COMPLETE FLAT MODULES

Let R be a commutative and noetherian ring. It is known tht if R is lo cal with maximal ideal M and F is a flat R-module, then the Hausdorff completion ($) over cap F of F with the M-adic topology is flat. We sh ow that if we assume that the Krull dimension of R is finite, then for any ideal I subset of 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 c omplete 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 defin ed by any non-zero ideal of R/I. As a tool in proving the above, we wi ll show that when R has finite Krull dimension, the I-adic Hausdorff c ompletion of a minimal pure injective resolution of a flat module F is a minimal pure injective resolution of its completion F. Then it wil l be shown that hat 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.