Stable cohomology of groups and algebraic varieties

The notion of stable cohomology of algebraic varieties and, based on it, the analogous concept for finite and profinite groups are introduced. It is proved that the ordinary and stable cohomology coincide for the Galois group of the algebraic closure of a function field with an algebraically closed constant field of characteristic zero, and also that the unramified cohomology of this Galois group with coefficients in a module with trivial group action coincides with the unramified cohomology groups of a variety having a given field of rational functions.