Lemma 58.14.2 (0BTW)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.14: Geometric and arithmetic fundamental groups
Lemma 58.14.2 (cite)

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Lemma 58.14.2. Let $k$ be a field with perfection $k^{perf}$ . Let $X$ be a connected scheme over $k$ . Then $X_{k^{perf}}$ is connected and $\pi _1(X_{k^{perf}}) \to \pi _1(X)$ is an isomorphism.

Proof. Special case of topological invariance of the fundamental group. See Proposition 58.8.4. To see that $\mathop{\mathrm{Spec}}(k^{perf}) \to \mathop{\mathrm{Spec}}(k)$ is a universal homeomorphism you can use Algebra, Lemma 10.46.10. $\square$

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