The Stacks project

62.2 Growing sections

In this section we discuss results of the following type.

Lemma 62.2.1. Let $X$ be a scheme. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $\varphi : U' \to U$ be a morphism of $X_{\acute{e}tale}$. Let $Z' \subset U'$ be a closed subscheme such that $Z' \to U' \to U$ is a closed immersion with image $Z \subset U$. Then there is a canonical bijection

\[ \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = \{ s' \in \mathcal{F}(U') \mid \text{Supp}(s') \subset Z'\} \]

which is given by restriction if $\varphi ^{-1}(Z) = Z'$.

Proof. Consider the closed subscheme $Z'' = \varphi ^{-1}(Z)$ of $U'$. Then $Z' \subset Z''$ is closed because $Z'$ is closed in $U'$. On the other hand, $Z' \to Z''$ is an étale morphism (as a morphism between schemes étale over $Z$) and hence open. Thus $Z'' = Z' \amalg T$ for some closed subset $T$. The open covering $U' = (U' \setminus T) \cup (U' \setminus Z')$ shows that

\[ \{ s' \in \mathcal{F}(U') \mid \text{Supp}(s') \subset Z'\} = \{ s' \in \mathcal{F}(U' \setminus T) \mid \text{Supp}(s') \subset Z'\} \]

and the étale covering $\{ U' \setminus T \to U, U \setminus Z \to U\} $ shows that

\[ \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = \{ s' \in \mathcal{F}(U' \setminus T) \mid \text{Supp}(s') \subset Z'\} \]

This finishes the proof. $\square$

Lemma 62.2.2. Let $X$ be a scheme. Let $Z \subset X$ be a locally closed subscheme. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Given $U, U' \subset X$ open containing $Z$ as a closed subscheme, there is a canonical bijection

\[ \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = \{ s \in \mathcal{F}(U') \mid \text{Supp}(s) \subset Z\} \]

which is given by restriction if $U' \subset U$.

Proof. Since $Z$ is a closed subscheme of $U \cap U'$, it suffices to prove the lemma when $U' \subset U$. Then it is a special case of Lemma 62.2.1. $\square$

Let us introduce a bit of nonstandard notation which will stand us in good stead later. Namely, in the situation of Lemma 62.2.2 above, let us denote

\[ H_ Z(\mathcal{F}) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} \]

where $U \subset X$ is any choice of open subscheme containing $Z$ as a closed subscheme. The reader who is troubled by the lack of precision this entails may choose $U = X \setminus \partial Z$ where $\partial Z = \overline{Z}\setminus Z$ is the “boundary” of $Z$ in $X$. However, in many of the arguments below the flexibility of choosing different opens will play a role. Here are some properties of this construction:

  1. If $Z \subset Z'$ are locally closed subschemes of $X$ and $Z$ is closed in $Z'$, then there is a natural injective map

    \[ H_ Z(\mathcal{F}) \to H_{Z'}(\mathcal{F}). \]
  2. If $f : Y \to X$ is a morphism of schemes and $Z \subset X$ is a locally closed subscheme, then there is a natural pullback map $f^* : H_ Z(\mathcal{F}) \to H_{f^{-1}Z}(f^{-1}\mathcal{F})$.

It will be convenient to extend our notation to the following situation: suppose that we have $W \in X_{\acute{e}tale}$ and a locally closed subscheme $Z \subset W$. Then we will denote

\[ H_ Z(\mathcal{F}) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset Z\} = H_ Z(\mathcal{F}|_{W_{\acute{e}tale}}) \]

where $U \subset W$ is any choice of open subscheme containing $Z$ as a closed subscheme, exactly as above1.

[1] In fact, Lemma 62.2.1 shows, given $Z$ over $X$ which is isomorphic to a locally closed subscheme of some object $W$ of $X_{\acute{e}tale}$, that the choice of $W$ is irrelevant.

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