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 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.Comment: 24 page

Homology of Artinian Modules Over Commutative Noetherian Rings

This work is primarily concerned with the study of artinian modules over commutative noetherian rings. We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules. In the last chapter we prove that if the Betti numbers of a finitely generated module over an equidimensional local ring are eventually non-decreasing, then the dimensions of sufficiently high syzygies are constant

Gröbner Finite Path Algebras

Let K be a field and Γ a finite directed multi-graph. In this paper I classify all path algebras KΓ and admissible orders with the property that all of their finitely generated ideals have finite Gröbner bases