Complete flat modules

Abstract

Let R be a commutative and noetherian ring. It is known tht if R is local with maximal ideal M and F is a flat R-module, then the Hausdorff completion F of F with the M-adic topology is flat. We show that if we assume that the Krull dimension of R is finite, then for any ideal I C R, the Hausdorff completion F* of a flat module F with the I-adic topology is flat. Furthermore, for a flat module F over such R, there is a largest ideal I such that F is Hausdorff and complete with the I-adic topology. For this I, the flat R/I-module F/IF will not be Hausdorff and complete with respect to the topology defined by any non-zero ideal of R/I. As a tool in proving the above, we will show that when R has finite Krull dimension, the I-adic Hausdorff completion of a minimal pure injective resolution of a flat module F is a minimal pure injective resolution of its completion F*. Then it will be shown that flat modules behave like finitely generated modules in the sense that on F* the I-adic and the completion topologies coincide, so F* is I-adically complete.