Advertisement

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

  • R. Y. Sharp
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 15)

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.

Keywords

Local Ring Maximal Ideal Commutative Ring Homomorphic Image Noetherian Ring 
These 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Brodmann, Asymptotic stability of Ass(M/I n M), Proc. AMS 74 (1979), 16–18.MathSciNetGoogle Scholar
  2. 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. 3.
    D. Kirby, Coprimary decomposition of Artinian modules, J. London Math. Soc. (2) 6 (1973), 571–576.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    D. Kirby, Artinian modules and Hilbert polynomials, Quart. J. Math. Oxford (2) 24 (1973), 47–57.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    I. G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica 11 (1973), 23–43.Google Scholar
  6. 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. 7.
    E. Matlis, Infective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528.MathSciNetMATHGoogle Scholar
  8. 8.
    E. Matlis, Modules with descending chain condition, Trans. AMS 97 (1960), 495–508.MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. McAdam, “Asymptotic prime divisors,” Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983.Google Scholar
  10. 10.
    S. McAdam and P. Eakin, The asymptotic Ass, J. Algebra 61 (1979), 71–81.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    D. G. Northcott, “AJI Introduction to Homological Algebra,” Cambridge University Press, 1960.Google Scholar
  12. 12.
    D. G. Northcott, Generalized Koszul complexes and Artinian modules, Quart. J. Math. Oxford (2) 23 (1972), 289–297.MathSciNetMATHCrossRefGoogle Scholar
  13. 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. 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. 15.
    O. Zariski and P. Samuel, “Commutative Algebra, Vol. II,” Graduate Texts in Mathematics, vol. 29, Springer-Verlag, Berlin, 1975.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • R. Y. Sharp
    • 1
  1. 1.Department of Pure MathematicsUniversity of SheffieldSheffieldEngland

Personalised recommendations