The Stacks project

Lemma 76.9.4. Let $S$ be a scheme. Let $X \subset X'$ be a thickening of algebraic spaces over $S$. Let $U$ be an affine object of $X_{spaces, {\acute{e}tale}}$. Then

\[ \Gamma (U, \mathcal{O}_{X'}) \to \Gamma (U, \mathcal{O}_ X) \]

is surjective where we think of $\mathcal{O}_{X'}$ as a sheaf on $X_{spaces, {\acute{e}tale}}$ via (76.9.1.2).

Proof. Let $U' \to X'$ be the ├ętale morphism of algebraic spaces such that $U = X \times _{X'} U'$, see Theorem 76.8.1. By Limits of Spaces, Lemma 70.15.1 we see that $U'$ is an affine scheme. Hence $\Gamma (U, \mathcal{O}_{X'}) = \Gamma (U', \mathcal{O}_{U'}) \to \Gamma (U, \mathcal{O}_ U)$ is surjective as $U \to U'$ is a closed immersion of affine schemes. Below we give a direct proof for finite order thickenings which is the case most used in practice. $\square$

Proof for finite order thickenings. We may assume that $X \subset X'$ is a first order thickening by the principle explained above. Denote $\mathcal{I}$ the kernel of the surjection $\mathcal{O}_{X'} \to \mathcal{O}_ X$. As $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_{X'}$-module and since $\mathcal{I}^2 = 0$ by the definition of a first order thickening we may apply Morphisms of Spaces, Lemma 67.14.1 to see that $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_ X$-module. Hence the lemma follows from the long exact cohomology sequence associated to the short exact sequence

\[ 0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0 \]

and the fact that $H^1_{\acute{e}tale}(U, \mathcal{I}) = 0$ as $\mathcal{I}$ is quasi-coherent, see Descent, Proposition 35.9.3 and Cohomology of Schemes, Lemma 30.2.2. $\square$


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