Nagoya Mathematical Journal

On modules of finite projective dimension

S. P. Dutta

Full-text: Open access

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 pI 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.

Article information

Source
Nagoya Math. J., Volume 219 (2015), 87-111.

Dates
Received: 4 March 2013
Revised: 5 March 2014
Accepted: 13 August 2014
First available in Project Euclid: 20 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1445345518

Digital Object Identifier
doi:10.1215/00277630-3140702

Mathematical Reviews number (MathSciNet)
MR3413574

Zentralblatt MATH identifier
1007.13006

Subjects
Primary: 13D02: Syzygies, resolutions, complexes 13D22: Homological conjectures (intersection theorems)
Secondary: 13C15: Dimension theory, depth, related rings (catenary, etc.) 13D25 13H05: Regular local rings

Keywords
projective dimension grade order ideal syzygy regular local ring almost-Cohen–Macaulay algebra

Citation

Dutta, S. P. On modules of finite projective dimension. Nagoya Math. J. 219 (2015), 87--111. doi:10.1215/00277630-3140702. https://projecteuclid.org/euclid.nmj/1445345518


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