On modules of finite projective dimension

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 $p\in I$ 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 $R$.