18.43 Locally constant sheaves

Here is the general definition.

Definition 18.43.1. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc.

1. We say $\mathcal{F}$ is a constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc if it is isomorphic as a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc to a constant sheaf $\underline{E}$ as in Section 18.42.

2. We say $\mathcal{F}$ is locally constant if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\}$ such that $\mathcal{F}|_{U_ i}$ is a constant sheaf.

3. If $\mathcal{F}$ is a sheaf of sets or groups, then we say $\mathcal{F}$ is finite locally constant if the constant values are finite sets or finite groups.

Lemma 18.43.2. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi. If $\mathcal{G}$ is a locally constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring $\Lambda$, etc on $\mathcal{D}$, the same is true for $f^{-1}\mathcal{G}$ on $\mathcal{C}$.

Proof. Omitted. $\square$

Lemma 18.43.3. Let $\mathcal{C}$ be a site with a final object $X$.

1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of sets on $\mathcal{C}$. If $\mathcal{F}$ is finite locally constant, there exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant sheaves associated to a map of sets.

2. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of abelian groups on $\mathcal{C}$. If $\mathcal{F}$ is finite locally constant, there exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant abelian sheaves associated to a map of abelian groups.

3. Let $\Lambda$ be a ring. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of $\Lambda$-modules on $\mathcal{C}$. If $\mathcal{F}$ is of finite type, then there exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant sheaves of $\Lambda$-modules associated to a map of $\Lambda$-modules.

Proof. Proof omitted. $\square$

Lemma 18.43.4. Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring. Let $M$, $N$ be $\Lambda$-modules. Let $\mathcal{F}, \mathcal{G}$ be a locally constant sheaves of $\Lambda$-modules.

1. If $M$ is of finite presentation, then

$\underline{\mathop{\mathrm{Hom}}\nolimits _\Lambda (M, N)} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\underline{M}, \underline{N})$
2. If $M$ and $N$ are both of finite presentation, then

$\underline{\text{Isom}_\Lambda (M, N)} = \mathit{Isom}_{\underline{\Lambda }}(\underline{M}, \underline{N})$
3. If $\mathcal{F}$ is of finite presentation, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\mathcal{F}, \mathcal{G})$ is a locally constant sheaf of $\Lambda$-modules.

4. If $\mathcal{F}$ and $\mathcal{G}$ are both of finite presentation, then $\mathit{Isom}_{\underline{\Lambda }}(\mathcal{F}, \mathcal{G})$ is a locally constant sheaf of sets.

Proof. Proof of (1). Set $E = \mathop{\mathrm{Hom}}\nolimits _\Lambda (M, N)$. We want to show the canonical map

$\underline{E} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\underline{M}, \underline{N})$

is an isomorphism. The module $M$ has a presentation $\Lambda ^{\oplus s} \to \Lambda ^{\oplus t} \to M \to 0$. Then $E$ sits in an exact sequence

$0 \to E \to \mathop{\mathrm{Hom}}\nolimits _\Lambda (\Lambda ^{\oplus t}, N) \to \mathop{\mathrm{Hom}}\nolimits _\Lambda (\Lambda ^{\oplus s}, N)$

and we have similarly

$0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\underline{M}, \underline{N}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\underline{\Lambda ^{\oplus t}}, \underline{N}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\underline{\Lambda ^{\oplus s}}, \underline{N})$

This reduces the question to the case where $M$ is a finite free module where the result is clear.

Proof of (3). The question is local on $\mathcal{C}$, hence we may assume $\mathcal{F} = \underline{M}$ and $\mathcal{G} = \underline{N}$ for some $\Lambda$-modules $M$ and $N$. By Lemma 18.42.5 the module $M$ is of finite presentation. Thus the result follows from (1).

Parts (2) and (4) follow from parts (1) and (3) and the fact that $\mathit{Isom}$ can be viewed as the subsheaf of sections of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\mathcal{F}, \mathcal{G})$ which have an inverse in $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\mathcal{G}, \mathcal{F})$. $\square$

Lemma 18.43.5. Let $\mathcal{C}$ be a site.

1. The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

2. The category of finite locally constant abelian sheaves is a weak Serre subcategory of $\textit{Ab}(\mathcal{C})$.

3. Let $\Lambda$ be a Noetherian ring. The category of finite type, locally constant sheaves of $\Lambda$-modules on $\mathcal{C}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{C}, \Lambda )$.

Proof. Proof of (1). We may work locally on $\mathcal{C}$. Hence by Lemma 18.43.3 we may assume we are given a finite diagram of finite sets such that our diagram of sheaves is the associated diagram of constant sheaves. Then we just take the limit or colimit in the category of sets and take the associated constant sheaf. Some details omitted.

To prove (2) and (3) we use the criterion of Homology, Lemma 12.10.3. Existence of kernels and cokernels is argued in the same way as above. Of course, the reason for using a Noetherian ring in (3) is to assure us that the kernel of a map of finite $\Lambda$-modules is a finite $\Lambda$-module. To see that the category is closed under extensions (in the case of sheaves $\Lambda$-modules), assume given an extension of sheaves of $\Lambda$-modules

$0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0$

on $\mathcal{C}$ with $\mathcal{F}$, $\mathcal{G}$ finite type and locally constant. Localizing on $\mathcal{C}$ we may assume $\mathcal{F}$ and $\mathcal{G}$ are constant, i.e., we get

$0 \to \underline{M} \to \mathcal{E} \to \underline{N} \to 0$

for some $\Lambda$-modules $M, N$. Choose generators $y_1, \ldots , y_ m$ of $N$, so that we get a short exact sequence $0 \to K \to \Lambda ^{\oplus m} \to N \to 0$ of $\Lambda$-modules. Localizing further we may assume $y_ j$ lifts to a section $s_ j$ of $\mathcal{E}$. Thus we see that $\mathcal{E}$ is a pushout as in the following diagram

$\xymatrix{ 0 \ar[r] & \underline{K} \ar[d] \ar[r] & \underline{\Lambda ^{\oplus m}} \ar[d] \ar[r] & \underline{N} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \underline{M} \ar[r] & \mathcal{E} \ar[r] & \underline{N} \ar[r] & 0 }$

By Lemma 18.43.3 again (and the fact that $K$ is a finite $\Lambda$-module as $\Lambda$ is Noetherian) we see that the map $\underline{K} \to \underline{M}$ is locally constant, hence we conclude. $\square$

Lemma 18.43.6. Let $\mathcal{C}$ be a site. Let $\Lambda$ be a ring. The tensor product of two locally constant sheaves of $\Lambda$-modules on $\mathcal{C}$ is a locally constant sheaf of $\Lambda$-modules.

Proof. Omitted. $\square$

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