Why a sheaf is coherent

Question

I am reading Ravi Vakil's Foundations of Algebraic Geometry, in the paragraph before theorem 18.1.4, Vakil mentions this:

For any coherent sheaf $\mathscr{F}$ on $\mathbb{P}_A^n$, we can find a surjection $\mathscr{O}(m)^{\oplus j}\to\mathscr{F}$, which yields the exact sequence $$0\to \mathscr{G} \to \mathscr{O}(m)^{\oplus j}\to\mathscr{F} \to 0$$ for some coherent sheaf $\mathscr{G}$.

Here, $A$ is a ring, and $\mathscr{O}(m)$ is the degree $m$ invertible sheaf on $\mathbb{P}^n_A$

And in the proof of theorem 18.1.4(ii), which says that $H^i(X,\mathscr{F}(m))=0$ for a coherent sheaf $\mathscr{F}$on the projective $A$-scheme $X$ and for all $i>0$, Vakil has used this result.

And my question is, why should the sheaf $\mathscr{G}$ be coherent? From the coherence of $\mathscr{F}$, I can only see that $\mathscr{G}$ is finitely generated.

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As Takumi Murayama mentioned, I need the coherence of $\mathscr{O}(m)$, which may not be true even $A$ is coherent itself. Actually I need this "result" to prove the following, which is the exercise 18.1.A in Vakil's note:

Assuming $A$ is a coherent ring, and $\mathscr{F}$ is a coherent sheaf on the projective $A$-scheme $\mathbb{P}^n_A$, show that $H^i(\mathbb{P}^n_A,\mathscr{F})$ is a coherent $A$-module.

My way to do this exercise: using the exact sequence of quasicoherent sheaf $$0\to \mathscr{G} \to \mathscr{O}(m)^{\oplus j}\to\mathscr{F} \to 0$$ we have an exact sequence of $A$-modules: $$H^n(\mathbb{P}^n_A,\mathscr{G})\to H^n(\mathbb{P}^n_A,\mathscr{O}(m)^{\oplus j}) \to H^n(\mathbb{P}^n_A,\mathscr{F})\to 0$$ Since $H^n(\mathbb{P}^n_A,\mathscr{O}(m)^{\oplus j})$ is finitely generated, so is $H^n(\mathbb{P}^n_A,\mathscr{F})$. And since $\mathscr{G}$ is finitely generated, we can used a similar argument to show that $H^n(\mathbb{P}^n_A,\mathscr{G})$ is finitely generated. Therefore $H^n(\mathbb{P}^n_A,\mathscr{F})$ is finitely presented, which is the same as coherent since $A$ is coherent over itself. Now, we just use the coherence of $\mathscr{F}$ to show coherence of $H^n(\mathbb{P}^n_A,\mathscr{F})$, if we know the coherence of $\mathscr{G}$, we can show the coherence of $H^n(\mathbb{P}^n_A,\mathscr{F})$. If we repeat the arguments, we can show that $H^{n-1}(\mathbb{P}^n_A,\mathscr{F})$ and $H^{n-1}(\mathbb{P}^n_A,\mathscr{G})$ are coherent, and similar the other cohomology groups.

In my proof, I assume that $\mathscr{G}$ is coherent, so is there any way to show the coherence of $H^i(\mathbb{P}^n_A,\mathscr{F})$ with the "result" I mentioned above?

Any help or hints are apprecaited, thank you so much.

Answer 1

It's a general fact that the kernel of a morphism of coherent $\mathcal{O}_X$-modules is in fact coherent. In Vakil's notes, this is Exercise 13.8.G. Thus, it suffices to note that $\mathcal{O}_{\mathbf{P}^n_A}(m)$ is coherent for any $m$. In Theorem 18.1.4, you assume that $A$ is noetherian, so it suffices to note that $\mathcal{O}_{\mathbf{P}^n_A}(m)$ is locally free, hence locally isomorphic to the structure sheaf, which is coherent by noetherianity.

On the other hand, if $A$ is not noetherian, funny things can occur: even if $A$ is coherent, $A[x]$ may not be coherent [Soublin, Prop. 18]. In this case, $\mathcal{O}_{\mathbf{P}^1_A}(m)$ would not be coherent, so you are right to be worried in the non-noetherian setting!