Flatness and Completion for Infinitely Generated Modules over
Noetherian Rings

Amnon Yekutieli, BGU

Abstract: Suppose A is a commutative ring, and \a is an ideal in it. This
talk is about properties of the \a-adic completion operation.

When A is noetherian, and we only consider finitely generated modules, the
situation is well-known to all of us: completion is an exact functor, and is
the same as tensoring with the completion of A.

I will begin the talk with examples of the strange behavior of completion when
the ring A is not noetherian, or when the modules are infinitely generated.

Now assume A is noetherian. I will describe the structure of the \a-adic
completion of a free A-module (of infinite rank of course). This completion is
best described in analytic terms, as a "module of decaying functions". It is a
theorem that a module of decaying functions is flat and complete. Any module
isomorphic to a module of decaying functions is called "\a-adically free".

Finally I will discuss the case of a complete noetherian local ring A with
maximal ideal \m. Here every flat \m-adically complete A-module is \m-adically
free. I will state a result about flat \m-adically complete sheaves of A-modules
on a topological space. This result is needed for deformation quantization, and
is one of the reasons for my interest in completions.

The material comes from my paper arXiv:0902.4378 (to appear in Comm. Algebra).

(October 2010)