The Stacks project

51.14 Finiteness of local cohomology, III

We extend the discussion of finiteness of local cohomology in Sections 51.7 and 51.11 to bounded below complexes with finite cohomology modules.

Lemma 51.14.1. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Let $K$ be an object of $D_{\textit{Coh}}^+(A)$. Let $n \in \mathbf{Z}$. The following are equivalent

  1. $H^ i_ T(K)$ is finite for $i \leq n$,

  2. there exists an ideal $J \subset A$ with $V(J) \subset T$ such that $J$ annihilates $H^ i_ T(K)$ for $i \leq n$.

If $T = V(I) = Z$ for an ideal $I \subset A$, then these are also equivalent to

  1. there exists an $e \geq 0$ such that $I^ e$ annihilates $H^ i_ Z(K)$ for $i \leq n$.

Proof. This lemma is the natural generalization of Lemma 51.7.1 whose proof the reader should read first. Assume (1) is true. Recall that $H^ i_ J(K) = H^ i_{V(J)}(K)$, see Dualizing Complexes, Lemma 47.10.1. Thus $H^ i_ T(K) = \mathop{\mathrm{colim}}\nolimits H^ i_ J(K)$ where the colimit is over ideals $J \subset A$ with $V(J) \subset T$, see Lemma 51.5.3. Since $H^ i_ T(K)$ is finitely generated for $i \leq n$ we can find a $J \subset A$ as in (2) such that $H^ i_ J(K) \to H^ i_ T(K)$ is surjective for $i \leq n$. Thus the finite list of generators are $J$-power torsion elements and we see that (2) holds with $J$ replaced by some power.

Let $a \in \mathbf{Z}$ be an integer such that $H^ i(K) = 0$ for $i < a$. We prove (2) $\Rightarrow $ (1) by descending induction on $a$. If $a > n$, then we have $H^ i_ T(K) = 0$ for $i \leq n$ hence both (1) and (2) are true and there is nothing to prove.

Assume we have $J$ as in (2). Observe that $N = H^ a_ T(K) = H^0_ T(H^ a(K))$ is finite as a submodule of the finite $A$-module $H^ a(K)$. If $n = a$ we are done; so assume $a < n$ from now on. By construction of $R\Gamma _ T$ we find that $H^ i_ T(N) = 0$ for $i > 0$ and $H^0_ T(N) = N$, see Remark 51.5.6. Choose a distinguished triangle

\[ N[-a] \to K \to K' \to N[-a + 1] \]

Then we see that $H^ a_ T(K') = 0$ and $H^ i_ T(K) = H^ i_ T(K')$ for $i > a$. We conclude that we may replace $K$ by $K'$. Thus we may assume that $H^ a_ T(K) = 0$. This means that the finite set of associated primes of $H^ a(K)$ are not in $T$. By prime avoidance (Algebra, Lemma 10.15.2) we can find $f \in J$ not contained in any of the associated primes of $H^ a(K)$. Choose a distinguished triangle

\[ L \to K \xrightarrow {f} K \to L[1] \]

By construction we see that $H^ i(L) = 0$ for $i \leq a$. On the other hand we have a long exact cohomology sequence

\[ 0 \to H^{a + 1}_ T(L) \to H^{a + 1}_ T(K) \xrightarrow {f} H^{a + 1}_ T(K) \to H^{a + 2}_ T(L) \to H^{a + 2}_ T(K) \xrightarrow {f} \ldots \]

which breaks into the identification $H^{a + 1}_ T(L) = H^{a + 1}_ T(K)$ and short exact sequences

\[ 0 \to H^{i - 1}_ T(K) \to H^ i_ T(L) \to H^ i_ T(K) \to 0 \]

for $i \leq n$ since $f \in J$. We conclude that $J^2$ annihilates $H^ i_ T(L)$ for $i \leq n$. By induction hypothesis applied to $L$ we see that $H^ i_ T(L)$ is finite for $i \leq n$. Using the short exact sequence once more we see that $H^ i_ T(K)$ is finite for $i \leq n$ as desired.

We omit the proof of the equivalence of (2) and (3) in case $T = V(I)$. $\square$

Proposition 51.14.2. Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Let $s \in \mathbf{Z}$. Let $K \in D_{\textit{Coh}}^+(A)$. The following are equivalent

  1. $H^ i_ T(K)$ is a finite $A$-module for $i \leq s$, and

  2. for all $\mathfrak p \not\in T$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have

    \[ \text{depth}_{A_\mathfrak p}(K_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > s \]

Proof. Formal consequence of Proposition 51.13.2 and Lemma 51.14.1. $\square$

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