From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5221 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Question on exact sequence Date: Mon, 9 Nov 2009 17:57:03 -0500 (EST) Message-ID: Reply-To: Michael Barr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1257813463 4716 80.91.229.12 (10 Nov 2009 00:37:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 10 Nov 2009 00:37:43 +0000 (UTC) To: Categories list Original-X-From: categories@mta.ca Tue Nov 10 01:37:36 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N7ejc-0001qo-MQ for gsmc-categories@m.gmane.org; Tue, 10 Nov 2009 01:37:24 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N7eJT-0006iE-0K for categories-list@mta.ca; Mon, 09 Nov 2009 20:10:23 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5221 Archived-At: I have recently discovered a curious fact about abelian categories. First, let me briefly describe the well-known snake lemma. If we have a commutative diagram with exact rows (there are variations without the 0 at the left end of the top and without the 0 at the right end of the bottom, but here is the strongest form) 0 ---> A ----> B ----> C ----> 0 | | | | | | |f |g |h | | | v v v 0 ---> A' ---> B' ---> C' ---> 0 then there is an exact sequence 0 --> ker f --> ker g --> ker h --> cok f --> cok g --> cok h --> 0 The curious discovery is that you have any pair of composable maps f: A --> B and h: B --> C and you form the diagram (with g = hf) 1 f A ----> A ----> B | | | | | | |f |g |h | | | v v v B ----> C ----> C h 1 you get the same exact sequence. So I would imagine that there must be a "master theorem" of which these are two cases. Does anyone know what it says? The connecting map here is just the inclusion of ker h into B followed by the projection on cok f. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]