# Base change for prime-to-$p$ fundamental group

Let $$k$$ be an algebraic closure of $$\mathbb{F}_p$$. Let $$X$$ be a connected smooth quasi-projective $$k$$-scheme. If $$K$$ is an algebraically closed field containing $$k$$, is the prime-to-$$p$$ etale fundamental group of $$X$$ isomorphic to that of its base change to $$K$$?

## 1 Answer

Yes, the canonical map $$X_K\rightarrow X$$ induces an isomorphism on $$\pi_1^{(p)}$$. You can find here a detailed proof of a slightly more general statement.

• for future use, in case the link gets broken, this is : Aaron Landesman, Invariance of the fundamental group under base change between algebraically closed fields. – Niels Apr 23 '19 at 8:14
• A friend contacted me asking for a permanent home for this write up, so it is now posted at arxiv.org/abs/2005.09690 Also see mathoverflow.net/questions/257722/… – Aaron Landesman May 21 at 1:43