From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5232 Path: news.gmane.org!not-for-mail From: Clemens.BERGER@unice.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on exact sequence Date: Wed, 11 Nov 2009 17:34:04 +0100 Message-ID: Reply-To: Clemens.BERGER@unice.fr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1257993858 29025 80.91.229.12 (12 Nov 2009 02:44:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Nov 2009 02:44:18 +0000 (UTC) To: Marco Grandis , Michael Barr , categories@mta.ca Original-X-From: categories@mta.ca Thu Nov 12 03:44:11 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 1N8PfO-0001Me-C8 for gsmc-categories@m.gmane.org; Thu, 12 Nov 2009 03:44:10 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8PDf-0007KI-JL for categories-list@mta.ca; Wed, 11 Nov 2009 22:15:31 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5232 Archived-At: Marco Grandis wrote: > 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 > Dear Michael and Marco, as addition to Marco's answer, I would propose the following notation: for any pair of composable arrows f:A-->B and g:B-->C denote by H(g,f) the cokernel of ker(gf)-->ker(g), or (what amounts to the same in an abelian category), the kernel of coker(f)-->coker(gf). Thus the object H(g,f) is precisely the one which allows one to glue together the short exact sequences 0-->ker(f)-->ker(gf)-->ker(g) and coker(f)-->coker(gf)-->coker(g)-->0. If gf=0 then H(g,f)=ker(g)/im(f) is precisely the homology object at B, but as Ross already mentioned, this object is well defined for any composable pair of arrows. Now, if we have three composable arrows f:A-->B, g:B-->C, h:C-->D, then there is a 4-term exact sequence 0-->H(g,f)-->H(hg,f)-->H(h,gf)-->H(h,g)-->0. The snake lemma can be derived from the special case hgf=0 of this 4-term exact sequence, where g corresponds precisely to the middle vertical arrow of Michael's diagram. It is interesting to observe that a proof (without elements !) of this 4-term exact sequence uses some nice composition properties of Hilton-exact squares (a common generalization of pullback and pushout squares). Some more details can be found at pg. 24 of http://math.unice.fr/~cberger/structure1. Several questions arise naturally: this 4-term exact sequence looks like a "cocycle". Are there generalizations to n composable arrows ? What about generalizations to non-abelian categories ? All the best, Clemens. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]