Modules of generalized fractions and balanced big Cohen-Macaulay modules

Summary

Introduction

It is well known that, for a Gorenstein ring A, the total ring of fractions of A provides the injective envelope of A. One of the motivations behind the work which led to our construction of modules of generalized fractions (which was outlined in a lecture at the Symposium, is reviewed in §2 below, and is described in detail in [9]) was a desire to find a similarly satisfactory description of the terms Eⁱ(A) for i > O in the minimal injective resolution for the Gorenstein ring A.

At the end of this paper it is shown that generalized fractions do provide such descriptions: whenever R is a commutative ring (with identity), M is an R–module, n is a positive integer and U is what is called a triangular subset of Rⁿ, a module U⁻ⁿM of generalized fractions may be constructed; in the case of a Gorenstein ring A, the set U_n of all poor A–sequences of length n forms a triangular subset of Aⁿ (we say that a sequence a₁, …, a_n of elements of A forms a poor A–sequence if

((Aa₁ + … + Aa_i−1) : a_i) = (Aa₁ + … + Aa_i−1)

for all i 1, …, n), and the module of generalized fractions U⁻ⁿ_nA turns out to be isomorphic to Eⁿ⁻¹ (A).