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Hartshorne, Algebraic Geometry, Theorem III.5.2, reads (in part)

Theorem 5.2

Let $X$ be a projective scheme over a Noetherian ring, and let $\mathcal{O}_X(1)$ be a very ample invertible sheaf on $X$ over $\operatorname{Spec} A$. Let $\mathscr{F}$ be a coherent sheaf on $X$. Then:

(1) for each $i \geq 0$, $H^u(X, \mathscr{F})$ is a finitely-generated $A$-module;

[...]

The proof proceeds by reducing to the case $X = \mathbb{P}^r_A$ and then checking things for $\mathcal{O}_X(q)$, which is fine. Then we need to establish things for arbitrary coherent sheaves. The proof here reads:

[...]

In general, given a coherent sheaf $\mathscr{F}$ on $X$, we can write $\mathscr{F}$ as a quotient of a sheaf $\mathscr{E}$, which is a finite direct sum of sheaves $\mathcal{O}(q_i)$ for various integers $q_i$. Let $\mathscr{R}$ be the kernel, $$0 \to \mathscr{R} \to \mathscr{E} \to \mathscr{F} \to 0,$$ Then $\mathscr{R}$ is also coherent. We get an exact sequence of $A$-modules $$\cdots \to H^i(X, \mathscr{E}) \to H^i(X, \mathscr{F}) \to H^{i+1}(X, \mathscr{R}) \to \cdots$$ Now the module on the left is finitely generated because $\mathscr{E}$ is a sum of $\mathcal{O}(q_i)$, as remarked above. The module on the right is finitely generated by the induction hypothesis.

[...]

This last sentence I don't understand. It doesn't explicitly state what the inductive hypothesis is, but I assume that it's that $H^j(X, \mathscr{F})$ is a finitely-generated $A$-module for $j > i$. In this case, what we know is that we have an exact sequence $$\cdots \to H^i(X, \mathscr{F}) \to H^{i+1}(X, \mathscr{R}) \to H^{i+1}(X, \mathscr{E}) \to H^{i+1}(X, \mathscr{F}) \to \cdots$$ where the two rightmost terms are finitely generated as $A$-modules. Since we don't seem to know anything about $H^i(X, \mathscr{F})$ on the left side yet, I don't see how this helps us establish anything about $H^{i+1}(X, \mathscr{R})$.

What have I missed?

Daniel McLaury
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    Your induction hypothesis shouldn't be on a specific $\mathcal{F}$, but all coherent sheaves at the same time. If you assume that $H^j$ is finitely generated for all $j > i$ for all coherent sheaves (in particular $\mathcal{F}$ and $\mathcal{R}$), then you can say something about $H^i(X,\mathcal{F})$, assuming that you already know the cohomology of $H^i(X,\mathcal{E})$, which amounts to cohomology of $O(n)$ on projective spaces. –  Nov 25 '13 at 05:47
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    Your first sentence alone completely answers my question. Feel free to post it as an answer and I'll accept it. Thanks! – Daniel McLaury Nov 25 '13 at 05:50

1 Answers1

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Your induction hypothesis shouldn't be on a specific $\mathcal{F}$, but all coherent sheaves at the same time. If you assume that $H^j$ is finitely generated for all $j > i$ for all coherent sheaves (in particular $\mathcal{F}$ and $\mathcal{R}$), then you can say something about $H^i(X,\mathcal{F})$, assuming that you already know the cohomology of $H^i(X,\mathcal{E})$, which amounts to cohomology of $O(n)$ on projective spaces.