Loading web-font TeX/Math/Italic

The Stacks project

Lemma 73.8.7. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{F} be a presheaf on (\textit{Spaces}/X)_{ph}. Then \mathcal{F} is a sheaf if and only if

  1. \mathcal{F} satisfies the sheaf condition for étale coverings, and

  2. if f : V \to U is a proper surjective morphism of (\textit{Spaces}/X)_{ph}, then \mathcal{F}(U) maps bijectively to the equalizer of the two maps \mathcal{F}(V) \to \mathcal{F}(V \times _ U V).

Proof. We will show that if (1) and (2) hold, then \mathcal{F} is sheaf. Let \{ T_ i \to T\} be a ph covering, i.e., a covering in (\textit{Spaces}/X)_{ph}. We will verify the sheaf condition for this covering. Let s_ i \in \mathcal{F}(T_ i) be sections which restrict to the same section over T_ i \times _ T T_{i'}. We will show that there exists a unique section s \in \mathcal{F} restricting to s_ i over T_ i. Let \{ U_ j \to T\} be an étale covering with U_ j affine. By property (1) it suffices to produce sections s_ j \in \mathcal{F}(U_ j) which agree on U_ j \cap U_{j'} in order to produce s. Consider the ph coverings \{ T_ i \times _ T U_ j \to U_ j\} . Then s_{ji} = s_ i|_{T_ i \times _ T U_ j} are sections agreeing over (T_ i \times _ T U_ j) \times _{U_ j} (T_{i'} \times _ T U_ j). Choose a proper surjective morphism V_ j \to U_ j and a finite affine open covering V_ j = \bigcup V_{jk} such that the standard ph covering \{ V_{jk} \to U_ j\} refines \{ T_ i \times _ T U_ j \to U_ j\} . If s_{jk} \in \mathcal{F}(V_{jk}) denotes the pullback of s_{ji} to V_{jk} by the implied morphisms, then we find that s_{jk} glue to a section s'_ j \in \mathcal{F}(V_ j). Using the agreement on overlaps once more, we find that s'_ j is in the equalizer of the two maps \mathcal{F}(V_ j) \to \mathcal{F}(V_ j \times _{U_ j} V_ j). Hence by (2) we find that s'_ j comes from a unique section s_ j \in \mathcal{F}(U_ j). We omit the verification that these sections s_ j have all the desired properties. \square


Comments (0)


Your email address will not be published.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.