From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4545 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Amusing fact Date: Fri, 5 Sep 2008 11:49:43 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241020017 13786 80.91.229.2 (29 Apr 2009 15:46:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:57 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Sun Sep 7 14:14:43 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Sep 2008 14:14:43 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KcNk4-0007Zb-Kg for categories-list@mta.ca; Sun, 07 Sep 2008 14:08:04 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 51 Xref: news.gmane.org gmane.science.mathematics.categories:4545 Archived-At: There is nothing deep in the following, but it is amusing and slightly surprising. In the well-known diagram below (the dots indicate spaces, which email doesn't handle well): ........0........0.......0.......... ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.......v.......... 0 ----> A' ----> A ----> A'' ----> 0 ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.... ..v.......... ........ ... f .. .. g .. .......... 0 ----> B' ----> B ----> B'' ----> 0 ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.......v.......... 0 ----> C' ----> C ----> C'' ----> 0 ........|........|.......|.......... ........|........|.......|.......... ........|........|.......|.......... ........v........v.......v.......... ........0........0.......0.......... it is widely known that if the three columns, middle row and one of the other two rows is exact, so is the remaining row. What if the upper and lower rows are exact (along with the three columns)? It might not be a complex, that is it might happen that gf \neq 0. Less widely known is that if ker(g) \inc im(f), then it is also exact. That is actually if and only if. That is, if either of K = ker(f) and I = im(f) contains the other, they are equal. Well, I got to wondering about that and eventually conjectured and proved that it is always the case that I/(I\cap K) is isomorphic to K/(I\cap K). This makes it transparent that if either contains the other, they have to be equal. There is just one point remaining. I did this by chasing elements around in Ab (so it is true in any abelian category). Does anyone see a clever proof using the snake lemma? I don't. Michael