Gorenstein modules

Cousin complexes and generalized fractions

“…Not only can they be used to characterize Cohen-Macaulay rings (as was mentioned in the introduction) and Gorenstein rings, but they also reflect some of the structure of the minimal injective resolution of a Gorenstein ring that is brought out in Bass's Fundamental Theorem [1]: see [6, (5.5)]. Their use in the theory of Gorenstein modules [7] means that they have connections with the canonical modules of Herzog and Kunz [3]. They can be used to characterize the (commutative Noetherian) rings that satisfy the conditions (S k ) [5, p. 125]: see [8].…”

Commutative Extensions by Canonical Modules are Gorenstein Rings

“…For further properties of canonical modules we refer to Sharp [8], Herzog and Kunz [6], Foxby [4] and Fossum, Griffith and Reiten [3] although the only results which are really needed are: (c) If Af is a canonical /1-module and if a is a regular element in A, then M/aM is a canonical A/aA-module. Also we note that when B^-A is a surjective ring homomorphism, when B is Gorenstein and when A is Cohen-Macaulay then the spectral sequence (Cartan and Eilenberg [2]) with El"=ExtA(X,ExtQB(A,B)) and abutment ExtB(X,B) degenerates to natural isomorphisms ExtA(X,Ext%(A, B))^ExtB+d(X, B) for all /1-modules X, for all integers p^.0 and for d=dim B-dim A.…”

“…If B is a Gorenstein local ring (see Bass [1]) and if B-*A is a surjection then A is a Cohen-Macaulay ring if and only if Ext'B(A, B)=0 for /'#dim B-dim A. If A is CohenMacaulay and if d=dim B-dim A, then ExtB(A, B) is a canonical Amodule (see Grothendieck [5] and Sharp [8]). Furthermore A is Gorenstein if and only if A^ExtB(A, B).…”

“…A more geometric term is module of dualizing differentials, which is very suggestive terminology but also rather cumbersome. Sharp [8] uses the term Gorenstein module of rank one. )…”

Comparison of graded and ungraded Cousin complexes

“…When L is non-zero and finitely generated, it is known that L is Cohen-Macaulay if and only if C(L)' is exact (see [7,Theorem (2.4…”