Counterexamples to the Eisenbud–Goto regularity conjecture

McCullough, Jason; Peeva, Irena

doi:10.1090/jams/891

Counterexamples to the Eisenbud–Goto regularity conjecture
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by Jason McCullough and Irena Peeva

J. Amer. Math. Soc. 31 (2018), 473-496

DOI: https://doi.org/10.1090/jams/891

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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