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The Stacks project

[III Cor 3.3.2, EGA]

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.

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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