On the projective dimension and the unmixed part of three cubics

Abstract

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.

23 pages; to appear in Journal of Algebra

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