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Counterexamples to the Eisenbud-Goto regularity conjecture

Authors: Jason McCullough and Irena Peeva
Journal: J. Amer. Math. Soc. 31 (2018), 473-496
MSC (2010): Primary 13D02
Published electronically: November 10, 2017
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Abstract: Our main theorem shows that the regularity of nondegenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field $ k$. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal $ I$, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of $ I$. The method is also related to producing bounds in the spirit of Stillman's Conjecture, recently solved by Ananyan and Hochster.

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Additional Information

Jason McCullough
Affiliation: Mathematics Department, Iowa State University, Ames, Iowa 50011

Irena Peeva
Affiliation: Mathematics Department, Cornell University, Ithaca, New York 14853

Keywords: Syzygies, free resolutions, Castelnuovo--Mumford regularity
Received by editor(s): September 21, 2016
Received by editor(s) in revised form: August 24, 2017
Published electronically: November 10, 2017
Additional Notes: The second author was partially supported by NSF grants DMS-1406062 and DMS-1702125.
Article copyright: © Copyright 2017 American Mathematical Society

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