Topological models for arithmetic
A ``good \(\text{mod } \ell\) model'' \(X^ A\) for the etale topological type of a commutative ring \(A\) is found in certain cases. Thus, (1) If \(A= \mathbb{Z}[1/ \ell]\) for an odd regular prime \(\ell\), \(X^ A= \mathbb{R} P^ \alpha\vee S^ 1\). (2) If \(A\) is a coordinate ring of a suitable affine curve over \(\mathbb{F}_ q\), then \(X^ A\) is a fibration over \(S^ 1\) with fiber the \(p\)- completion of a finite wedge of circles. (3) If \(A\) is a generalized local field of transcendence degree \(r\) over \(\mathbb{F}_ q\), \(X^ A\) is a fibration over \(S^ 1\) with fibre the \(p\)- completion of a product of \(r\) circles. Secondly, the etale \(K\)-theory space of \(X^ A\) in case (1) has a cohomology which injects into the cohomology of the algebraic \(K\)-theory space of \(A\). Corollaries depending on the validity of the Lichtenbaum- Quillen conjecture are then derived.
scientific article; zbMATH DE number 527435