Homology of artinian and Matlis reflexive modules, I

Abstract

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

Some homological properties of ideals with cohomological dimension one

“…So, it follows from Lemma 2.1 and adjointness, that the R-modules Ext i R (R/I, T ) ≃ D(Tor R i (R/I, M)) are Artinian for all i ≥ 0. Now, it follows from [19,Lemma 1.15(a)] that, the R-modules Tor R i (R/I, M) are finitely generated for all i ≥ 0, as required.…”

On top local cohomology modules, Matlis duality and tensor products

“…(1) As H d a (R) is an Artinian and m-torsion R-module (cf. Lemma 2.3), then it follows by [12,Proposition 5.8] that H d a (R) ⊗ R H d a (R) = 0. To this end note that from Proposition 2.4, we have depth…”

Codualizing modules

“…all terms in the above long exact sequence are finitely generated by[13, Theorem 2.2]. Now, a simple use of Nakayama's lemma shows that pdR (T ) < ∞ if and only if pd R (Hom R (R/xR, T )) < ∞.…”