The Stacks project

59.46 Closed immersions and pushforward

Before stating and proving Proposition 59.46.4 in its correct generality we briefly state and prove it for closed immersions. Namely, some of the preceding arguments are quite a bit easier to follow in the case of a closed immersion and so we repeat them here in their simplified form.

In the rest of this section $i : Z \to X$ is a closed immersion. The functor

\[ \mathit{Sch}/X \longrightarrow \mathit{Sch}/Z, \quad U \longmapsto U_ Z = Z \times _ X U \]

will be denoted $U \mapsto U_ Z$ as indicated. Since being a closed immersion is preserved under arbitrary base change the scheme $U_ Z$ is a closed subscheme of $U$.

Lemma 59.46.1. Let $i : Z \to X$ be a closed immersion of schemes. Let $U, U'$ be schemes étale over $X$. Let $h : U_ Z \to U'_ Z$ be a morphism over $Z$. Then there exists a diagram

\[ \xymatrix{ U & W \ar[l]_ a \ar[r]^ b & U' } \]

such that $a_ Z : W_ Z \to U_ Z$ is an isomorphism and $h = b_ Z \circ (a_ Z)^{-1}$.

Proof. Consider the scheme $M = U \times _ Y U'$. The graph $\Gamma _ h \subset M_ Z$ of $h$ is open. This is true for example as $\Gamma _ h$ is the image of a section of the étale morphism $\text{pr}_{1, Z} : M_ Z \to U_ Z$, see Étale Morphisms, Proposition 41.6.1. Hence there exists an open subscheme $W \subset M$ whose intersection with the closed subset $M_ Z$ is $\Gamma _ h$. Set $a = \text{pr}_1|_ W$ and $b = \text{pr}_2|_ W$. $\square$

Lemma 59.46.2. Let $i : Z \to X$ be a closed immersion of schemes. Let $V \to Z$ be an étale morphism of schemes. There exist étale morphisms $U_ i \to X$ and morphisms $U_{i, Z} \to V$ such that $\{ U_{i, Z} \to V\} $ is a Zariski covering of $V$.

Proof. Since we only have to find a Zariski covering of $V$ consisting of schemes of the form $U_ Z$ with $U$ étale over $X$, we may Zariski localize on $X$ and $V$. Hence we may assume $X$ and $V$ affine. In the affine case this is Algebra, Lemma 10.143.10. $\square$

If $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ is a geometric point of $X$, then either $\overline{x}$ factors (uniquely) through the closed subscheme $Z$, or $Z_{\overline{x}} = \emptyset $. If $\overline{x}$ factors through $Z$ we say that $\overline{x}$ is a geometric point of $Z$ (because it is) and we use the notation “$\overline{x} \in Z$” to indicate this.

Lemma 59.46.3. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{G}$ be a sheaf of sets on $Z_{\acute{e}tale}$. Let $\overline{x}$ be a geometric point of $X$. Then

\[ (i_{small, *}\mathcal{G})_{\overline{x}} = \left\{ \begin{matrix} * & \text{if} & \overline{x} \not\in Z \\ \mathcal{G}_{\overline{x}} & \text{if} & \overline{x} \in Z \end{matrix} \right. \]

where $*$ denotes a singleton set.

Proof. Note that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} = *$ is the final object in the category of étale sheaves on $U$, i.e., the sheaf which associates a singleton set to each scheme étale over $U$. This explains the value of $(i_{small, *}\mathcal{G})_{\overline{x}}$ if $\overline{x} \not\in Z$.

Next, suppose that $\overline{x} \in Z$. Note that

\[ (i_{small, *}\mathcal{G})_{\overline{x}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{G}(U_ Z) \]

and on the other hand

\[ \mathcal{G}_{\overline{x}} = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v})} \mathcal{G}(V). \]

Let $\mathcal{C}_1 = \{ (U, \overline{u})\} ^{opp}$ be the opposite of the category of étale neighbourhoods of $\overline{x}$ in $X$, and let $\mathcal{C}_2 = \{ (V, \overline{v})\} ^{opp}$ be the opposite of the category of étale neighbourhoods of $\overline{x}$ in $Z$. The canonical map

\[ \mathcal{G}_{\overline{x}} \longrightarrow (i_{small, *}\mathcal{G})_{\overline{x}} \]

corresponds to the functor $F : \mathcal{C}_1 \to \mathcal{C}_2$, $F(U, \overline{u}) = (U_ Z, \overline{x})$. Now Lemmas 59.46.2 and 59.46.1 imply that $\mathcal{C}_1$ is cofinal in $\mathcal{C}_2$, see Categories, Definition 4.17.1. Hence it follows that the displayed arrow is an isomorphism, see Categories, Lemma 4.17.2. $\square$

Proposition 59.46.4. Let $i : Z \to X$ be a closed immersion of schemes.

  1. The functor

    \[ i_{small, *} : \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]

    is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and

  2. the functor

    \[ i_{small, *} : \textit{Ab}(Z_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]

    is fully faithful and its essential image is those abelian sheaves on $X_{\acute{e}tale}$ whose support is contained in $Z$.

In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.

Proof. Let's discuss the case of sheaves of sets. For any sheaf $\mathcal{G}$ on $Z$ the morphism $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism by Lemma 59.46.3 (and Theorem 59.29.10). This implies formally that $i_{small, *}$ is fully faithful, see Sites, Lemma 7.41.1. It is clear that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} \cong *$ where $U = X \setminus Z$. Conversely, suppose that $\mathcal{F}$ is a sheaf of sets on $X$ such that $\mathcal{F}|_{U_{\acute{e}tale}} \cong *$. Consider the adjunction mapping

\[ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F} \]

Combining Lemmas 59.46.3 and 59.36.2 we see that it is an isomorphism. This finishes the proof of (1). The proof of (2) is identical. $\square$


Comments (2)

Comment #3246 by Hadi Hedayatzadeh on

I believe there is a typo in Lemma 53.46.3.: the stalk of the pushforward at a point in Z should be (i.e., F should be changed to G).


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