### Abstract

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p ∈ I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

Original language | English (US) |
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Pages (from-to) | 87-111 |

Number of pages | 25 |

Journal | Nagoya Mathematical Journal |

Volume | 219 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)