The Stacks project

Lemma 76.9.7. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Consider a short exact sequence

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

of sheaves on $X_{\acute{e}tale}$ where $\mathcal{A}$ is a sheaf of $f^{-1}\mathcal{O}_ B$-algebras, $\mathcal{A} \to \mathcal{O}_ X$ is a surjection of sheaves of $f^{-1}\mathcal{O}_ B$-algebras, and $\mathcal{I}$ is its kernel. If

  1. $\mathcal{I}$ is an ideal of square zero in $\mathcal{A}$, and

  2. $\mathcal{I}$ is quasi-coherent as an $\mathcal{O}_ X$-module

then there exists a first order thickening $X \subset X'$ over $B$ and an isomorphism $\mathcal{O}_{X'} \to \mathcal{A}$ of $f^{-1}\mathcal{O}_ B$-algebras compatible with the surjections to $\mathcal{O}_ X$.

Proof. In this proof we redo some of the arguments used in the proofs of Lemmas 76.9.4 and 76.9.5. We first handle the case $B = S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $U$ be an affine scheme, and let $U \to X$ be étale. Then

\[ 0 \to \mathcal{I}(U) \to \mathcal{A}(U) \to \mathcal{O}_ X(U) \to 0 \]

is exact as $H^1(U_{\acute{e}tale}, \mathcal{I}) = 0$ as $\mathcal{I}$ is quasi-coherent, see Descent, Proposition 35.9.3 and Cohomology of Schemes, Lemma 30.2.2. If $V \to U$ is a morphism of affine objects of $X_{spaces, {\acute{e}tale}}$ then

\[ \mathcal{I}(V) = \mathcal{I}(U) \otimes _{\mathcal{O}_ X(U)} \mathcal{O}_ X(V) \]

since $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_ X$-module, see Descent, Proposition 35.8.9. Hence $\mathcal{A}(U) \to \mathcal{A}(V)$ is an étale ring map, see Algebra, Lemma 10.143.11. Hence we see that

\[ U \longmapsto U' = \mathop{\mathrm{Spec}}(\mathcal{A}(U)) \]

is a functor from $X_{affine, {\acute{e}tale}}$ to the category of affine schemes and étale morphisms. In fact, we claim that this functor can be extended to a functor $U \mapsto U'$ on all of $X_{\acute{e}tale}$. To see this, if $U$ is an object of $X_{\acute{e}tale}$, note that

\[ 0 \to \mathcal{I}|_{U_{Zar}} \to \mathcal{A}|_{U_{Zar}} \to \mathcal{O}_ X|_{U_{Zar}} \to 0 \]

and $\mathcal{I}|_{U_{Zar}}$ is a quasi-coherent sheaf on $U$, see Descent, Proposition 35.9.4. Hence by More on Morphisms, Lemma 37.2.2 we obtain a first order thickening $U \subset U'$ of schemes such that $\mathcal{O}_{U'}$ is isomorphic to $\mathcal{A}|_{U_{Zar}}$. It is clear that this construction is compatible with the construction for affines above.

Choose a presentation $X = U/R$, see Spaces, Definition 65.9.3 so that $s, t : R \to U$ define an étale equivalence relation. Applying the functor above we obtain an étale equivalence relation $s', t' : R' \to U'$ in schemes. Consider the algebraic space $X' = U'/R'$ (see Spaces, Theorem 65.10.5). The morphism $X = U/R \to U'/R' = X'$ is a first order thickening. Consider $\mathcal{O}_{X'}$ viewed as a sheaf on $X_{\acute{e}tale}$. By construction we have an isomorphism

\[ \gamma : \mathcal{O}_{X'}|_{U_{\acute{e}tale}} \longrightarrow \mathcal{A}|_{U_{\acute{e}tale}} \]

such that $s^{-1}\gamma $ agrees with $t^{-1}\gamma $ on $R_{\acute{e}tale}$. Hence by Properties of Spaces, Lemma 66.18.14 this implies that $\gamma $ comes from a unique isomorphism $\mathcal{O}_{X'} \to \mathcal{A}$ as desired.

To handle the case of a general base algebraic space $B$, we first construct $X'$ as an algebraic space over $\mathbf{Z}$ as above. Then we use the isomorphism $\mathcal{O}_{X'} \to \mathcal{A}$ to define $f^{-1}\mathcal{O}_ B \to \mathcal{O}_{X'}$. According to Lemma 76.9.2 this defines a morphism $X' \to B$ compatible with the given morphism $X \to B$ and we are done. $\square$

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