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Order ideals and a generalized Krull height theorem

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Abstract.

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N*(x) of an element x ∈ N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x ∈ mN. We show that the inequality even holds if the hypothesis becomes true after first extending scalars to some local domain and then factoring out torsion. We give other conditions in terms of residual intersections and integral closures of modules. We derive information about order ideals that leads to bounds on the heights of trace ideals of modules—even in circumstances where we do not have the expected bounds for the heights of the order ideals!

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Authors and Affiliations

  1. Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA, 94720, USA

    David Eisenbud

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Craig Huneke

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Bernd Ulrich

Authors
  1. David Eisenbud
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  2. Craig Huneke
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  3. Bernd Ulrich
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Corresponding author

Correspondence to Bernd Ulrich.

Additional information

All three authors were partially supported by the NSF.

Revised version: 24 November 2003

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Eisenbud, D., Huneke, C. & Ulrich, B. Order ideals and a generalized Krull height theorem. Math. Ann. 330, 417–439 (2004). https://doi.org/10.1007/s00208-004-0513-6

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  • DOI: https://doi.org/10.1007/s00208-004-0513-6

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