classification of quasi-coherent sheaves on an affine scheme

Question

I am having trouble with Q. Liu's proof of the classification of quasi-coherent sheaves on an affine scheme. Specifically, I don't understand the first step of his argument (proposition 1.6., section 5.1).

He starts by saying that, by definition of quasi-coherent sheaf one always has an affine open and a sequence $$\mathcal{O}^{J}_V\to\mathcal{O}^{I}_V\xrightarrow{\alpha}\mathcal{F}_{|V} \to 0$$

He does not prove this claim, but this should be easy: by definition of quasi-coherent sheaf one has a generic open $U$ and an exact sequence as above, and then one can restrict to an affine open $V \subset U$ and check the exactness of the sequence on the stalks.

question What I don't understand is what he does next: he claims that considering the $\mathcal{O}_V(V)$-module $\operatorname{Im}\alpha_V$ and $\tilde{\operatorname{Im}\alpha_V}$, one has an exact sequence $$\mathcal{O}^{J}_V\to\mathcal{O}^{I}_V\xrightarrow{}\tilde{\operatorname{Im}\alpha_V}\to 0$$

I do not see why he has that exact sequence.

First of all, by the right exactness one has $\operatorname{Im}\alpha_V \cong \mathcal{F}(V)$, so he might as well call it $\mathcal{F}(V)$.

Next, I think he uses, tacitly, the fact that $\mathcal{O}_V \cong \tilde{\mathcal{O}_V(V)}$, which can be proved by observing that $$\mathcal{O}_V(D(f)) \cong (\mathcal{O}_V)_f \cong \tilde{\mathcal{O}_V(V)}(D(f))$$ and hence they are isomorphic on stalks.

specific question

I don't see how does he get the sequence $$\mathcal{O}^{J}_V(V)\to\mathcal{O}^{I}_V(V)\xrightarrow{} \mathcal{F}(V)\to 0$$

because I know that exactness on the right of a sequence of sheaves does not in general imply that the sequence must be right exact on each open subset. So why can he infer this in that case?

Answer 1

First, note that the exponents here correspond to direct sums, not products. Let $\beta: O_V^{J} \rightarrow O_V^{I}$ map: then by the above exact sequence, $\alpha$ (or $\mathcal{F}_{|V}$) is the cokernel of $\beta$.

Let $M$ be the cokernel of $\beta(V): O_V^J(V) \rightarrow O_V^I(V)$: it’s an $O_V(V)$-module, and the sequence $$O_V^J(V) \overset{\beta(V)}{\rightarrow} O_V^I(V) \rightarrow M \rightarrow 0$$ is exact.

Apply the tilde functor: $O_V^J(V)$ becomes $O_V^J$, $O_V^I(V)$ becomes $O_V^I$, and $\beta(V)$ becomes $\beta$, so that we have an exact sequence

$$O_V^J \overset{\beta}{\rightarrow} O_V^I \rightarrow \tilde{M} \rightarrow 0,$$

and thus $\tilde{M} \cong \mathcal{F}_{|V}$.