The Stacks project

The method of proof follows roughly the method of proof of [Theorem 1, Faltings-algebraisation] and [Satz 2, Faltings-uber]. The result is almost the same as [Theorem 1.1, MRaynaud-paper] (affine complement case) and [Theorem 3.9, MRaynaud-book] (complement is union of few affines).

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. $\square$

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