Base Change Of Fundamental Group Schemes

Abstract

Let $K/k$ be a field extension, $X$ a connected quasi-compact and quasi-separated scheme over $k$ with a rational point $x$, $p:X_K\to X$ the natural projection, ${\mathcal{C}}_X$ and ${\mathcal{C}}_{X_K}$ the Tannakian categories whose objects consist of vector bundles on $X$ and $X_K$ respectively. We give criteria for the natural homomorphism $\pi ({\mathcal{C}}_{X_K},x_K)\to \pi ({\mathcal{C}}_X,x{)}_K$ to be an isomorphism. Suppose further $X$ is geometrically reduced connected scheme proper over $k$. As applications, we prove base change isomorphisms for ${\pi }^{\acute{e}t},{\pi }^{E\acute{e}t}$ under arbitrary extensions and for ${\pi }^S,{\pi }^N,{\pi }^{EN}$ under arbitrary algebraic extensions; a cuspidal cubic curve example shows that it can fail for purely transcendental extensions. In positive characteristic, base change isomorphism holds for ${\pi }^{Loc}$ and ${\pi }^{ELoc}$ under separable algebraic extensions; an elliptic curve example shows that it can fail for finite purely inseparable extensions.

arXiv comment23 pages, Comments welcome!