# Is the perfection (perfect closure) presheaf a sheaf?

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.

• 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 $\Gamma(U, \mathcal{O}_X)_{\rm pf}$. Mar 16, 2020 at 20:51
• @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).
– A.G
Mar 17, 2020 at 10:32
• 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. Mar 17, 2020 at 10:48
• It was obvious. Sorry, and thank you.
– A.G
Mar 17, 2020 at 11:01

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 $$X$$.