From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5223 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Tue, 10 Nov 2009 15:44:57 +0100 Message-ID: Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1257952973 27216 80.91.229.12 (11 Nov 2009 15:22:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 11 Nov 2009 15:22:53 +0000 (UTC) To: Michael Barr , categories@mta.ca Original-X-From: categories@mta.ca Wed Nov 11 16:22:46 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 1N8F1w-0003I7-HT for gsmc-categories@m.gmane.org; Wed, 11 Nov 2009 16:22:44 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8EYa-0004nF-GA for categories-list@mta.ca; Wed, 11 Nov 2009 10:52:24 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5223 Archived-At: Dear Michael, The following lemma extends both results. We have a sequence of consecutive morphisms indexed on the integers ... ----> An ----> An+1 ----> An+2 ----> ... (if your sequence is finite, you extend by zero's). Call f(n,m) the composite from An to Am (n < m). Then, writing H/K a subquotient H/(H intersection K), there is an unbounded exact sequence of induced morphisms: ... ----> Ker f(n,n+2) / Im f(n-1,n) ----> Ker f(n+1,n+2) / Im f(n-1,n+1) ----> Ker f(n+1,n+3) / Im f(n,n+1) ----> ... where morphisms are alternatively induced by an 'elementary' morphism (say An --> An+1) or by an identity. At each step, one increases of one unit the first index in the numerator and the second index in the denominator, or the opposite (alternatively); after two steps, all indices are increased of one unit, and we go along in the same way. - Your lemma comes out of a sequence A ----> B ----> C (extended with zeros). - Snake's lemma, with your letters, comes out of a sequence of three morphisms whose total composite is 0 A ----> B ----> B' ----> C taking into account that A' = Ker(B' ----> C') and C = Cok(A ----> B). I like your lemma (and the Snake's). The form above does not look really nice. Perhaps someone else will find a nicer solution? However, if one looks at the universal model of a sequence of consecutive morphisms, in my third paper on Distributive Homological Algebra, Cahiers 26, 1985, p.186, the exact sequence above is obvious. (Much in the same way as for the sequence of the Snake Lemma, p. 188, diagrams (10) and (11).) This is how I found it. Best wishes Marco [For admin and other information see: http://www.mta.ca/~cat-dist/ ]