The Stacks project

95.9 Restriction

A trivial but useful observation is that the localization of a category fibred in groupoids at an object is equivalent to the big site of the scheme it lies over.

Lemma 95.9.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over $U = p(x)$. The functor $p$ induces an equivalence of sites $\mathcal{X}_\tau /x \to (\mathit{Sch}/U)_\tau $.

Proof. Special case of Stacks, Lemma 8.10.4. $\square$

We use the lemma above to talk about the pullback and the restriction of a (pre)sheaf to a scheme.

Definition 95.9.2. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over $U = p(x)$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$.

  1. The pullback $x^{-1}\mathcal{F}$ of $\mathcal{F}$ is the restriction $\mathcal{F}|_{(\mathcal{X}/x)}$ viewed as a presheaf on $(\mathit{Sch}/U)_{fppf}$ via the equivalence $\mathcal{X}/x \to (\mathit{Sch}/U)_{fppf}$ of Lemma 95.9.1.

  2. The restriction of $\mathcal{F}$ to $U_{\acute{e}tale}$ is $x^{-1}\mathcal{F}|_{U_{\acute{e}tale}}$, abusively written $\mathcal{F}|_{U_{\acute{e}tale}}$.

This notation makes sense because to the object $x$ the $2$-Yoneda lemma, see Algebraic Stacks, Section 93.5 associates a $1$-morphism $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}/x$ which is quasi-inverse to $p : \mathcal{X}/x \to (\mathit{Sch}/U)_{fppf}$. Hence $x^{-1}\mathcal{F}$ truly is the pullback of $\mathcal{F}$ via this $1$-morphism. In particular, by the material above, if $\mathcal{F}$ is a sheaf (or a Zariski, étale, smooth, syntomic sheaf), then $x^{-1}\mathcal{F}$ is a sheaf on $(\mathit{Sch}/U)_{fppf}$ (or on $(\mathit{Sch}/U)_{Zar}$, $(\mathit{Sch}/U)_{\acute{e}tale}$, $(\mathit{Sch}/U)_{smooth}$, $(\mathit{Sch}/U)_{syntomic}$).

Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\varphi : x \to y$ be a morphism of $\mathcal{X}$ lying over the morphism of schemes $a : U \to V$. Recall that $a$ induces a morphism of small étale sites $a_{small} : U_{\acute{e}tale}\to V_{\acute{e}tale}$, see Étale Cohomology, Section 59.34. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. Let $\mathcal{F}|_{U_{\acute{e}tale}}$ and $\mathcal{F}|_{V_{\acute{e}tale}}$ be the restrictions of $\mathcal{F}$ via $x$ and $y$. There is a natural comparison map

95.9.2.1
\begin{equation} \label{stacks-sheaves-equation-comparison-push} c_\varphi : \mathcal{F}|_{V_{\acute{e}tale}} \longrightarrow a_{small, *}(\mathcal{F}|_{U_{\acute{e}tale}}) \end{equation}

of presheaves on $U_{\acute{e}tale}$. Namely, if $V' \to V$ is étale, set $U' = V' \times _ V U$ and define $c_\varphi $ on sections over $V'$ via

\[ \xymatrix{ a_{small, *}(\mathcal{F}|_{U_{\acute{e}tale}})(V') & \mathcal{F}|_{U_{\acute{e}tale}}(U') \ar@{=}[l] & \mathcal{F}(x') \ar@{=}[l] \\ \mathcal{F}|_{V_{\acute{e}tale}}(V') \ar@{=}[rr] \ar[u]^{c_\varphi } & & \mathcal{F}(y') \ar[u]_{\mathcal{F}(\varphi ')} } \]

Here $\varphi ' : x' \to y'$ is a morphism of $\mathcal{X}$ fitting into a commutative diagram

\[ \vcenter { \xymatrix{ x' \ar[r] \ar[d]_{\varphi '} & x \ar[d]^\varphi \\ y' \ar[r] & y } } \quad \text{lying over}\quad \vcenter { \xymatrix{ U' \ar[r] \ar[d] & U \ar[d]^ a \\ V' \ar[r] & V } } \]

The existence and uniqueness of $\varphi '$ follow from the axioms of a category fibred in groupoids. We omit the verification that $c_\varphi $ so defined is indeed a map of presheaves (i.e., compatible with restriction mappings) and that it is functorial in $\mathcal{F}$. In case $\mathcal{F}$ is a sheaf for the étale topology we obtain a comparison map

95.9.2.2
\begin{equation} \label{stacks-sheaves-equation-comparison} c_\varphi : a_{small}^{-1}(\mathcal{F}|_{V_{\acute{e}tale}}) \longrightarrow \mathcal{F}|_{U_{\acute{e}tale}} \end{equation}

which is also denoted $c_\varphi $ as indicated (this is the customary abuse of notation in not distinguishing between adjoint maps).

Lemma 95.9.3. Let $\mathcal{F}$ be an étale sheaf on $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$.

  1. If $\varphi : x \to y$ and $\psi : y \to z$ are morphisms of $\mathcal{X}$ lying over $a : U \to V$ and $b : V \to W$, then the composition

    \[ a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_{\acute{e}tale}})) \xrightarrow {a_{small}^{-1}c_\psi } a_{small}^{-1}(\mathcal{F}|_{V_{\acute{e}tale}}) \xrightarrow {c_\varphi } \mathcal{F}|_{U_{\acute{e}tale}} \]

    is equal to $c_{\psi \circ \varphi }$ via the identification

    \[ (b \circ a)_{small}^{-1}(\mathcal{F}|_{W_{\acute{e}tale}}) = a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_{\acute{e}tale}})). \]
  2. If $\varphi : x \to y$ lies over an étale morphism of schemes $a : U \to V$, then (95.9.2.2) is an isomorphism.

  3. Suppose $f : \mathcal{Y} \to \mathcal{X}$ is a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and $y$ is an object of $\mathcal{Y}$ lying over the scheme $U$ with image $x = f(y)$. Then there is a canonical identification $f^{-1}\mathcal{F}|_{U_{\acute{e}tale}} = \mathcal{F}|_{U_{\acute{e}tale}}$.

  4. Moreover, given $\psi : y' \to y$ in $\mathcal{Y}$ lying over $a : U' \to U$ the comparison map $c_\psi : a_{small}^{-1}(f^{-1}\mathcal{F}|_{U_{\acute{e}tale}}) \to f^{-1}\mathcal{F}|_{U'_{\acute{e}tale}}$ is equal to the comparison map $c_{f(\psi )} : a_{small}^{-1}\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{F}|_{U'_{\acute{e}tale}}$ via the identifications in (3).

Proof. The verification of these properties is omitted. $\square$

Next, we turn to the restriction of (pre)sheaves of modules.

Lemma 95.9.4. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over $U = p(x)$. The equivalence of Lemma 95.9.1 extends to an equivalence of ringed sites $(\mathcal{X}_\tau /x, \mathcal{O}_\mathcal {X}|_ x) \to ((\mathit{Sch}/U)_\tau , \mathcal{O})$.

Proof. This is immediate from the construction of the structure sheaves. $\square$

Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{F}$ be a (pre)sheaf of modules on $\mathcal{X}$ as in Definition 95.7.1. Let $x$ be an object of $\mathcal{X}$ lying over $U$. Then Lemma 95.9.4 guarantees that the restriction $x^{-1}\mathcal{F}$ is a (pre)sheaf of modules on $(\mathit{Sch}/U)_{fppf}$. We will sometimes write $x^*\mathcal{F} = x^{-1}\mathcal{F}$ in this case. Similarly, if $\mathcal{F}$ is a sheaf for the Zariski, étale, smooth, or syntomic topology, then $x^{-1}\mathcal{F}$ is as well. Moreover, the restriction $\mathcal{F}|_{U_{\acute{e}tale}} = x^{-1}\mathcal{F}|_{U_{\acute{e}tale}}$ to $U$ is a presheaf of $\mathcal{O}_{U_{\acute{e}tale}}$-modules. If $\mathcal{F}$ is a sheaf for the étale topology, then $\mathcal{F}|_{U_{\acute{e}tale}}$ is a sheaf of modules. Moreover, if $\varphi : x \to y$ is a morphism of $\mathcal{X}$ lying over $a : U \to V$ then the comparison map (95.9.2.2) is compatible with $a_{small}^\sharp $ (see Descent, Remark 35.8.4) and induces a comparison map

95.9.4.1
\begin{equation} \label{stacks-sheaves-equation-comparison-modules} c_\varphi : a_{small}^*(\mathcal{F}|_{V_{\acute{e}tale}}) \longrightarrow \mathcal{F}|_{U_{\acute{e}tale}} \end{equation}

of $\mathcal{O}_{U_{\acute{e}tale}}$-modules. Note that the properties (1), (2), (3), and (4) of Lemma 95.9.3 hold in the setting of étale sheaves of modules as well. We will use this in the following without further mention.

Lemma 95.9.5. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. The site $\mathcal{X}_\tau $ has enough points.

Proof. By Sites, Lemma 7.38.5 we have to show that there exists a family of objects $x$ of $\mathcal{X}$ such that $\mathcal{X}_\tau /x$ has enough points and such that the sheaves $h_ x^\# $ cover the final object of the category of sheaves. By Lemma 95.9.1 and Étale Cohomology, Lemma 59.30.1 we see that $\mathcal{X}_\tau /x$ has enough points for every object $x$ and we win. $\square$


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