale K-groups, and showed that in general this map is surjective for arithmetic rings. The well-known Lichtenbaum-Quillen Conjectures essentially assert that this map is an isomorphism. The basic results of class field theory and arithmetic duality can be interpreted as statements about the etale cohomology of arithmetic rings A [lo] or equivalently as statements about the ordinary cohomology of A,,. In [S] we exploited this fact to find, for some specific arithmetic rings A, an elementary topological space XA mapping to A,, by a cohomology isomorphism. Finding XA allowed us not only to compute the etale K-groups of A explicitly, but also to identify the underlying local coefficient topological K-theory space and compute its cohomology (which, by the Lichtenbaum-Quillen conjecture, should be isomorphic to the cohomology of GL(A)). We do two things in the present paper. First of all, we extend the above program of finding “models” for A,, to some other rings. In each case, for a specific prime I invertible in A, we find a “good mod 1 model” XA for A; this is an explicit space or pro-space XA which in an appropriate sense captures the mod 1 cohomology of A,,. Here are the main examples: