The Stacks project

Remark 52.16.12. In Lemma 52.16.11 if $A$ is universally catenary with Cohen-Macaulay formal fibres (for example if $A$ has a dualizing complex), then the condition that $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules, is equivalent with

\[ \text{depth}((A/f)_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > 2 \]

for all $\mathfrak p \in V(f) \setminus V(\mathfrak a)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ by Local Cohomology, Theorem 51.11.6.

For example, if $A/fA$ is $(S_2)$ and if every irreducible component of $Z = V(\mathfrak a)$ has codimension $\geq 3$ in $Y = \mathop{\mathrm{Spec}}(A/fA)$, then we get the finiteness of $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$. This should be contrasted with the slightly weaker conditions found in Lemma 52.20.1 (see also Remark 52.20.2).

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Your email address will not be published.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



View Remark 52.16.12 as pdf