Lemma 48.30.8 (0G4R)—The Stacks project

Lemma 48.30.8. Let $X$ be a Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}^\bullet $ be a complex of coherent $\mathcal{O}_ X$-modules. Let $p \in \mathbf{Z}$. Set $\mathcal{H} = H^ p(\mathcal{F}^\bullet )$ and $\mathcal{H}_ n = H^ p(\mathcal{I}^ n\mathcal{F}^\bullet )$. Then there are canonical $\mathcal{O}_ X$-module maps

\[ \ldots \to \mathcal{H}_3 \to \mathcal{H}_2 \to \mathcal{H}_1 \to \mathcal{H} \]

There exists a $c > 0$ such that for $n \geq c$ the image of $\mathcal{H}_ n \to \mathcal{H}$ is contained in $\mathcal{I}^{n - c}\mathcal{H}$ and there is a canonical $\mathcal{O}_ X$-module map $\mathcal{I}^ n\mathcal{H} \to \mathcal{H}_{n - c}$ such that the compositions

\[ \mathcal{I}^ n \mathcal{H} \to \mathcal{H}_{n - c} \to \mathcal{I}^{n - 2c}\mathcal{H} \quad \text{and}\quad \mathcal{H}_ n \to \mathcal{I}^{n - c}\mathcal{H} \to \mathcal{H}_{n - 2c} \]

are the canonical ones. In particular, the inverse systems $(\mathcal{H}_ n)$ and $(\mathcal{I}^ n\mathcal{H})$ are isomorphic as pro-objects of $\textit{Mod}(\mathcal{O}_ X)$.

Proof. If $X$ is affine, translated into algebra this is More on Algebra, Lemma 15.101.1. In the general case, argue exactly as in the proof of that lemma replacing the reference to Artin-Rees in algebra with a reference to Cohomology of Schemes, Lemma 30.10.3. Details omitted. $\square$

The Stacks project

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