The Stacks project

[Lemma 3, Faltings-annulators]

Lemma 51.7.1. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Let $M$ be a finite $A$-module. Let $n \geq 0$. The following are equivalent

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

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

Proof. We prove the equivalence of (1) and (2) by induction on $n$. For $n = 0$ we have $H^0_ T(M) \subset M$ is finite. Hence (1) is true. Since $H^0_ T(M) = \mathop{\mathrm{colim}}\nolimits H^0_{V(J)}(M)$ with $J$ as in (2) we see that (2) is true. Assume that $n > 0$.

Assume (1) is true. Recall that $H^ i_ J(M) = H^ i_{V(J)}(M)$, see Dualizing Complexes, Lemma 47.10.1. Thus $H^ i_ T(M) = \mathop{\mathrm{colim}}\nolimits H^ i_ J(M)$ where the colimit is over ideals $J \subset A$ with $V(J) \subset T$, see Lemma 51.5.3. Since $H^ i_ T(M)$ is finitely generated for $i \leq n$ we can find a $J \subset A$ as in (2) such that $H^ i_ J(M) \to H^ i_ T(M)$ 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.

Assume we have $J$ as in (2). Let $N = H^0_ T(M)$ and $M' = M/N$. 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. Thus we find that $H^0_ T(M') = 0$ and $H^ i_ T(M') = H^ i_ T(M)$ for $i > 0$. We conclude that we may replace $M$ by $M'$. Thus we may assume that $H^0_ T(M) = 0$. This means that the finite set of associated primes of $M$ 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 $M$. Then the long exact local cohomology sequence associated to the short exact sequence

\[ 0 \to M \to M \to M/fM \to 0 \]

turns into short exact sequences

\[ 0 \to H^ i_ T(M) \to H^ i_ T(M/fM) \to H^{i + 1}_ T(M) \to 0 \]

for $i < n$. We conclude that $J^2$ annihilates $H^ i_ T(M/fM)$ for $i < n$. By induction hypothesis we see that $H^ i_ T(M/fM)$ is finite for $i < n$. Using the short exact sequence once more we see that $H^{i + 1}_ T(M)$ is finite for $i < n$ as desired.

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

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AW8. Beware of the difference between the letter 'O' and the digit '0'.