Homology of artinian and mini-max modules, II

TL;DR

This paper studies the finiteness properties of Ext and Tor modules over commutative rings, establishing conditions under which these modules are Matlis reflexive and related duality results.

Contribution

It provides new results on the finiteness and duality properties of Ext and Tor modules for modules satisfying certain finiteness conditions over noetherian rings.

Findings

Ext and Tor modules are Matlis reflexive under specified conditions

Duality isomorphisms between Ext and Tor modules are established

Results apply to modules with noetherian, artinian, and mini-max properties

Abstract

Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules Ext^i_R(L,L') and Tor_i^R(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that Ext^i_R(M,M'), Ext^i_R(M',M), and Tor_i^R(M,M') are Matlis reflexive over R for all i\geq 0 and that Ext^i_R(M,M')^\vee\cong Tor_i^R(M,M'^\vee) and Ext^i_R(M',M)^\vee\cong Tor_i^R(M',M^\vee).