Skip to main content
SpringerLink
Log in

Cohen-Macaulay properties of the Koszul homology

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

The Koszul homology H.(y,N) which is constructed with respect to a sequencey and a maximal Cohen-Macaulay (CM) module N over a local CM ring A admitting a canonical module ωA will be compared with the Koszul homology H. (y, HomA(N, ωA)).

If R:=A/I with I=(y) is a CM ring, then the canonical module ωR of R exists and we will mainly show the existence of a natural isomorphism H. (y, HomA(N, ωA)≃HomR(H. (y, N), ωR, if H. (y, N) is a maximal CM module over R. This generalizes a result of Herzog in [2]. Using this isomorphism we are able to compute the graded canonical module of the graded ring grI (A) in a certain case.

In the last part of this paper we define a polynominal UN (y,x) associated with the Koszul homology H. (y, N) similar to Huneke in [7]. Huneke proved that Hj (y, N) is CM, if j<minding UN (y,x). We will proceed to show that Hj (y, N) is CM if j>deg UN (y,x).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cartan, H., Eilenberg, S.: Homological Algebra, Princeton University Press, 1956

  2. Herzog, J.: Komplexe, Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift, Universität Regensburg, 1974

  3. Herzog, J., Kunz, E.: Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, 238, Heidelberg, Springer, 1971

    Google Scholar 

  4. Herzog, J., Simis, A., Vasconcelos, W.V.: Koszul Homology and Blowing-Up Rings in Commutative Algebra, Proceedings of the Trento Conference, Lecture Notes in Pure and Applied Mathematics, 84, New York, Marcel Dekker, 79–169 (1983)

    Google Scholar 

  5. Herzog, J., Simis, A., Vasconcelos, W.V.: On the canonical module of the Rees algebra and the associated graded ring of an ideal, to appear in Journal of Algebra

  6. Huneke, C: Strongly Cohen-Macaulay schemes and residual intersections, Transactions of the American Mathematical Society 277, 739–763 (1983)

    Google Scholar 

  7. Huneke, C: Numerical invariants of liaison classes, Invent. Math. 75, 301–325 (1984)

    Google Scholar 

  8. Sanders, H.: Approximationskomplexe und Koszulhomologie, Dissertationsschrift, Universität Essen, 1985

Download references

Author information

Authors and Affiliations

  1. FB 6/ Mathematik, Universität GHS Essen, Universitätsstr. 3, D-4300, Essen 1

    Herbert Sanders

Authors
  1. Herbert Sanders
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The material presented in this paper constitutes part of the author's thesis submitted to Universität Essen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sanders, H. Cohen-Macaulay properties of the Koszul homology. Manuscripta Math 55, 343–357 (1986). https://doi.org/10.1007/BF01186650

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01186650

Keywords

Search

Navigation