18.36 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 18.36.1. Let \mathcal{C} be a site. Let p be a point of \mathcal{C}.
We have (\mathcal{F}^\# )_ p = \mathcal{F}_ p for any presheaf of sets on \mathcal{C}.
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.
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 \mathop{\mathit{Sh}}\nolimits (\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 18.36.2. Let \mathcal{C} be a site. Let p be a point of \mathcal{C}.
The functor \textit{Ab}(\mathcal{C}) \to \textit{Ab}, \mathcal{F} \mapsto \mathcal{F}_ p is exact.
The stalk functor \textit{PAb}(\mathcal{C}) \to \textit{Ab}, \mathcal{F} \mapsto \mathcal{F}_ p is exact.
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 18.36.1 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 18.36.3. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let p be a point of \mathcal{C}.
The functor \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p), \mathcal{F} \mapsto \mathcal{F}_ p is exact.
The stalk functor \textit{PMod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ p), \mathcal{F} \mapsto \mathcal{F}_ p is exact.
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 18.36.2, the construction of the stalk functor above, and Lemma 18.14.1.
\square
Lemma 18.36.4. 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 18.26.2 we win by the relation between the stalk of pullbacks at p and stalks at q explained in Sites, Lemma 7.34.2 or Sites, Lemma 7.34.3.
\square
Comments (2)
Comment #3408 by Mike Paluch on
Comment #3467 by Johan on