Lemma 30.24.2. Let $A$ be Noetherian ring and $I \subset A$ and ideal. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism and let $\mathcal{L}$ be an $f$-ample invertible sheaf. Let $\mathcal{I} = I\mathcal{O}_ X$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. Then there exists an integer $d_0$ such that

$H^1(X, \mathop{\mathrm{Ker}}(\mathcal{F}_{n + 1} \to \mathcal{F}_ n) \otimes \mathcal{L}^{\otimes d} ) = 0$

for all $n \geq 0$ and all $d \geq d_0$.

Proof. Set $B = \bigoplus I^ n/I^{n + 1}$ and $\mathcal{B} = \bigoplus \mathcal{I}^ n/\mathcal{I}^{n + 1} = f^*\widetilde{B}$. By Lemma 30.23.8 the graded quasi-coherent $\mathcal{B}$-module $\mathcal{G} = \bigoplus \mathop{\mathrm{Ker}}(\mathcal{F}_{n + 1} \to \mathcal{F}_ n)$ is of finite type. Hence the lemma follows from Lemma 30.19.3 part (2). $\square$

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