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 num bers mu(i)(m, M) are finite for all i. The class of modules with finit e 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 d uality. We prove that Matlis reflexive modules satisfy local duality i n general, where R is any local ring with a dualizing complex.