The Stacks project

Lemma 52.8.1. Let $I, J$ be ideals of a Noetherian ring $A$. Let $M$ be a finite $A$-module. Let $\mathfrak p \subset A$ be a prime. Let $s$ and $d$ be integers. Assume

  1. $A$ has a dualizing complex,

  2. $\mathfrak p \not\in V(J) \cap V(I)$,

  3. $\text{cd}(A, I) \leq d$, and

  4. for all primes $\mathfrak p' \subset \mathfrak p$ we have $\text{depth}_{A_{\mathfrak p'}}(M_{\mathfrak p'}) + \dim ((A/\mathfrak p')_\mathfrak q) > d + s$ for all $\mathfrak q \in V(\mathfrak p') \cap V(J) \cap V(I)$.

Then there exists an $f \in A$, $f \not\in \mathfrak p$ which annihilates $H^ i(R\Gamma _ J(M)^\wedge )$ for $i \leq s$ where ${}^\wedge $ indicates $I$-adic completion.

Proof. We will use that $R\Gamma _ J = R\Gamma _{V(J)}$ and similarly for $I + J$, see Dualizing Complexes, Lemma 47.10.1. Observe that $R\Gamma _ J(M)^\wedge = R\Gamma _ I(R\Gamma _ J(M))^\wedge = R\Gamma _{I + J}(M)^\wedge $, see Dualizing Complexes, Lemmas 47.12.1 and 47.9.6. Thus we may replace $J$ by $I + J$ and assume $I \subset J$ and $\mathfrak p \not\in V(J)$. Recall that

\[ R\Gamma _ J(M)^\wedge = R\mathop{\mathrm{Hom}}\nolimits _ A(R\Gamma _ I(A), R\Gamma _ J(M)) \]

by the description of derived completion in More on Algebra, Lemma 15.91.10 combined with the description of local cohomology in Dualizing Complexes, Lemma 47.10.2. Assumption (3) means that $R\Gamma _ I(A)$ has nonzero cohomology only in degrees $\leq d$. Using the canonical truncations of $R\Gamma _ I(A)$ we find it suffices to show that

\[ \text{Ext}^ i(N, R\Gamma _ J(M)) \]

is annihilated by an $f \in A$, $f \not\in \mathfrak p$ for $i \leq s + d$ and any $A$-module $N$. In turn using the canonical truncations for $R\Gamma _ J(M)$ we see that it suffices to show $H^ i_ J(M)$ is annihilated by an $f \in A$, $f \not\in \mathfrak p$ for $i \leq s + d$. This follows from Local Cohomology, Lemma 51.10.2. $\square$

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Your email address will not be published.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 0EFH. Beware of the difference between the letter 'O' and the digit '0'.