The Stacks project

47.18 The local duality theorem

The main result in this section is due to Grothendieck.

Lemma 47.18.1. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $\omega _ A^\bullet $ be a normalized dualizing complex. Let $Z = V(\mathfrak m) \subset \mathop{\mathrm{Spec}}(A)$. Then $E = R^0\Gamma _ Z(\omega _ A^\bullet )$ is an injective hull of $\kappa $ and $R\Gamma _ Z(\omega _ A^\bullet ) = E[0]$.

Proof. By Lemma 47.10.1 we have $R\Gamma _{\mathfrak m} = R\Gamma _ Z$. Thus

\[ R\Gamma _ Z(\omega _ A^\bullet ) = R\Gamma _{\mathfrak m}(\omega _ A^\bullet ) = \text{hocolim}\ R\mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, \omega _ A^\bullet ) \]

by Lemma 47.8.2. Let $E'$ be an injective hull of the residue field. By Lemma 47.16.4 we can find isomorphisms

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, \omega _ A^\bullet ) \cong \mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, E')[0] \]

compatible with transition maps. Since $E' = \bigcup E'[\mathfrak m^ n] = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(A/\mathfrak m^ n, E')$ by Lemma 47.7.3 we conclude that $E \cong E'$ and that all other cohomology groups of the complex $R\Gamma _ Z(\omega _ A^\bullet )$ are zero. $\square$

Remark 47.18.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with a normalized dualizing complex $\omega _ A^\bullet $. By Lemma 47.18.1 above we see that $R\Gamma _ Z(\omega _ A^\bullet )$ is an injective hull of the residue field placed in degree $0$. In fact, this gives a “construction” or “realization” of the injective hull which is slightly more canonical than just picking any old injective hull. Namely, a normalized dualizing complex is unique up to isomorphism, with group of automorphisms the group of units of $A$, whereas an injective hull of $\kappa $ is unique up to isomorphism, with group of automorphisms the group of units of the completion $A^\wedge $ of $A$ with respect to $\mathfrak m$.

Here is the main result of this section.

Theorem 47.18.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $\omega _ A^\bullet $ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $Z = V(\mathfrak m) \subset \mathop{\mathrm{Spec}}(A)$. Denote ${}^\wedge $ derived completion with respect to $\mathfrak m$. Then

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )^\wedge \cong R\mathop{\mathrm{Hom}}\nolimits _ A(R\Gamma _ Z(K), E[0]) \]

for $K$ in $D(A)$.

Proof. Observe that $E[0] \cong R\Gamma _ Z(\omega _ A^\bullet )$ by Lemma 47.18.1. By More on Algebra, Lemma 15.84.12 completion on the left hand side goes inside. Thus we have to prove

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K^\wedge , (\omega _ A^\bullet )^\wedge ) = R\mathop{\mathrm{Hom}}\nolimits _ A(R\Gamma _ Z(K), R\Gamma _ Z(\omega _ A^\bullet )) \]

This follows from the equivalence between $D_{comp}(A, \mathfrak m)$ and $D_{\mathfrak m^\infty \text{-torsion}}(A)$ given in Proposition 47.12.2. More precisely, it is a special case of Lemma 47.12.3. $\square$

Here is a special case of the theorem above.

Lemma 47.18.4. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $\omega _ A^\bullet $ be a normalized dualizing complex. Let $E$ be an injective hull of the residue field. Let $K \in D_{\textit{Coh}}(A)$. Then

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

where ${}^\wedge $ denotes $\mathfrak m$-adic completion.

Proof. By Lemma 47.15.2 we see that $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ is an object of $D_{\textit{Coh}}(A)$. It follows that the cohomology modules of the derived completion of $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ are equal to the usual completions $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, \omega _ A^\bullet )^\wedge $ by More on Algebra, Lemma 15.86.4. On the other hand, we have $R\Gamma _{\mathfrak m} = R\Gamma _ Z$ for $Z = V(\mathfrak m)$ by Lemma 47.10.1. Moreover, the functor $\mathop{\mathrm{Hom}}\nolimits _ A(-, E)$ is exact hence factors through cohomology. Hence the lemma is consequence of Theorem 47.18.3. $\square$


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