The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

18.35 Stalks of modules

We have to be a bit careful when taking stalks at points, since the colimit defining a stalk (see Sites, Equation 7.32.1.1) may not be filtered1. On the other hand, by definition of a point of a site the stalk functor is exact and commutes with arbitrary colimits. In other words, it behaves exactly as if the colimit were filtered.

Lemma 18.35.1. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$.

  1. We have $(\mathcal{F}^\# )_ p = \mathcal{F}_ p$ for any presheaf of sets on $\mathcal{C}$.

  2. The stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact (see Categories, Definition 4.23.1) and commutes with arbitrary colimits.

  3. The stalk functor $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact (see Categories, Definition 4.23.1) and commutes with arbitrary colimits.

Proof. By Sites, Lemma 7.32.5 we have (1). By Sites, Lemmas 7.32.4 we see that $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is a left adjoint, and by Sites, Lemma 7.32.5 we see the same thing for $\textit{Sh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$. Hence the stalk functor commutes with arbitrary colimits (see Categories, Lemma 4.24.5). It follows from the definition of a point of a site, see Sites, Definition 7.32.2 that $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact. Since sheafification is exact (Sites, Lemma 7.10.14) it follows that $\textit{PSh}(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact. $\square$

In particular, since the stalk functor $\mathcal{F} \mapsto \mathcal{F}_ p$ on presheaves commutes with all finite limits and colimits we may apply the reasoning of the proof of Sites, Proposition 7.44.3. The result of such an argument is that if $\mathcal{F}$ is a (pre)sheaf of algebraic structures listed in Sites, Proposition 7.44.3 then the stalk $\mathcal{F}_ p$ is naturally an algebraic structure of the same kind. Let us explain this in detail when $\mathcal{F}$ is an abelian presheaf. In this case the addition map $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ induces a map

\[ + : \mathcal{F}_ p \times \mathcal{F}_ p = (\mathcal{F} \times \mathcal{F})_ p \longrightarrow \mathcal{F}_ p \]

where the equal sign uses that stalk functor on presheaves of sets commutes with finite limits. This defines a group structure on the stalk $\mathcal{F}_ p$. In this way we obtain our stalk functor

\[ \textit{PAb}(\mathcal{C}) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \mathcal{F}_ p \]

By construction the underlying set of $\mathcal{F}_ p$ is the stalk of the underlying presheaf of sets. This also defines our stalk functor for sheaves of abelian groups by precomposing with the inclusion $\textit{Ab}(\mathcal{C}) \subset \textit{PAb}(\mathcal{C})$.

Lemma 18.35.2. Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$.

  1. The functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

  2. The stalk functor $\textit{PAb}(\mathcal{C}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

  3. For $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PAb}(\mathcal{C}))$ we have $\mathcal{F}_ p = \mathcal{F}^\# _ p$.

Proof. This is formal from the results of Lemma 18.35.1 and the construction of the stalk functor above. $\square$

Next, we turn to the case of sheaves of modules. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. (It suffices for the discussion that $\mathcal{O}$ be a presheaf of rings.) Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $p$ be a point of $\mathcal{C}$. In this case we get a map

\[ \cdot : \mathcal{O}_ p \times \mathcal{O}_ p = (\mathcal{O} \times \mathcal{O})_ p \longrightarrow \mathcal{O}_ p \]

which is the stalk of the multiplication map and

\[ \cdot : \mathcal{O}_ p \times \mathcal{F}_ p = (\mathcal{O} \times \mathcal{F})_ p \longrightarrow \mathcal{F}_ p \]

which is the stalk of the multiplication map. We omit the verification that this defines a ring structure on $\mathcal{O}_ p$ and an $\mathcal{O}_ p$-module structure on $\mathcal{F}_ p$. In this way we obtain a functor

\[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}_ p), \quad \mathcal{F} \longmapsto \mathcal{F}_ p \]

By construction the underlying set of $\mathcal{F}_ p$ is the stalk of the underlying presheaf of sets. This also defines our stalk functor for sheaves of $\mathcal{O}$-modules by precomposing with the inclusion $\textit{Mod}(\mathcal{O}) \subset \textit{PMod}(\mathcal{O})$.

Lemma 18.35.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $p$ be a point of $\mathcal{C}$.

  1. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p)$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

  2. The stalk functor $\textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p)$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is exact.

  3. For $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PMod}(\mathcal{O}))$ we have $\mathcal{F}_ p = \mathcal{F}^\# _ p$.

Proof. This is formal from the results of Lemma 18.35.2, the construction of the stalk functor above, and Lemma 18.14.1. $\square$

Lemma 18.35.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites. Let $p$ be a point of $\mathcal{C}$ or $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and set $q = f \circ p$. Then

\[ (f^*\mathcal{F})_ p = \mathcal{F}_ q \otimes _{\mathcal{O}_{\mathcal{D}, q}} \mathcal{O}_{\mathcal{C}, p} \]

for any $\mathcal{O}_\mathcal {D}$-module $\mathcal{F}$.

Proof. We have

\[ f^*\mathcal{F} = f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} \mathcal{O}_\mathcal {C} \]

by definition. Since taking stalks at $p$ (i.e., applying $p^{-1}$) commutes with $\otimes $ by Lemma 18.26.1 we win by the relation between the stalk of pullbacks at $p$ and stalks at $q$ explained in Sites, Lemma 7.34.1 or Sites, Lemma 7.34.2. $\square$

[1] Of course in almost any naturally occurring case the colimit is filtered and some of the discussion in this section may be simplified.

Comments (2)

Comment #3408 by Mike Paluch on

I believe the proof of Lemma 04EM ought to read by Sites, Lemma 00Y8 we see the same thing for \textit{Sh}(\mathcal{C}) \to \textit{Sets} rather than \textit{PSh}(\mathcal{C}) \to \textit{Sets}.


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