The Stacks project

Lemma 38.38.7. Let $p$ be a prime number.

  1. If $A$ is an $\mathbf{F}_ p$-algebra, then $\mathop{\mathrm{colim}}\nolimits _ F A = A^{awn}$.

  2. If $S$ is a scheme over $\mathbf{F}_ p$, then the h sheafification of $\mathcal{O}$ sends a quasi-compact and quasi-separated $X$ to $\mathop{\mathrm{colim}}\nolimits _ F \Gamma (X, \mathcal{O}_ X)$.

Proof. Proof of (1). Observe that $A \to \mathop{\mathrm{colim}}\nolimits _ F A$ induces a universal homeomorphism on spectra by Algebra, Lemma 10.46.7. Thus it suffices to show that $B = \mathop{\mathrm{colim}}\nolimits _ F A$ is absolutely weakly normal, see Morphisms, Lemma 29.47.6. Note that the ring map $F : B \to B$ is an automorphism, in other words, $B$ is a perfect ring. Hence Lemma 38.38.6 applies.

Proof of (2). This follows from (1) and Lemmas 38.38.2 and 38.38.5 by looking affine locally. $\square$

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