Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields

Abstract

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an essential way on several finiteness results for unramified cohomology.

Besides minor changes in exposition, we have added a new adelic proof of Theorem 1.2. This, in particular, allows us to remove any restrictions on the characteristic of the field