The Stacks project

Lemma 51.9.4. Let $(A, \mathfrak m)$ be a Noetherian local ring with normalized dualizing complex $\omega _ A^\bullet $. Let $M$ be a finite $A$-module. Set $E^ i = \text{Ext}_ A^{-i}(M, \omega _ A^\bullet )$. Then

  1. $E^ i$ is a finite $A$-module nonzero only for $0 \leq i \leq \dim (\text{Supp}(M))$,

  2. $\dim (\text{Supp}(E^ i)) \leq i$,

  3. $\text{depth}(M)$ is the smallest integer $\delta \geq 0$ such that $E^\delta \not= 0$,

  4. $\mathfrak p \in \text{Supp}(E^0 \oplus \ldots \oplus E^ i) \Leftrightarrow \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim (A/\mathfrak p) \leq i$,

  5. the annihilator of $E^ i$ is equal to the annihilator of $H^ i_\mathfrak m(M)$.

Proof. Parts (1), (2), and (3) are copies of the statements in Dualizing Complexes, Lemma 47.16.5. For a prime $\mathfrak p$ of $A$ we have that $(\omega _ A^\bullet )_\mathfrak p[-\dim (A/\mathfrak p)]$ is a normalized dualzing complex for $A_\mathfrak p$. See Dualizing Complexes, Lemma 47.17.3. Thus

\[ E^ i_\mathfrak p = \text{Ext}^{-i}_ A(M, \omega _ A^\bullet )_\mathfrak p = \text{Ext}^{-i + \dim (A/\mathfrak p)}_{A_\mathfrak p} (M_\mathfrak p, (\omega _ A^\bullet )_\mathfrak p[-\dim (A/\mathfrak p)]) \]

is zero for $i - \dim (A/\mathfrak p) < \text{depth}_{A_\mathfrak p}(M_\mathfrak p)$ and nonzero for $i = \dim (A/\mathfrak p) + \text{depth}_{A_\mathfrak p}(M_\mathfrak p)$ by part (3) over $A_\mathfrak p$. This proves part (4). If $E$ is an injective hull of the residue field of $A$, then we have

\[ \mathop{\mathrm{Hom}}\nolimits _ A(H^ i_\mathfrak m(M), E) = \text{Ext}^{-i}_ A(M, \omega _ A^\bullet )^\wedge = (E^ i)^\wedge = E^ i \otimes _ A A^\wedge \]

by the local duality theorem (in the form of Dualizing Complexes, Lemma 47.18.4). Since $A \to A^\wedge $ is faithfully flat, we find (5) is true by Matlis duality (Dualizing Complexes, Proposition 47.7.8). $\square$

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Your email address will not be published.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