On finite weak and injective dimension

Abstract

An extension is given of the theorem relating projective dimen- sion to depth for a finitely generated module of finite projective dimension over a commutative Noetherian local ring. This extension is dualized to relate injective dimension to a concept of codepth when the injective dimension is known to be finite. 1. Introduction and notation. A classical result on the projective dimension of modules over a commutative Noetherian local ring says that if the module is finitely generated and of finite projective dimension, then that dimension is the difference between the depth of the ring and that of the module (1, Theorem 3.7). We extend this result by changing the context to weak dimension. We then dualize to injective dimension, and the result we get seems to be known only in the special case of finitely generated modules. For the duration of the paper, all rings are commutative Noetherian with unit, and all modules are unitary. If M is an R-module, then we denote the weak, projective, and injective dimensions of M over R by w.d.RM, p.d.RM, and i.d.RM respectively, and we use ER(M) to denote the injective envelope of M over R. If R is local with residue field k, then we use Mv to stand for the Matlis dual HomR(M, ER(k)) of M. If M is any R-module, and I is an ideal in R, then we define the depth of M over I as the lowest degree in which ExtR (R/I, M) is nonzero, and the codepth of M over I as the lowest degree in which TorR(R/I, M) is nonzero; we denote these by depth, M and codepth, M respectively. If ExtR(R/I, M) is 0 in all degrees, we say depth, M = ox, and likewise for codepth. We adopt the convention that if we ask for the largest dimension in which a positively graded object is nonzero, and that object is 0 in all degrees, then the requested dimension is - ox. Thus, for example, the projective dimension of 0 over R will be - oo, since it is the supremum over all R-modules A of the largest dimension in which ExtR (0, A) is nonzero. The author wishes to thank Professor Everett L. Lady for referring him to the article (5) which appears to be the origin of the spectral sequence needed here, and Professor Melvin Hochster for pointing out nonhomological proofs of two of the following corollaries.