The homological grade of a module over a commutative ring

Alshaniafi, Yousef Saleh (1990) The homological grade of a module over a commutative ring. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

Abstract

Throughout the thesis `a ring' means a `commutative ring with identity', and `R-module' means a `unitary R-module'. The category of all R-modules is denoted by Mod-R and the annihilator of an R-module N is denoted by ann_RN. For an ideal A of a ring R and an R-module M, David Kirby and Hefzi A. Mehran, in a recent paper, define the homological grade of M in A, hgr_R(A;M), to be inf(n PROB*LEMExtⁿ/_R(R/A,M) PROB*LEM 0). In the thesis, we consider the homological grade of M in A as the homological grade of M in the cyclic R-module R/A. We denote it by hgr_R(R/A;M) and extend this definition to any arbitrary R-module N replacing R/A. The notion of an M-sequence in the work of D. Kirby and H.A. Mehran is adapted for use in new situations. For an R-module M and an exact sequence 0 → N_1 → N_{2 → N}_3 → 0 of R-modules with N₂ finitely generated, it is proved that, hgr_R(R/ann_R N₂; M) = hgr_R(N₂; M) = min(hgr_R(N₁; M), hgr_R(N₃; M)) = the supremum of lengths of M-sequences in ann_RN₂. For R-modules N, M where N is finitely generated and for r ≤ hgr_R(N; M), it is shown that Ext^r/_R(N, M) PROB*LEM Hom_R(N, ker d^r), where ker d^r is the r^th cosyzygy in the minimal injective resolution of M. If, in addition, ann_RN is finitely generated, a relation between hgr_R(N;M), proj dim_RM and hgr_R(N; R) is given. Dual notions of the homological grade and an M-sequence is given and most of the results concerning the homological grade are dualized. For an artinian R-module M and a finitely generated R-module N, it is proved that for every n ≤ cohgr_R(N; M) there exists a regular M-consequence in ann_RN of length n. Finally, if (R,PROB*LEM) is a local (noetherian) ring, it is shown that for every r ≤ hgr_R(R/PROB*LEM;R) the contravariant left exact functor Ext^r/_R (-, Hom_R(ker d^r, E(R/PROB*LEM))) is an exact functor in the category of all R-modules of finite length. And if, in addition, R is complete and n = hgr_R(R/PROB*LEM;R) then, for a finitely generated R-module N PROB*LEM 0 of finite injective dimension, the R-module Hom_R(PROB*LEM_n,N) is finitely generated with finite projective dimension equal to n - hgr_R(R/PROB*LEM;N), where PROB*LEM_n = Hom_R(ker d^n, E(R/PROB*LEM)).