From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6836 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Goursat's lemma Date: Thu, 25 Aug 2011 19:03:28 -0400 (EDT) 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: dough.gmane.org 1314385649 32606 80.91.229.12 (26 Aug 2011 19:07:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 26 Aug 2011 19:07:29 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Fri Aug 26 21:07:25 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Qx1kS-0004m9-27 for gsmc-categories@m.gmane.org; Fri, 26 Aug 2011 21:07:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:34655) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1Qx1ho-0006c7-Fb; Fri, 26 Aug 2011 16:04:40 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Qx1hn-0006ZR-QF for categories-list@mlist.mta.ca; Fri, 26 Aug 2011 16:04:39 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6836 Archived-At: This is something that Lambek used for diagram chasing in his book on Rings and Modules. I have discovered a very simple proof of a slight generalization. The lemma states (as least Jim stated it this way) that if G is a submodule of A x B and C = image of G --> A, C' the image of G --> B, D the kernel of G --> B and D' the kernel of G --> A, then C/D is isomorphic to C'/D'. Forget that G is a subgroup of A x B and just suppose you have two exact sequences 0 --> D' --> G --> C --> 0 and 0 --> D --> G --> C' --> 0, then the composites D --> G --> C and D' --> G --> C' have the same cokernel. The dual claim is that they have the same kernel. But thinking of D and D' are submodules of G, then the kernels are quite obviously D \cap D' and the original conclusion follows by duality. However, the same thing is true for groups (both D and D', being kernels, are normal in G and hence in C, C'). I see no such simple duality argument there. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]