A family of ideals with few generators in low degree and large projective dimension
Author:
Jason McCullough
Journal:
Proc. Amer. Math. Soc. 139 (2011), 20172023
MSC (2010):
Primary 13D05; Secondary 13D02
Published electronically:
December 15, 2010
MathSciNet review:
2775379
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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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 Lindsay Burch, A note on the homology of ideals generated by three elements in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 949952. MR 0230718 (37:6278)
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 Yi Zhang, A Property of Local Cohomology Modules of Polynomial Rings, preprint, arXiv:1001.3363v1 [math.AC], January 2010.
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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:
http://dx.doi.org/10.1090/S00029939201010792X
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.
