Galois coinvariants and fundamental groups of arithmetic schemes

Abstract

We study Galois coinvariants of the geometric étale fundamental group and étale cohomology of varieties over a finitely generated field $k$. We establish finiteness properties of these coinvariants for (possibly infinite) Galois extensions $L/k$ such that $\operatorname{Gal}(L/k)^{\mathrm{ab}}$ has finite exponent. As an application, we obtain a finiteness result for étale fundamental groups of arithmetic schemes over number fields.