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
E^ i is a finite A-module nonzero only for 0 \leq i \leq \dim (\text{Supp}(M)),
\dim (\text{Supp}(E^ i)) \leq i,
\text{depth}(M) is the smallest integer \delta \geq 0 such that E^\delta \not= 0,
\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,
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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