The Stacks project

Lemma 52.21.1. In the situation above assume $X$ locally has a dualizing complex. Let $T \subset Y$ be a subset stable under specialization. Assume for $y \in T$ and for a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $V(\mathfrak p) \cap V(\mathcal{I}^\wedge _ y) = \{ \mathfrak m_ y^\wedge \} $ we have

\[ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) > 0 \]

Then there exists a canonical map $(\mathcal{F}_ n) \to (\mathcal{F}_ n')$ of inverse systems of coherent $\mathcal{O}_ X$-modules with the following properties

  1. for $y \in T$ we have $\text{depth}(\mathcal{F}'_{n, y}) \geq 1$,

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

  3. the induced morphism $(\mathcal{F}_ n) \to (\mathcal{G}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ is surjective with kernel annihilated by a power of $\mathcal{I}$.

Proof. For every $n$ we let $\mathcal{F}_ n \to \mathcal{F}'_ n$ be the surjection constructed in Local Cohomology, Lemma 51.15.1. Since this is the quotient of $\mathcal{F}_ n$ by the subsheaf of sections supported on $T$ we see that we get canonical maps $\mathcal{F}'_{n + 1} \to \mathcal{F}'_ n$ such that we obtain a map $(\mathcal{F}_ n) \to (\mathcal{F}_ n')$ of inverse systems of coherent $\mathcal{O}_ X$-modules. Property (1) holds by construction.

To prove properties (2) and (3) we may assume that $X = \mathop{\mathrm{Spec}}(A_0)$ is affine and $A_0$ has a dualizing complex. Let $I_0 \subset A_0$ be the ideal corresponding to $Y$. Let $A, I$ be the $I$-adic completions of $A_0, I_0$. For later use we observe that $A$ has a dualizing complex (Dualizing Complexes, Lemma 47.22.4). Let $M$ be the finite $A$-module corresponding to $(\mathcal{F}_ n)$, see Cohomology of Schemes, Lemma 30.23.1. Then $\mathcal{F}_ n$ corresponds to $M_ n = M/I^ nM$. Recall that $\mathcal{F}'_ n$ corresponds to the quotient $M'_ n = M_ n / H^0_ T(M_ n)$, see Local Cohomology, Lemma 51.15.1 and its proof.

Set $s = 0$ and $d = \text{cd}(A, I)$. We claim that $A, I, T, M, s, d$ satisfy assumptions (1), (3), (4), (6) of Situation 52.10.1. Namely, (1) and (3) are immediate from the above, (4) is the empty condition as $s = 0$, and (6) is the assumption we made in the statement of the lemma.

By Theorem 52.10.8 we see that $\{ H^0_ T(M_ n)\} $ is Mittag-Leffler, that $\mathop{\mathrm{lim}}\nolimits H^0_ T(M_ n) = H^0_ T(M)$, and that $H^0_ T(M)$ is killed by a power of $I$. Thus the limit of the short exact sequences $0 \to H^0_ T(M_ n) \to M_ n \to M'_ n \to 0$ is the short exact sequence

\[ 0 \to H^0_ T(M) \to M \to \mathop{\mathrm{lim}}\nolimits M'_ n \to 0 \]

Setting $M' = \mathop{\mathrm{lim}}\nolimits M'_ n$ we find that $\mathcal{G}_ n$ corresponds to the finite $A_0$-module $M'/I^ nM'$. To finish the prove we have to show that the canonical map $\{ M'/I^ nM'\} \to \{ M'_ n\} $ is a pro-isomorphism. This is equivalent to saying that $\{ H^0_ T(M) + I^ nM\} \to \{ \ker (M \to M'_ n)\} $ is a pro-isomorphism. Which in turn says that $\{ H^0_ T(M)/H^0_ T(M) \cap I^ nM\} \to \{ H^0_ T(M_ n)\} $ is a pro-isomorphism. This is true because $\{ H^0_ T(M_ n)\} $ is Mittag-Leffler, $\mathop{\mathrm{lim}}\nolimits H^0_ T(M_ n) = H^0_ T(M)$, and $H^0_ T(M)$ is killed by a power of $I$ (so that Artin-Rees tells us that $H^0_ T(M) \cap I^ nM = 0$ for $n$ large enough). $\square$

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