A Method for the Study of Artinian Modules, With an Application to Asymptotic Behavior

Abstract

If N is a Noetherian module over the commutative ring R (throughout the paper, R will denote a commutative ring with identity), then the study of N in many contexts can be reduced to the study of a finitely generated module over a commutative Noetherian ring, because N has a natural structure as a module over R/(0 : N) and the latter ring is Noetherian. For a long time it has been a source of irritation to me that I did not know of any method which would reduce the study of an Artinian module A over the commutative ring R to the study of an Artinian module over a commutative Noetherian ring. However, during the MSRI Microprogram on Commutative Algebra, my attention was drawn to a result of W. Heinzer and D. Lantz [2, Proposition 4.3]; this proposition proves that if A is a faithful Artinian module over a quasi-local ring (R, M) which is (Hausdorff) complete in the M-adic topology, then R is Noetherian. It turns out that a generalization of this result provides a missing link to complete a chain of reductions by which one can, for some purposes, reduce the study of an Artinian module over an arbitrary commutative ring R to the study of an Artinian module over a complete (Noetherian) local ring; in the latter situation we have Matlis’s duality available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.KeywordsLocal RingMaximal IdealCommutative RingHomomorphic ImageNoetherian RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.