Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L′ be R-modules. We investigate the properties of the functors and . For instance, we show the following: (a)if L and L′ are artinian, then is artinian, and is noetherian over the completion ;(b)if L is artinian and L′ is Matlis reflexive, then , , and are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.