Lemma 30.20.1. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Set $B = \bigoplus _{n \geq 0} I^ n$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then for every $p \geq 0$ the graded $B$-module $\bigoplus _{n \geq 0} H^ p(X, I^ n\mathcal{F})$ is a finite $B$-module.

[III Cor 3.3.2, EGA]

**Proof.**
Let $\mathcal{B} = \bigoplus I^ n\mathcal{O}_ X = f^*\widetilde{B}$. Then $\bigoplus I^ n\mathcal{F}$ is a finite type graded $\mathcal{B}$-module. Hence the result follows from Lemma 30.19.3 part (1).
$\square$

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