The Stacks project

52.11 Algebraization of formal sections, I

In this section we study the problem of algebraization of formal sections in the local case. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let

\[ X = \mathop{\mathrm{Spec}}(A) \supset U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} \]

and denote $Y = V(I)$ the closed subscheme corresponding to $I$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. In this section we consider the limits

\[ \mathop{\mathrm{lim}}\nolimits _ n H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \]

This is closely related to the cohomology of the pullback of $\mathcal{F}$ to the formal completion of $U$ along $Y$; however, since we have not yet introduced formal schemes, we cannot use this terminology here.

Lemma 52.11.1. Let $U$ be the punctured spectrum of a Noetherian local ring $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Let $I \subset A$ be an ideal. Then

\[ H^ i(R\Gamma (U, \mathcal{F})^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \]

for all $i$ where $R\Gamma (U, \mathcal{F})^\wedge $ denotes the derived $I$-adic completion.

Proof. By Lemmas 52.6.20 and 52.7.2 we have

\[ R\Gamma (U, \mathcal{F})^\wedge = R\Gamma (U, \mathcal{F}^\wedge ) = R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \]

Thus we obtain short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{i - 1}(U, \mathcal{F}/I^ n\mathcal{F}) \to H^ i(R\Gamma (U, \mathcal{F})^\wedge ) \to \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

by Cohomology, Lemma 20.35.1. The $R^1\mathop{\mathrm{lim}}\nolimits $ terms vanish because the inverse systems of groups $H^ i(U, \mathcal{F}/I^ n\mathcal{F})$ satisfy the Mittag-Leffler condition by Lemma 52.5.2. $\square$

reference

Theorem 52.11.2. Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \mathop{\mathrm{Spec}}(A)$, $Y = V(I)$, and $U = X \setminus \{ \mathfrak m\} $. Let $\mathcal{F}$ be a coherent sheaf on $U$. Assume

  1. $\text{cd}(A, I) \leq d$, i.e., $H^ i(X \setminus Y, \mathcal{G}) = 0$ for $i \geq d$ and quasi-coherent $\mathcal{G}$ on $X$,

  2. for any $x \in X \setminus Y$ whose closure $\overline{\{ x\} }$ in $X$ meets $U \cap Y$ we have

    \[ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) \geq s \quad \text{or}\quad \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \dim (\overline{\{ x\} }) > d + s \]

Then there exists an open $V_0 \subset U$ containing $U \cap Y$ such that for any open $V \subset V_0$ containing $U \cap Y$ the map

\[ H^ i(V, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \]

is an isomorphism for $i < s$. If in addition $ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \dim (\overline{\{ x\} }) > s $ for all $x \in U \cap Y$, then these cohomology groups are finite $A$-modules.

Proof. Choose a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction to $U$ of the coherent $\mathcal{O}_ X$-module associated to $M$, see Local Cohomology, Lemma 51.8.2. Then the assumptions of Lemma 52.9.5 are satisfied. Pick $J_0$ as in that lemma and set $V_0 = X \setminus V(J_0)$. Then opens $V \subset V_0$ containing $U \cap Y$ correspond $1$-to-$1$ with ideals $J \subset J_0$ with $V(J) \cap V(I) = \{ \mathfrak m\} $. Moreover, for such a choice we have a distinguished triangle

\[ R\Gamma _ J(M) \to M \to R\Gamma (V, \mathcal{F}) \to R\Gamma _ J(M)[1] \]

We similarly have a distinguished triangle

\[ R\Gamma _\mathfrak m(M)^\wedge \to M \to R\Gamma (U, \mathcal{F})^\wedge \to R\Gamma _\mathfrak m(M)^\wedge [1] \]

involving derived $I$-adic completions. The cohomology groups of $R\Gamma (U, \mathcal{F})^\wedge $ are equal to the limits in the statement of the theorem by Lemma 52.11.1. The canonical map between these triangles and some easy arguments show that our theorem follows from the main Lemma 52.9.5 (note that we have $i < s$ here whereas we have $i \leq s$ in the lemma; this is because of the shift). The finiteness of the cohomology groups (under the additional assumption) follows from Lemma 52.9.3. $\square$

Lemma 52.11.3. Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal $I$. Set $X = \mathop{\mathrm{Spec}}(A)$, $Y = V(I)$, and $U = X \setminus \{ \mathfrak m\} $. Let $\mathcal{F}$ be a coherent sheaf on $U$. Assume for any associated point $x \in U$ of $\mathcal{F}$ we have $\dim (\overline{\{ x\} }) > \text{cd}(A, I) + 1$ where $\overline{\{ x\} }$ is the closure in $X$. Then the map

\[ \mathop{\mathrm{colim}}\nolimits H^0(V, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F}) \]

is an isomorphism of finite $A$-modules where the colimit is over opens $V \subset U$ containing $U \cap Y$.

Proof. Apply Theorem 52.11.2 with $s = 1$ (we get finiteness too). $\square$


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