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 �crit : > 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. -- Jo�l 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. -- Jo�l Riou - Joel.Riou@ens.fr