A doubt on Proposition 5.1.12 of Liu's Algebraic geometry and arithmetic curves.

Question

Let $X$ be a scheme. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be an exact sequence of $\mathcal{O}_X$-modules. If two of them are quasi-coherent, then so is the third.

This is the point $(d)$ of Proposition 5.1.12 of Liu's Algebraic Geometry and Arithmetic curves. I need only to show the case in which $\mathcal{F}$ and $\mathcal{H}$ are quasi-coherent. By local nature of quasi-coherent sheaves, we may assume that $X$ is affine. We know by Proposition 1.8 of the same book that the sequence

$0 \to \mathcal{F}(X) \to \mathcal{G}(X) \to \mathcal{H}(X) \to 0$

is exact. Now it is said that we have a commutative diagram

$\require{AMScd}$ \begin{CD} 0 @>>> \mathcal{F}(X)' @>>> \mathcal{G}(X)' @>>> \mathcal{H}(X)' @>>>0\\ & @VVV @VVV @VVV \\ 0 @>>>\mathcal{F} @>>> \mathcal{G} @>>> \mathcal{F} @>>>0 \end{CD}

With $\mathcal{F}(X)',\mathcal{G}(X)',\mathcal{H}(X)'$ i indicate the sheaves of $\mathcal{O}_X$-modules induced by the modules $\mathcal{F}(X),\mathcal{G}(X),\mathcal{H}(X)$. The first and the last vertical arrow are the isomorphisms we have by hypothesis. I don't understand how is the vertical arrow in the middle of the diagram above defined and so i cannot understand why the diagram is commutative.

Answer 1

Let $X = \operatorname{Spec}A$ be an affine scheme. If $M$ is an $A$-module, there is an associated sheaf of $\mathcal O_X$-modules $\widetilde{M}$ defined in the usual way by defining $\widetilde{M}(U)$ to be a certain $\mathcal O_X(U)$-submodule of

$$\prod\limits_{\mathfrak p \in X} M_{\mathfrak p}$$ Let $\mathcal F$ be a sheaf of $\mathcal O_X$-modules, so that $\mathcal F(X)$ is an $\mathcal O_X(X) = A$-module. Then $\widetilde{\mathcal F(X)}$ is a sheaf of $\mathcal O_X$-modules as defined above.

There is an associated morphism of sheaves of $\mathcal O_X$-modules $\theta = \theta_{\mathcal F}:\widetilde{\mathcal F(X)} \rightarrow \mathcal F$. I think your question comes down to asking:

1 . How is $\theta$ defined?

2 . Why is $\theta$ natural? (this will imply the commutativity of the diagram) In other words, if $\alpha: \mathcal F \rightarrow \mathcal G$ is a morphism of sheaves of $\mathcal O_X$-modules, we should get another morphism $\widetilde{\alpha}: \widetilde{\mathcal F(X)} \rightarrow \widetilde{\mathcal G(X)}$ as in your diagram, and it should satisfy $\alpha \circ \theta_{\mathcal F} = \widetilde{\alpha} \circ \theta_{\mathcal G}$.