Invariant factors and two criteria for projectivity of modules

Abstract

Let 7? be a commutative ring, and E an 7?-module.For each ^1, we define the ideal ap(E) to be the annihilator of the pth exterior product of E (denoted by A" E), and ap(E) is called the pth invariant factor of E. If is a finitely generated 7?-module, then Ap = 0 (or ,,() =7?) for all but a finite number of p, and the largest integer p for which l\p ^0 is called the exterior rank of (denoted by ext rank ).The fact that the invariant factors of a module give some insight into the structure of the module(2) is shown in 1, where we prove Theorem A. Let Rbe a local domain, and E a finitely generated R-module.If ap(E) is a principal ideal for all p, then = @pR/ap(E) (direct sum).By investigating some of the properties of ,(), where s = ext rank , we obtain, in 3, proofs of the following two theorems: Theorem B. Let R be a noetherian ring(3), and E a finitely generated Rmodule.If the dimension of the R/m vector space /m 75 the same for each maximal ideal m in R, and if r denotes the radical of R, then E/xE is R/xprojective.If R/x is a domain, then this sufficient condition is also necessary.Theorem C. If R is a local domain, E a finitely generated R-module, and p a prime ideal in R such that /p is R/p-free and Ep is R^-free, then E is Rfree.In 3 we consider another ideal, (E), associated with an 7?-module , and indicate its connection with the invariant factors p().We also indicate how (E) may be used to give an alternate proof to Theorem C.1. Proof of Theorem A. In this section, we are assuming that 7? is a local ring, and that is a finitely generated 7?-module.We recall that if xi, , xn is a basis for /m over R/m (where m is the maximal ideal of 7?), then xi, , x" is a minimal generating set of over 7? if x, in represents the coset of XiEE/mE.Moreover, since Ap /m is the same whether con-Received by the editors October 11, 1961.(') This work was one with the partial support of NSF Grant G-5864.(2) S. Kertzner is systematically studying the invariant factors of modules over regular local rings to see the extent to which they determine the structure of such modules.(3) Throughout this note our rings will be commutative with identity, and the radical will be the intersection of the maximal ideals.Modules will be unitary ; if E is an i?-module and p a prime ideal, we denote by Ap _ r^ " respectively the pth exterior product, the ring of quotients of R with respect to p, and the modules of quotients of E with respect to p.