On Modules of Finite Projective Dimension

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 non-zero-divisor, if I is an ideal of finite projective dimension over R and p ∈ I or p is a non-zero-divisor on R/I, then every minimal generator of I is a non-zero-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 non-zero-divisor in R. In this note we would like to consider two aspects of finitely generated modules of finite projective dimension over any local ring: embeddability and grade of order ideals of minimal generators of its syzygies (minimal). In regard to embeddability Auslander and Buchweitz ([A-B]) proved that any finitely generated module on a Gorenstein local ring can be embedded in a module of finite projective dimension such that the cokernel is Cohen-Macaulay. This result has several applications in solving homological questions and conjectures in commutative algebra including Serre’s χi-conjectures and its generalizations on intersection multiplicity ([D3]). For all these conjectures one is usually concerned with finitely generated modules M which are of finite projective dimension over a local ring R, but are not so over R/xR, x being a non-zero-divisor in the annihilator of M in R. However, a similar result has been absent for non-Gorenstein local rings until now. In this note we prove the following with respect to embeddability for modules of finite projective dimension: AMS Subject Classification: Primary 13D02, 13D22; Secondary 13C15, 13D25, 13H05