A family of ideals with few generators in low degree and large projective dimension

Author:
Jason McCullough

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2017-2023

MSC (2010):
Primary 13D05; Secondary 13D02

Published electronically:
December 15, 2010

MathSciNet review:
2775379

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Abstract | References | Similar Articles | Additional Information

Abstract: Stillman posed a question as to whether the projective dimension of a homogeneous ideal in a polynomial ring over a field can be bounded by some formula depending only on the number and degrees of the minimal generators of . More recently, motivated by work on local cohomology modules in characteristic , Zhang asked more specifically if the projective dimension of is bounded by the sum of the degrees of the generators. We define a family of homogeneous ideals in a polynomial ring over a field of arbitrary characteristic whose projective dimension grows exponentially if the number and degrees of the generators are allowed to grow linearly. We therefore answer Zhang's question in the negative and provide a lower bound to any answer to Stillman's question. We also describe some explicit counterexamples to Zhang's question including an ideal generated by 7 quadrics with projective dimension 15.

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Additional Information

**Jason McCullough**

Affiliation:
Department of Mathematics, University of California, Riverside, 202 Surge Hall, 900 University Avenue, Riverside, California 92521

Email:
jmccullo@math.ucr.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10792-X

Keywords:
Projective dimension,
homogeneous ideal,
polynomial ring

Received by editor(s):
June 10, 2010

Published electronically:
December 15, 2010

Communicated by:
Irena Peeva

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.