The Stacks project

Lemma 59.82.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $i : Z \to X$ be a closed immersion. Assume that

  1. for any sheaf $\mathcal{F}$ on $X_{Zar}$ the map $\Gamma (X, \mathcal{F}) \to \Gamma (Z, i^{-1}\mathcal{F})$ is bijective, and

  2. for any finite morphism $X' \to X$ assumption (1) holds for $Z \times _ X X' \to X'$.

Then for any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (Z, i^{-1}_{small}\mathcal{F})$.

Proof. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. There is a canonical (base change) map

\[ i^{-1}(\mathcal{F}|_{X_{Zar}}) \longrightarrow (i_{small}^{-1}\mathcal{F})|_{Z_{Zar}} \]

of sheaves on $Z_{Zar}$. We will show this map is injective by looking at stalks. The stalk on the left hand side at $z \in Z$ is the stalk of $\mathcal{F}|_{X_{Zar}}$ at $z$. The stalk on the right hand side is the colimit over all elementary étale neighbourhoods $(U, u) \to (X, z)$ such that $U \times _ X Z \to Z$ has a section over a neighbourhood of $z$. As étale morphisms are open, the image of $U \to X$ is an open neighbourhood $U_0$ of $z$ in $X$. The map $\mathcal{F}(U_0) \to \mathcal{F}(U)$ is injective by the sheaf condition for $\mathcal{F}$ with respect to the étale covering $U \to U_0$. Taking the colimit over all $U$ and $U_0$ we obtain injectivity on stalks.

It follows from this and assumption (1) that the map $\Gamma (X, \mathcal{F}) \to \Gamma (Z, i^{-1}_{small}\mathcal{F})$ is injective. By (2) the same thing is true on all $X'$ finite over $X$.

Let $s \in \Gamma (Z, i^{-1}_{small}\mathcal{F})$. By construction of $i^{-1}_{small}\mathcal{F}$ there exists an étale covering $\{ V_ j \to Z\} $, étale morphisms $U_ j \to X$, sections $s_ j \in \mathcal{F}(U_ j)$ and morphisms $V_ j \to U_ j$ over $X$ such that $s|_{V_ j}$ is the pullback of $s_ j$. Observe that every nonempty closed subscheme $T \subset X$ meets $Z$ by assumption (1) applied to the sheaf $(T \to X)_*\underline{\mathbf{Z}}$ for example. Thus we see that $\coprod U_ j \to X$ is surjective. By More on Morphisms, Lemma 37.45.7 we can find a finite surjective morphism $X' \to X$ such that $X' \to X$ Zariski locally factors through $\coprod U_ j \to X$. It follows that $s|_{Z'}$ Zariski locally comes from a section of $\mathcal{F}|_{X'}$. In other words, $s|_{Z'}$ comes from $t' \in \Gamma (X', \mathcal{F}|_{X'})$ by assumption (2). By injectivity we conclude that the two pullbacks of $t'$ to $X' \times _ X X'$ are the same (after all this is true for the pullbacks of $s$ to $Z' \times _ Z Z'$). Hence we conclude $t'$ comes from a section of $\mathcal{F}$ over $X$ by Remark 59.55.6. $\square$

Comments (5)

Comment #3638 by Brian Conrad on

In the final paragraph of the proof, replace "every closed subscheme " with "every non-empty closed subscheme ".

Comment #5558 by Harry Gindi on

In the injectivity party of the lemma, it is probably worth knowing that if U is étale over X and U_0 c X is the open image, then U→U_0 is an étale surjection and therefore since F is an étale sheaf, we have F(U_0)→F(U) is injective, as it is the inclusion of the equalizer

F(U_0)→F(U)=>F(U×_{U_0} U).

Comment #9307 by Fiasco on

In the end of proof of lemma, why does Zariski locally come from a section of ? And why are these local sections compatible such that they can glue altogether?

For the first question, let's be more careful(with the same notations). Let be an open subscheme which factors through for some . So defines a section of named , we want to show its restriction is equal to . Note that has an etale cover , So we only need to check each piece . We look at the stalks and choose a geometric point .

On the one hand, the stalk of restriction of is just the stalk of at pt, which is considered as a geometric point of .

On the other hand, the stalk of is just the stalk of at pt, which is considered as a geometric point of .

Now the key point is we only know pt factors through . But by definition and coincide at some scheme which is etale over such that and both factor through . So can be strictly "smaller" than .

There are also:

  • 7 comment(s) on Section 59.82: Affine analog of proper base change

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