From: jriou@clipper.ens.fr (Joel Riou)
Subject: Re: Finitely generated, flat => projective?
Date: 5 Feb 2001 12:03:05 GMT
Newsgroups: sci.math
Summary: Finitely generated but not finitely presented flat modules

jhonglat@my-deja.com, dans le message (sci.math:390995), a écrit :
> Are there finitely generated flat modules that aren't projective?

A projective module M that is finitely generated must be finitely
presented and flat, and a finitely presented flat module is projective, so
the problem is equivalent to finding a finitely generated flat module that
is not finitely presented. Then the ring A mustn't be Noetherian.

-- 
Joël Riou - Joel.Riou@ens.fr
==============================================================================

From: jriou@clipper.ens.fr (Joel Riou)
Subject: Re: Finitely generated, flat => projective?
Date: 5 Feb 2001 13:24:50 GMT
Newsgroups: sci.math

I think I got such a counterexample.

Let A be the ring F_2^\N where F_2 is the field with 2 elements.
In this ring, it is easy to see that any ideal of finite type is generated
by an idempotent. It implies that for any ideal I of finite type, A/I is
projective, and then A/I is flat. So, any A/J, where J is any
ideal, is flat, because A/J is the inductive limit of the filtrant system
{A/I, I of finite type}. So any A-module generated by one element is flat.

A is not noetherian, so there exists an ideal J that is not of finite
type. A/J is finitely generated and flat. But, if A/J was projective, the
following exact sequence would split :
0 -> J -> A -> A/J -> 0 It would mean that J is generated by one element,
contradiction.

So A/J is finitely generated, plat and not projective.

-- 
Joël Riou - Joel.Riou@ens.fr