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The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius. It does not commute with products, as was shown by YCor in the answer to this question. So, as it was suggested in that answer, I ask as a separate question the following one:

Why in a scheme $X$ (not necessarily reduced), the presheaf associated to $U\to \Gamma (U,\mathcal{O_\rm{X}})_\rm{pf}$ is a sheaf?

This is stated in section 6 in Greenberg: Perfect closures of rings and schemes but since perfection does not commute with products, I do not understand the proof.

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    $\begingroup$ I think YCor's example in the linked question shows that this is not quite a sheaf unless you disallow infinite coverings. However, if you sheafify your presheaf, then it will still be true for affine (or quasi-compact) $U$ that sections over $U$ of that sheaf will equal $\Gamma(U, \mathcal{O}_X)_{\rm pf}$. $\endgroup$
    – Piotr Achinger
    Commented Mar 16, 2020 at 20:51
  • $\begingroup$ @Piotr Achinger Why that example shows that it is not a sheaf? Is it something easy that I have to think more? (sorry if it is). $\endgroup$
    – A.G
    Commented Mar 17, 2020 at 10:32
  • $\begingroup$ Let $(A_i)$ be an infinite sequence of rings such that $(\prod A_i)_{\rm pf} \neq \prod (A_i)_{\rm pf}$, and let $X$ be the disjoint union of $U_i = \operatorname{Spec} A_i$. Note that $X$ is not affine since it is not quasi-compact. Still, $\Gamma(X, \mathcal{O}_X) = \prod A_i$, so $\Gamma(X, \mathcal{O}_X)_{\rm pf} = (\prod A_i)_{\rm pf}$. Consider sheaf condition the covering of $X$ by $U_i$, it simply says $\mathcal{F}(X) = \prod \mathcal{F}(U_i)$, which is not true for your presheaf. $\endgroup$
    – Piotr Achinger
    Commented Mar 17, 2020 at 10:48
  • $\begingroup$ It was obvious. Sorry, and thank you. $\endgroup$
    – A.G
    Commented Mar 17, 2020 at 11:01

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Let $p$ be a prime number. Let $X$ be a scheme over $\mathbf{F}_p$. The correct statement is that there is a sheaf of $\mathcal{O}_X$-algebras $\mathcal{A}$ on $X$ such that $\mathcal{A}(U) = \mathcal{O}_X(U)_{\text{pf}}$ for every quasi-compact and quasi-separated open subscheme $U$ of $X$.

Namely, we take the following colimit in the category of sheaves $$ \mathcal{A} = \text{colim}\ (\mathcal{O}_X \to \mathcal{O}_X \to \mathcal{O}_X \to \ldots\ ) $$ where the transition maps are the Frobenius map and we may use the very general Lemma 009F to get the statement about sections over qcqs opens of $X$.

Bonus fact: this construction gives a sheaf in the h topology.

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