Flat Covers and Flat Cotorsion Modules

Abstract

It is not known whether modules over an arbitrary ring have flat covers, however for certain modules over commutative noetherian rings they can be shown to exist.These covers, in turn, have an interesting connection with flat cotorsion modules.A complete description of flat cotorsion modules analogous to that given by Harrison for torsion free, cotorsion abelian groups will be given.In this article, R will denote a commutative noetherian ring.Notation.If R is a local ring, m(R) will denote its maximal ideal.For p g Spec(), Rp will denote the completion of the local ring Rp, and for any R-module, M will denote the (separated) completion of the Rp module Mp with the m( )-adic topology.k(p) denotes the residue field of R (= Rp/m{Rp)).E(M) will be an injective envelope of M. If R is local, M" denotes the Maths dual, Hom(M, E{R/m(R))), of M, and M is said to be reflexive if Mvv = M naturally.For a set X, Mx is the module of all functions X -* M and M(X) the submodule of those functions with finite support.Soc(M) will denote the socle of M.