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). $\square$

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