A note on local duality
Let (R, m) be a local Noetherian ring. We show that if R is complete, then an R-module M satisfies local duality if and only if the Bass numbers μi(m, M) are finite for all i. The class of modules with finite Bass numbers includes all finitely generated, all Artinian, and all Matlis reflexive R-modules. If the ring R is not complete, we show by example that modules with finite Bass numbers need not satisfy local duality. We prove that Matlis reflexive modules satisfy local duality in general, where R is any local ring with a dualizing complex.
Belshoff, Richard, and Cameron Wickham. "A note on local duality." Bulletin of the London Mathematical Society 29, no. 1 (1997): 25-31.
Bulletin of the London Mathematical Society