On an application of the Fitting invariants

Abstract

Suppose that R is any commutative ring with unity element and that M is a finitely generated R-module. The usual way in which one investigates the extent to which M deviates from being free is to map a free module of appropriate rank onto M and to examine the module of relations. This point of view was exploited by H. Fitting in [4] when he defined an ascending sequence of invariant ideals for M which generalized the classical invariant factors for a finite abelian group. For our purposes, the smallest ideal in this sequence is the one of greatest interest. This is the invariant which generalizes the order of a finite abelian group. Let us denote this invariant by F(M) (see Section 2 for definitions). The most important formal properties of the function F may be summarized as follows. If S is a multiplicatively closed set in R not containing 0, then F(M), =F(M,) where the subscript S denotes extension to the ring of quotients R, . If 0 + A -+ B + M + 0 is a short exact sequence of finitely generated R-modules and M satisfies the conditions (i) the annihilator of M contains a nondivisor of zero and (ii) the homological dimension of M is at most one, then F(A)F(M) = F(B). This is analogous to the fact that the order of a finite group divided by the order of an invariant subgroup is equal to the order of the quotient group. Finally, if M satisfies (i) and (ii) above and R is a noetherian ring, then F(M) is an invertible ideal of R. This generalizes the fact that the order of a finite abelian group is a nonzero integer. The details of these assertions are treated in Section 2. Now the multiplicative property of F on short exact sequences whose cokernel satisfies certain conditions shows that F bears a formal similarity to the notion of the rank of a module. This similarity turns out to be strong enough to define an Euler characteristic for long exact sequences