The Stacks project

52.13 Algebraization of formal sections, III

The next section contains a nonexhaustive list of applications of the material on completion of local cohomology to higher cohomology of coherent modules on quasi-affine schemes and their completion with respect to an ideal.

Proposition 52.13.1. Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $\mathcal{F}$ be a coherent module on $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Let $s \geq 0$. Assume

  1. $A$ is $I$-adically complete and has a dualizing complex,

  2. if $x \in U \setminus V(I)$ then $\text{depth}(\mathcal{F}_ x) > s$ or

    \[ \text{depth}(\mathcal{F}_ x) + \dim (\mathcal{O}_{\overline{\{ x\} }, z}) > \text{cd}(A, I) + s + 1 \]

    for all $z \in V(\mathfrak a) \cap \overline{\{ x\} }$,

  3. one of the following conditions holds:

    1. the restriction of $\mathcal{F}$ to $U \setminus V(I)$ is $(S_{s + 1})$, or

    2. the dimension of $V(\mathfrak a)$ is at most $2$1.

Then the maps

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

are isomorphisms for $i < s$. Moreover we have an isomorphism

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

where the colimit is over opens $V \subset U$ containing $U \cap V(I)$.

Proof. We may assume $s > 0$ as the case $s = 0$ was done in Proposition 52.12.2.

Choose a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction to $U$ of the coherent module associated to $M$, see Local Cohomology, Lemma 51.8.2. Set $d = \text{cd}(A, I)$. Let $\mathfrak p$ be a prime of $A$ not contained in $V(I)$ and let $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$. Then either $\text{depth}(M_\mathfrak p) \geq s + 1 > s$ or we have $\dim ((A/\mathfrak p)_\mathfrak q) > d + s + 1$ by (2). By Lemma 52.10.5 we conclude that the assumptions of Situation 52.10.1 are satisfied for $A, I, V(\mathfrak a), M, s, d$. On the other hand, the hypotheses of Lemma 52.8.5 are satisfied for $s + 1$ and $d$; this is where condition (3) is used.

Applying Lemma 52.8.5 we find there exists an ideal $J_0 \subset \mathfrak a$ with $V(J_0) \cap V(I) = V(\mathfrak a)$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = V(\mathfrak a)$ the maps

\[ H^ i_ J(M) \longrightarrow H^ i(R\Gamma _\mathfrak a(M)^\wedge ) \]

is an isomorphism for $i \leq s + 1$.

For $i \leq s$ the map $H^ i_\mathfrak a(M) \to H^ i_ J(M)$ is an isomorphism by Lemmas 52.10.3 and 52.8.4. Using the comparison of cohomology and local cohomology (Local Cohomology, Lemma 51.2.2) we deduce $H^ i(U, \mathcal{F}) \to H^ i(V,\mathcal{F})$ is an isomorphism for $V = \mathop{\mathrm{Spec}}(A) \setminus V(J)$ and $i < s$.

By Theorem 52.10.8 we have $H^ i_\mathfrak a(M) = \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak a(M/I^ nM)$ for $i \leq s$. By Lemma 52.10.9 we have $H^{s + 1}_\mathfrak a(M) = \mathop{\mathrm{lim}}\nolimits H^{s + 1}_\mathfrak a(M/I^ nM)$.

The isomorphism $H^0(U, \mathcal{F}) = H^0(V, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F})$ follows from the above and Proposition 52.12.2. For $0 < i < s$ we get the desired isomorphisms $H^ i(U, \mathcal{F}) = H^ i(V, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F})$ in the same manner using the relation between local cohomology and cohomology; it is easier than the case $i = 0$ because for $i > 0$ we have

\[ H^ i(U, \mathcal{F}) = H^{i + 1}_\mathfrak a(M), \quad H^ i(V, \mathcal{F}) = H^{i + 1}_ J(M), \quad H^ i(R\Gamma (U, \mathcal{F})^\wedge ) = H^{i + 1}(R\Gamma _\mathfrak a(M)^\wedge ) \]

Similarly for the final statement. $\square$

Lemma 52.13.2. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $M$ be a finite $A$-module. Let $s \geq 0$. Assume

  1. $A$ is $f$-adically complete,

  2. $H^ i_\mathfrak a(M)$ is annihilated by a power of $f$ for $i \leq s + 1$.

Then with $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$ the map

\[ H^ i(U, \widetilde{M}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^ i(U, \widetilde{M/f^ nM}) \]

is an isomorphism for $i < s$.

Proof. The proof is the same as the proof of Lemma 52.12.6. We may apply Lemma 52.3.6 to $U$ and $\mathcal{F} = \widetilde{M}|_ U$ because $\mathcal{F}$ is a Noetherian object in the category of coherent $\mathcal{O}_ U$-modules. Since $H^ i(U, \mathcal{F}) = H^{i + 1}_\mathfrak a(M)$ (Local Cohomology, Lemma 51.8.2) is annihilated by a power of $f$ for $i \leq s$, we see that its $f$-adic Tate module is zero. Hence the lemma shows $\mathop{\mathrm{lim}}\nolimits H^{i - 1}(U, \mathcal{F}/f^ n \mathcal{F})$ is the $0$th cohomology group of the derived $f$-adic completion of $H^{i - 1}(U, \mathcal{F})$. However, if $s \geq i > 1$, then this equal to the $f$-power torsion module $H^ i_\mathfrak a(M)$ and hence equal to its own (derived) completion. For $i = 0$, we refer to Lemma 52.12.6. $\square$

[1] In the sense that the difference of the maximal and minimal values on $V(\mathfrak a)$ of a dimension function on $\mathop{\mathrm{Spec}}(A)$ is at most $2$.

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