Lemma 30.23.8. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. If $(\mathcal{F}_ n)$ is an object of $\textit{Coh}(X, \mathcal{I})$ then $\bigoplus \mathop{\mathrm{Ker}}(\mathcal{F}_{n + 1} \to \mathcal{F}_ n)$ is a finite type, graded, quasi-coherent $\bigoplus \mathcal{I}^ n/\mathcal{I}^{n + 1}$-module.

Proof. The question is local on $X$ hence we may assume $X$ is affine, i.e., we have a situation as in Lemma 30.23.1. In this case, if $(\mathcal{F}_ n)$ corresponds to the finite $A^\wedge$ module $M$, then $\bigoplus \mathop{\mathrm{Ker}}(\mathcal{F}_{n + 1} \to \mathcal{F}_ n)$ corresponds to $\bigoplus I^ nM/I^{n + 1}M$ which is clearly a finite module over $\bigoplus I^ n/I^{n + 1}$. $\square$

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