Ideals with Larger Projective Dimension and Regularity

Beder, Jesse S.; McCullough, Jason; Núñez-Betancourt, Luis; Seceleanu, Alexandra; Snapp, Bart; Stone, Branden

doi:10.48550/arxiv.1101.3368

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2012

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“…B. Engheta showed the existence of a bound on the projective dimension for three cubic forms (see [5], [6]); he has shown that pd(R/I) ≤ 36 (even though in all known examples of three cubics pd(R/I) ≤ 5). In general, the projective dimension can grow relatively fast as one increases the number of generators and the degrees (see [1], [3], and [9] for specific examples). In this paper the authors prove Conjecture 1 for the case when all of the d i are at most 2 and n is arbitrary.…”

Section: Introductionmentioning

confidence: 99%

Ideals Generated by Quadratic Polynomials

Ananyan

Hochster

2012

Mathematical Research Letters

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Let R be a polynomial ring in N variables over an arbitrary field K and let I be an ideal of R generated by n polynomials of degree at most 2. We show that there is a bound on the projective dimension of R/I that depends only on n, and not on N . The proof depends on showing that if K is infinite and n is a positive integer, there exists a positive integer C(n), independent of N , such that any n forms of degree at most 2 in R are contained in a subring of R generated over K by at most t ≤ C(n) forms G1, . . . , Gt of degree 1 or 2 such that G1, . . . , Gt is a regular sequence in R. C(n) is asymptotic to 2n2n .

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