The Stacks project

Lemma 62.3.2. Let $k$ be a field of characteristic $p > 0$ with perfect closure $k^{perf}$. Let $X$ be an algebraic scheme over $k$. Let $r \geq 0$ be an integer. The cokernel of the injective map $Z_ r(X) \to Z_ r(X_{k^{perf}})$ is a $p$-power torsion module (More on Algebra, Definition 15.88.1).

Proof. Since $X$ is quasi-compact, the abelian group $Z_ r(X)$ is free with basis given by the integral closed subschemes of dimension $r$. Similarly for $Z_ r(X_{k^{perf}})$. Since $X_{k^{perf}} \to X$ is a homeomorphism, it follows that $Z_ r(X) \to Z_ r(X_{k^{perf}})$ is injective with torsion cokernel. Every element in the cokernel is $p$-power torsion by Lemma 62.3.1. $\square$

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