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.
If $M$ is of finite presentation, then
\[ \underline{\mathop{\mathrm{Hom}}\nolimits _\Lambda (M, N)} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\underline{M}, \underline{N}) \]
If $M$ and $N$ are both of finite presentation, then
\[ \underline{\text{Isom}_\Lambda (M, N)} = \mathit{Isom}_{\underline{\Lambda }}(\underline{M}, \underline{N}) \]
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.
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 $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\underline{\Lambda }}(\mathcal{G}, \mathcal{F})$.
$\square$
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