Commutative Algebra pp 443-465 | Cite as

# 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.

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### References

- 1.M. Brodmann,
*Asymptotic stability of Ass*(*M/I**n**M*), Proc. AMS**74**(1979), 16–18.MathSciNetGoogle Scholar - 2.W. Heinzer and D. Lantz,
*Artinian modules and modules of which all proper submodules are finitely generated*, J. Algebra**95**(1985), 201–216.MathSciNetMATHCrossRefGoogle Scholar - 3.D. Kirby,
*Coprimary decomposition of Artinian modules*, J. London Math. Soc. (2)**6**(1973), 571–576.MathSciNetMATHCrossRefGoogle Scholar - 4.D. Kirby,
*Artinian modules and Hilbert polynomials*, Quart. J. Math. Oxford (2)**24**(1973), 47–57.MathSciNetMATHCrossRefGoogle Scholar - 5.I. G. Macdonald,
*Secondary representation of modules over a commutative ring*, Symposia Mathematica**11**(1973), 23–43.Google Scholar - 6.I. G. Macdonald and R. Y. Sharp,
*An elementary proof of the non-vanishing of certain local cohomology modules*, Quart. J. Math. Oxford (2)**23**(1972), 197–204.MathSciNetMATHCrossRefGoogle Scholar - 7.E. Matlis,
*Infective modules over Noetherian rings*, Pacific J. Math.**8**(1958), 511–528.MathSciNetMATHGoogle Scholar - 8.E. Matlis,
*Modules with descending chain condition*, Trans. AMS**97**(1960), 495–508.MathSciNetCrossRefGoogle Scholar - 9.S. McAdam, “Asymptotic prime divisors,” Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983.Google Scholar
- 10.S. McAdam and P. Eakin,
*The asymptotic Ass*, J. Algebra**61**(1979), 71–81.MathSciNetMATHCrossRefGoogle Scholar - 11.D. G. Northcott, “AJI Introduction to Homological Algebra,” Cambridge University Press, 1960.Google Scholar
- 12.D. G. Northcott,
*Generalized Koszul complexes and Artinian modules*, Quart. J. Math. Oxford (2)**23**(1972), 289–297.MathSciNetMATHCrossRefGoogle Scholar - 13.R. Y. Sharp,
*Asymptotic behaviour of certain sets of attached prime ideals*, J. London Math. Soc. (2)**34**(1986), 212–218.MathSciNetMATHCrossRefGoogle 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