On the projective dimension and the unmixed part of three cubics

Details On the projective dimension and the unmixed part of three cubics On the projective dimension and the unmixed part of three cubics

2006-11-19

arxiv.org/abs/math/0611552v1

Journal of Algebra 316 (2007), 715-734

Mathematics

Commutative Algebra

23 pages; to appear in Journal of Algebra

Scientific paper

10.1016/j.jalgebra.2006.11.018

Let $R$ be a polynomial ring over a field in an unspecified number of variables. We prove that if $J \subset R$ is an ideal generated by three cubic forms, and the unmixed part of $J$ contains a quadric, then the projective dimension of $R/J$ is at most 4. To this end, we show that if $K \subset R$ is a three-generated ideal of height two and $L \subset R$ an ideal linked to the unmixed part of $K$, then the projective dimension of $R/K$ is bounded above by the projective dimension of $R/L$ plus one.

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