The Stacks project

Remark 51.11.7. The astute reader will have realized that we can get away with a slightly weaker condition on the formal fibres of the local rings of $A$. Namely, in the situation of Theorem 51.11.6 assume $A$ is universally catenary but make no assumptions on the formal fibres. Suppose we have an $n$ and we want to prove that $H^ i_ Z(M)$ are finite for $i \leq n$. Then the exact same proof shows that it suffices that $s_{A, I}(M) > n$ and that the formal fibres of local rings of $A$ are $(S_ n)$. On the other hand, if we want to show that $H^ s_ Z(M)$ is not finite where $s = s_{A, I}(M)$, then our arguments prove this if the formal fibres are $(S_{s - 1})$.

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Your email address will not be published.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