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On modules of finite projective dimension

Part of: Local rings and semilocal rings Theory of modules and ideals Commutative algebra: Homological methods

Published online by Cambridge University Press:  11 January 2016

S. P. Dutta
S. P. Dutta*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, s-dutta@illinois.edu
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Abstract

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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 𝜖 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.

MSC classification

Secondary: 13D02: Syzygies, resolutions, complexes 13D22: Homological conjectures (intersection theorems) 13C15: Dimension theory, depth, related rings (catenary, etc.) 13H05: Regular local rings
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 87 - 111
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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