The Stacks project

Lemma 38.37.8. Let $\mathcal{F}$ be a sheaf on one of the sites $(\mathit{Sch}/S)_ h$ constructed in Definition 38.34.12. Let $X \to X'$ be a morphism of $(\mathit{Sch}/S)_ h$ which is a thickening and of finite presentation. Then $\mathcal{F}(X') \to \mathcal{F}(X)$ is bijective.

Proof. First proof. Observe that $X \to X'$ is a proper surjective morphism of finite presentation and $X \times _{X'} X = X$. By the sheaf property for the h covering $\{ X \to X'\} $ (Lemma 38.34.6) we conclude.

Second proof (silly). The blow up of $X'$ in $X$ is the empty scheme. The reason is that the affine blowup algebra $A[\frac{I}{a}]$ (Algebra, Section 10.70) is zero if $a$ is a nilpotent element of $A$. Details omitted. Hence we get an almost blow up square of the form

\[ \xymatrix{ \emptyset \ar[r] \ar[d] & \emptyset \ar[d] \\ X \ar[r] & X' } \]

Since $\mathcal{F}$ is a sheaf we have that $\mathcal{F}(\emptyset )$ is a singleton. Applying Lemma 38.37.7 we get the conclusion. $\square$

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