Order ideals, annihilator ideals and pathological behavior

Abstract

This article establishes a concrete relation between order ideals of minimal generators and annihilator ideals. For a regular local ring $R$ and ideal $I$ the authors construct an $R$-module $M$ with minimal generator having $I$ as order ideal. Further, it is shown that most variability in ideal–theoretic behavior of such order ideals is exhibited by modules of projective dimension~one. The authors ``introduce'' the concept of $*$-orthogonality and use their syzygy theorem to show constraints on the size and height of a $*$-orthogonal set in a given finitely generated non-free module. The paper contains an application of the theory of order ideals to the binomial behavior of syzygy rank.