The Stacks project

Lemma 52.21.3. In Situation 52.16.1 assume that $A$ has a dualizing complex. Let $d \geq \text{cd}(A, I)$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume $(\mathcal{F}_ n)$ satisfies the $(2, 2 + d)$-inequalities, see Definition 52.19.1. Then there exists a canonical map $(\mathcal{F}_ n) \to (\mathcal{F}_ n'')$ of inverse systems of coherent $\mathcal{O}_ U$-modules with the following properties

  1. if $\text{depth}(\mathcal{F}''_{n, y}) + \delta ^ Y_ Z(y) \geq 3$ for all $y \in U \cap Y$,

  2. $(\mathcal{F}''_ n)$ is isomorphic as a pro-system to an object $(\mathcal{H}_ n)$ of $\textit{Coh}(U, I\mathcal{O}_ U)$,

  3. the induced morphism $(\mathcal{F}_ n) \to (\mathcal{H}_ n)$ of $\textit{Coh}(U, I\mathcal{O}_ U)$ has kernel and cokernel annihilated by a power of $I$,

  4. the modules $H^0(U, \mathcal{F}''_ n)$ and $H^1(U, \mathcal{F}''_ n)$ are finite $A$-modules for all $n$.

Proof. The existence and properties (2), (3), (4) follow immediately from Lemma 52.21.2 applied to $U$, $U \cap Y$, $T = \{ y \in U \cap Y : \delta ^ Y_ Z(y) \leq 2\} $, $T' = \{ y \in U \cap Y : \delta ^ Y_ Z(y) \leq 1\} $, and $(\mathcal{F}_ n)$. The finiteness of the modules $H^0(U, \mathcal{F}''_ n)$ and $H^1(U, \mathcal{F}''_ n)$ follows from Local Cohomology, Lemma 51.12.1 and the elementary properties of the function $\delta ^ Y_ Z(-)$ proved in Lemma 52.18.1. $\square$

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