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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