Homology of artinian and Matlis reflexive modules, I

TL;DR

This paper studies the homological properties of artinian and Matlis reflexive modules over local noetherian rings, revealing finite length and reflexivity properties of Tor and Ext functors, with detailed vanishing behavior analysis.

Contribution

It provides new results on the finiteness, reflexivity, and vanishing of Tor and Ext functors involving artinian and Matlis reflexive modules over local noetherian rings.

Findings

Hom_R(L,L') has finite length when L is artinian and L' is noetherian

Tensor product of two artinian modules has finite length

Tor and Ext groups involving artinian modules are artinian or noetherian and Matlis reflexive

Abstract

Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors Tor_i^R(L,-) and Ext^i_R(L,-). For instance, we show the following: (a) if L is artinian and L' is noetherian, then Hom_R(L,L') has finite length; (b) if L and L' are artinian, then the tensor product L \otimes_R L' has finite length; (c) if L and L' are artinian, then Tor_i^R(L,L') is artinian, and Ext^i_R(L,L') is noetherian over the completion \hat R; and (d) if L is artinian and L' is Matlis reflexive, then Ext^i_R(L,L'), Ext^i_R(L',L), and Tor_i^R(L,L') are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.