The Stacks project

Proof. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. By (87.34.1.1) it suffices to construct $\mathcal{O}_ X$ as a sheaf of topological rings on $X_{affine, {\acute{e}tale}}$. Denote $\mathcal{C}$ the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $U$ is an affine formal algebraic space and $\varphi $ is representable by algebraic spaces and étale. By Lemma 87.34.7 the functor $U \mapsto U_{red}$ is an equivalence of categories $\mathcal{C} \to X_{affine, {\acute{e}tale}}$. Hence by the rule given above the lemma, we already have $\mathcal{O}_ X$ as a presheaf of topological rings on $X_{affine, {\acute{e}tale}}$. Thus it suffices to check the sheaf condition.

By definition of $X_{affine, {\acute{e}tale}}$ a covering corresponds to a finite family $\{ g_ i : U_ i \to U\} _{i = 1, \ldots , n}$ of morphisms of $\mathcal{C}$ such that $\{ U_{i, red} \to U_{red}\} $ is an étale covering. The morphisms $g_ i$ are representably by algebraic spaces (Lemma 87.19.3) hence affine (Lemma 87.19.7). Then $g_ i$ is étale (follows formally from Properties of Spaces, Lemma 66.16.6 as $U_ i$ and $U$ are étale over $X$ in the sense of Bootstrap, Section 80.4). Finally, write $U = \mathop{\mathrm{colim}}\nolimits U_\lambda $ as in Definition 87.9.1.

With these preparations out of the way, we can prove the sheaf property as follows. For each $\lambda $ we set $U_{i, \lambda } = U_ i \times _ U U_\lambda $ and $U_{ij, \lambda } = (U_ i \times _ U U_ j) \times _ U U_\lambda $. By the above, these are affine schemes, $\{ U_{i, \lambda } \to U_\lambda \} $ is an étale covering, and $U_{ij, \lambda } = U_{i, \lambda } \times _{U_\lambda } U_{j, \lambda }$. Also we have $U_ i = \mathop{\mathrm{colim}}\nolimits U_{i, \lambda }$ and $U_ i \times _ U U_ j = \mathop{\mathrm{colim}}\nolimits U_{ij, \lambda }$. For each $\lambda $ we have an exact sequence

\[ 0 \to \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \to \prod \nolimits _ i \Gamma (U_{i, \lambda }, \mathcal{O}_{U_{i, \lambda }}) \to \prod \nolimits _{i, j} \Gamma (U_{ij, \lambda }, \mathcal{O}_{U_{ij, \lambda }}) \]

as we have the sheaf condition for the structure sheaf on $U_\lambda $ and the étale topology (see Étale Cohomology, Proposition 59.17.1). Since limits commute with limits, the inverse limit of these exact sequences is an exact sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \to \prod \nolimits _ i \mathop{\mathrm{lim}}\nolimits \Gamma (U_{i, \lambda }, \mathcal{O}_{U_{i, \lambda }}) \to \prod \nolimits _{i, j} \mathop{\mathrm{lim}}\nolimits \Gamma (U_{ij, \lambda }, \mathcal{O}_{U_{ij, \lambda }}) \]

which exactly means that

\[ 0 \to \mathcal{O}_ X(U_{red}) \to \prod \nolimits _ i \mathcal{O}_ X(U_{i, red}) \to \prod \nolimits _{i, j} \mathcal{O}_ X((U_ i \times _ U U_ j)_{red}) \]

is exact and hence the sheaf property holds as desired. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DEJ. Beware of the difference between the letter 'O' and the digit '0'.