From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5251 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Question on exact sequence Date: Thu, 12 Nov 2009 14:58:52 -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 1258079423 20030 80.91.229.12 (13 Nov 2009 02:30:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 02:30:23 +0000 (UTC) To: Categories list Original-X-From: categories@mta.ca Fri Nov 13 03:30:15 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 1N8lvT-0004BS-04 for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 03:30:15 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8lRJ-0005lY-Gi for categories-list@mta.ca; Thu, 12 Nov 2009 21:59:05 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5251 Archived-At: I have finally worked a way of reducing my observation to an instance of the snake lemma. Begin with f: A ---> B and g: C ---> D and form the diagram (I have made the background dots so that no demented mailer will collapse the spaces ................................................... ...............(1)................................. ...............(f)............(-f 1)............... 0 -----> A ----------> A + B ----------> B -----> 0 .........|...............|...............|......... .........|...............|...............|......... .........|...............|(0 1).........|......... .........|f..............|(gf 0).........|-g....... .........|...............|...............|......... .........|...............|...............|......... .........|.....(1).......|...............|......... .........v.....(g).......v....(-g 1).....v......... 0 -----> B ----------> B + C ----------> C -----> 0 Exactness of the rows is easily checked, as is the fact that the kernel of the middle column is ker(gf) + 0. Dually, its cokernel is essentially cok(gf). The snake lemma associated to this is the exact sequence I wrote originally. Thanks to all who responded. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]