Structure of the flat covers of Artinian modules

Abstract

The aim of the paper is to obtain information about the flat coversand minimal flat resolutions of Artinian modules over a Noetherian ring.Let R be a commutative Noetherian ring and let A be anArtinian R-module. We prove that the flat cover of A is of the formprod_{pin {rm Att}_R(A)} T_p, where T_p is the completion of afree R_p-module. Also, we construct a minimal flat resolution forR/xR-module 0:_Ax from a given minimal flat resolution of A,when x is a non-unit and non-zero divisor of R such that A=xA.This result leads to a description of the structure of a minimal flatresolution for {rm H}^n_{underline m}(R), nth local cohomologymodule of R with respect to the ideal underline m,over a local Cohen-Macaulay ring (R, underline m) of dimension n.