The Stacks project

Lemma 47.11.2. Let $A$ be a Noetherian ring. Let $0 \to N' \to N \to N'' \to 0$ be a short exact sequence of finite $A$-modules. Let $I \subset A$ be an ideal.

  1. $\text{depth}_ I(N) \geq \min \{ \text{depth}_ I(N'), \text{depth}_ I(N'')\} $

  2. $\text{depth}_ I(N'') \geq \min \{ \text{depth}_ I(N), \text{depth}_ I(N') - 1\} $

  3. $\text{depth}_ I(N') \geq \min \{ \text{depth}_ I(N), \text{depth}_ I(N'') + 1\} $

Proof. Assume $IN \not= N$, $IN' \not= N'$, and $IN'' \not= N''$. Then we can use the characterization of depth using the Ext groups $\mathop{\mathrm{Ext}}\nolimits ^ i(A/I, N)$, see Lemma 47.11.1, and use the long exact cohomology sequence

\[ \begin{matrix} 0 \to \mathop{\mathrm{Hom}}\nolimits _ A(A/I, N') \to \mathop{\mathrm{Hom}}\nolimits _ A(A/I, N) \to \mathop{\mathrm{Hom}}\nolimits _ A(A/I, N'') \\ \phantom{0\ } \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(A/I, N') \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(A/I, N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(A/I, N'') \to \ldots \end{matrix} \]

from Algebra, Lemma 10.70.6. This argument also works if $IN = N$ because in this case $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(A/I, N) = 0$ for all $i$. Similarly in case $IN' \not= N'$ and/or $IN'' \not= N''$. $\square$


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Your email address will not be published.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