The Stacks project

47.19 Dualizing modules

If $(A, \mathfrak m, \kappa )$ is a Noetherian local ring and $\omega _ A^\bullet $ is a normalized dualizing complex, then we say the module $\omega _ A = H^{-\dim (A)}(\omega _ A^\bullet )$, described in Lemma 47.17.5, is a dualizing module for $A$. This module is a canonical module of $A$. It seems generally agreed upon to define a canonical module for a Noetherian local ring $(A, \mathfrak m, \kappa )$ to be a finite $A$-module $K$ such that

\[ \mathop{\mathrm{Hom}}\nolimits _ A(K, E) \cong H^{\dim (A)}_\mathfrak m(A) \]

where $E$ is an injective hull of the residue field. A dualizing module is canonical because

\[ \mathop{\mathrm{Hom}}\nolimits _ A(H^{\dim (A)}_\mathfrak m(A), E) = (\omega _ A)^\wedge \]

by Lemma 47.18.4 and hence applying $\mathop{\mathrm{Hom}}\nolimits _ A(-, E)$ we get

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A, E) & = \mathop{\mathrm{Hom}}\nolimits _ A((\omega _ A)^\wedge , E) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(\mathop{\mathrm{Hom}}\nolimits _ A(H^{\dim (A)}_\mathfrak m(A), E), E) \\ & = H^{\dim (A)}_\mathfrak m(A) \end{align*}

the first equality because $E$ is $\mathfrak m$-power torsion, the second by the above, and the third by Matlis duality (Proposition 47.7.8). The utility of the definition of a canonical module given above lies in the fact that it makes sense even if $A$ does not have a dualizing complex.

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