A Method for the Study of Artinian Modules, With an Application to Asymptotic Behavior
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.
Unable to display preview. Download preview PDF.
- 5.I. G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica 11 (1973), 23–43.Google Scholar
- 9.S. McAdam, “Asymptotic prime divisors,” Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983.Google Scholar
- 11.D. G. Northcott, “AJI Introduction to Homological Algebra,” Cambridge University Press, 1960.Google Scholar
- 14.D. W. Sharpe and P. Vamos, “Injective modules,” Cambridge Tracts in Mathematics and Mathematical Physics, vol. 62, Cambridge University Press, 1972.Google Scholar
- 15.O. Zariski and P. Samuel, “Commutative Algebra, Vol. II,” Graduate Texts in Mathematics, vol. 29, Springer-Verlag, Berlin, 1975.Google Scholar