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)