Grothendieck's local duality

Remarks

If $R$ is Gorenstein, then $\omega_R = R$ and the formula reads $H^i_{\mathfrak{m}}(M) \cong \Hom_R(\operatorname{Ext}^{d-i}_R(M, R), E)$. For $M = R$ and $i = d$ it gives $H^d_{\mathfrak{m}}(R) \cong \Hom_R(\omega_R, E)$. Example: for $R = K[\![x]\!]$, one has $d = 1$, $\omega_R = R$ and $E = K[\![x]\!]_x / K[\![x]\!]$, and indeed $H^1_{\mathfrak{m}}(R) = R_x / R \cong E = \Hom_R(R, E)$. When $R$ is complete, Matlis duality turns the formula around: $\operatorname{Ext}^{d-i}_R(M, \omega_R) \cong \Hom_R(H^i_{\mathfrak{m}}(M), E)$.

Going deeper

For a student

Combined with the vanishing pattern of local cohomology, local duality shows that $M$ is Cohen–Macaulay of dimension $t$ exactly when $\operatorname{Ext}^{d-i}_R(M, \omega_R) = 0$ for all $i \ne t$. It also identifies the canonical module as the Matlis dual of the top local cohomology $H^d_{\mathfrak{m}}(R)$ when $R$ is complete.