The Stacks project

52.21 Improving coherent formal modules

Let $X$ be a Noetherian scheme. Let $Y \subset X$ be a closed subscheme with quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. In this section we construct maps $(\mathcal{F}_ n) \to (\mathcal{F}'_ n)$ similar to the maps constructed in Local Cohomology, Section 51.15 for coherent modules. For a point $y \in Y$ we set

\[ \mathcal{O}_{X, y}^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{O}_{X, y}/\mathcal{I}^ n_ y, \quad \mathcal{I}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{I}_ y/\mathcal{I}^ n_ y \quad \text{and}\quad \mathfrak m_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathfrak m_ y/\mathcal{I}_ y^ n \]

Then $\mathcal{O}_{X, y}^\wedge $ is a Noetherian local ring with maximal ideal $\mathfrak m_ y^\wedge $ complete with respect to $\mathcal{I}_ y^\wedge = \mathcal{I}_ y\mathcal{O}_{X, y}^\wedge $. We also set

\[ \mathcal{F}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, y} \]

Then $\mathcal{F}_ y^\wedge $ is a finite module over $\mathcal{O}_{X, y}^\wedge $ with $\mathcal{F}_ y^\wedge /(\mathcal{I}_ y^\wedge )^ n\mathcal{F}_ y^\wedge = \mathcal{F}_{n, y}$ for all $n$, see Algebra, Lemmas 10.98.2 and 10.96.12.

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). $\square$

Lemma 52.21.2. In the situation above assume $X$ locally has a dualizing complex. Let $T' \subset T \subset Y$ be subsets stable under specialization. Let $d \geq 0$ be an integer. Assume

  1. affine locally we have $X = \mathop{\mathrm{Spec}}(A_0)$ and $Y = V(I_0)$ and $\text{cd}(A_0, I_0) \leq d$,

  2. for $y \in T$ and a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $V(\mathfrak p) \cap V(\mathcal{I}_ y^\wedge ) = \{ \mathfrak m_ y^\wedge \} $ we have

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

  3. for $y \in T'$ and for a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(\mathcal{I}_ y^\wedge )$ and $V(\mathfrak p) \cap V(\mathcal{I}_ y^\wedge ) \not= \{ \mathfrak m_ y^\wedge \} $ we have

    \[ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) \geq 1 \quad \text{or}\quad \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) > 1 + d \]

  4. for $y \in T'$ and a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $V(\mathfrak p) \cap V(\mathcal{I}_ y^\wedge ) = \{ \mathfrak m_ y^\wedge \} $ we have

    \[ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) > 1 \]

  5. if $y \leadsto y'$ is an immediate specialization and $y' \in T'$, then $y \in T$.

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. for $y' \in T'$ we have $\text{depth}(\mathcal{F}''_{n, y'}) \geq 2$,

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

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

Proof. As in Lemma 52.21.1 and its proof for every $n$ we let $\mathcal{F}_ n \to \mathcal{F}'_ n$ be the surjection constructed in Local Cohomology, Lemma 51.15.1. Next, we let $\mathcal{F}'_ n \to \mathcal{F}''_ n$ be the injection constructed in Local Cohomology, Lemma 51.15.5 and its proof. The constructions show that we get canonical maps $\mathcal{F}''_{n + 1} \to \mathcal{F}''_ n$ such that we obtain maps

\[ (\mathcal{F}_ n) \longrightarrow (\mathcal{F}_ n') \longrightarrow (\mathcal{F}''_ n) \]

of inverse systems of coherent $\mathcal{O}_ X$-modules. Properties (1) and (2) hold by construction.

To prove properties (3) and (4) 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 $M'_ n = M_ n / H^0_ T(M_ n)$. Also, recall that $M' = \mathop{\mathrm{lim}}\nolimits M'_ n$ is the quotient of $M$ by $H^0_ T(M)$ and that $\{ M'_ n\} $ and $\{ M'/I^ nM'\} $ are isomorphic as pro-systems. Finally, we see that $\mathcal{F}''_ n$ corresponds to an extension

\[ 0 \to M'_ n \to M''_ n \to H^1_{T'}(M'_ n) \to 0 \]

see proof of Local Cohomology, Lemma 51.15.5.

Set $s = 1$. 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, (4) is implied by (c), and (6) follows from (d). We omit the details of the verification (c) $\Rightarrow $ (4).

By Theorem 52.10.8 we see that $\{ H^1_{T'}(M'/I^ nM')\} $ is Mittag-Leffler, that $H^1_{T'}(M') = \mathop{\mathrm{lim}}\nolimits H^1_{T'}(M'/I^ nM')$, and that $H^1_{T'}(M')$ is killed by a power of $I$. We deduce $\{ H^1_{T'}(M'_ n)\} $ is Mittag-Leffler and $H^1_{T'}(M') = \mathop{\mathrm{lim}}\nolimits H^1_{T'}(M'_ n)$. Thus the limit of the short exact sequences displayed above is the short exact sequence

\[ 0 \to M' \to \mathop{\mathrm{lim}}\nolimits M''_ n \to H^1_{T'}(M') \to 0 \]

Set $M'' = \mathop{\mathrm{lim}}\nolimits M''_ n$. It follows from Local Cohomology, Proposition 51.11.1 that $H^1_{T'}(M')$ and hence $M''$ are finite $A$-modules. Thus we find that $\mathcal{H}_ n$ 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. Since we already know that $\{ M'/I^ nM'\} $ is pro-isomorphic to $\{ M'_ n\} $ the reader verifies (omitted) this is equivalent to asking $\{ H^1_{T'}(M')/I^ nH^1_{T'}(M')\} \to \{ H^1_{T'}(M'_ n)\} $ to be a pro-isomorphism. This is true because $\{ H^1_{T'}(M'_ n)\} $ is Mittag-Leffler, $H^1_{T'}(M') = \mathop{\mathrm{lim}}\nolimits H^1_{T'}(M'_ n)$, and $H^1_{T'}(M')$ is killed by a power of $I$. $\square$

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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