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Let $(X,\mathscr O)$ be a locally ringed space, and let $\mathscr F$, $\mathscr G$ be $\mathscr O$-modules. Consider the following facts: $\newcommand{\sF}{\mathscr F}\newcommand{\sG}{\mathscr G}\newcommand{\sO}{\mathscr O}\newcommand{\sHom}{\mathscr Hom}$

Proposition A If $\sF$ is finitely presented and $\sG$ is coherent, then $\sHom_{\sO}(\sF,\sG)$ is a coherent $\sO$-module.

Proposition B If $\sF$ is finitely presented, then $\sHom_\sO(\sF,\sG)_x \to \operatorname{Hom}_{\sO_x}(\sF_x,\sG_x)$ is an isomorphism for all $x\in X$.

I have a minor confusion about the proof of Proposition A. The proof is standard and goes basically as follows:

The question is local, so we may assume that we have an exact sequence $\mathscr O^p \to \mathscr O^q \to \mathscr F \to 0$. Applying $\sHom_\sO(-,\sG)$ yields an exact sequence

\begin{equation}\label{confused}\tag{$\star$} 0 \to \sHom_\sO(\sF,\sG) \to \sHom_\sO(\sO^q, \sG) \to \sHom_\sO(\sO^p, \sG)\end{equation}

which may be identified with $0 \to \sHom_\sO(\sF,\sG) \to \sG^q \to \sG^p$. Since the category of coherent sheaves is closed under direct sums and kernels, $\sHom_\sO(\sF,\sG)$ must be coherent.

My confusion is that both FAC (Paragraph 14, Proposition 6) and EGA I (Chapter 0, Corollary 5.3.7) cite Proposition B in justifying the exactness of $\eqref{confused}$, and I don't see why this is necessary or relevant.

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  • $\begingroup$ How do you show your sequence $(\star)$ is exact without using Proposition B? $\endgroup$ – Takumi Murayama Jun 8 '16 at 1:06
  • $\begingroup$ @Takumi If $\mathscr F' \to \mathscr F \to \mathscr F'' \to 0$ is exact, then so is $0 \to \mathscr Hom_{\mathscr O}(\mathscr F'', \mathscr G) \to \mathscr Hom_{\mathscr O}(\mathscr F, \mathscr G) \to \mathscr Hom_{\mathscr O}(\mathscr F', \mathscr G)$ [EGA I, Chapter 0, 4.1.5] $\endgroup$ – Yuri Sulyma Jun 8 '16 at 1:24
  • $\begingroup$ I don't doubt that there are other ways of showing this. Certainly it's relevant, though? $\endgroup$ – Hoot Jun 8 '16 at 2:02
  • $\begingroup$ @YuriSulyma I agree with you that it is true that it is left-exact; I am just wondering what your proof of it is. The easiest proof is to use Proposition B and argue by taking stalks, which is why I think they quoted it. $\endgroup$ – Takumi Murayama Jun 8 '16 at 2:04
  • $\begingroup$ @TakumiMurayama I was quoting the general fact that $\mathscr Hom$ is left-exact (in its first variable), without any hypotheses needed on the sheaves involved $\endgroup$ – Yuri Sulyma Jun 8 '16 at 2:17

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