# Adic Topology and Completion

## Abstract

In the first three sections of this chapter we summarize the basic results on the subject, include results for which there is no reference in the literature and complete the picture with some new observations. A few of these facts hold in full generality; but most of them require that the adic topology is taken with respect to a finitely generated ideal. The case when the ring is Noetherian is easier to handle, though far from being obvious when dealing with infinitely generated modules. In Sect. 2.4 we provide an extension of Bartijn’s result stating that the adic completion of a flat module over a Noetherian ring is flat. Section 2.5 contains the first information on the left-derived functors of the adic completion functor with respect to an ideal \(\mathfrak {a}\). Writing them, we have been astonished by how much can be said assuming only the hypothesis that the ideal \(\mathfrak {a}\) is finitely generated. As usual there are finer results when the ring is Noetherian, and it is helpful in Part II to see that some of these hold in greater generality and in a more general setting. In Sect. 2.6 we introduce a notion of relative flatness, which is helpful for the study of local homology. Section 2.7 contains some remarks on torsion modules and relatively injective modules.