Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 421–435 | Cite as

Regularity of prime ideals

  • Giulio Caviglia
  • Marc Chardin
  • Jason McCullough
  • Irena PeevaEmail author
  • Matteo Varbaro


We answer several natural questions which arise from a recent paper of McCullough and Peeva providing counterexamples to the Eisenbud–Goto Regularity Conjecture. We give counterexamples using Rees algebras, and also construct counterexamples that do not rely on the Mayr–Meyer construction. Furthermore, examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given. Using a result of Ananyan-Hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone.


Syzygies Free resolutions Castelnuovo–Mumford Regularity 

Mathematics Subject Classification

Primary 13D02 



We are very grateful to David Eisenbud for useful discussions.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Giulio Caviglia
    • 1
  • Marc Chardin
    • 2
  • Jason McCullough
    • 3
  • Irena Peeva
    • 4
    Email author
  • Matteo Varbaro
    • 5
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Institut de mathématiques de JussieuCNRS & Sorbonne UniversitéParisFrance
  3. 3.Mathematics DepartmentIowa State UniversityAmesUSA
  4. 4.Mathematics DepartmentCornell UniversityIthacaUSA
  5. 5.Dipartimento di MatematicaUniversità degli Studi di GenovaGenoaItaly

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