From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4479 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Term used in spectral sequences Date: Sun, 17 Aug 2008 14:57:37 -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 1241019972 13496 80.91.229.2 (29 Apr 2009 15:46:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:12 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Sun Aug 17 17:30:57 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 17 Aug 2008 17:30:57 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KUorZ-0007gN-Fa for categories-list@mta.ca; Sun, 17 Aug 2008 17:28:33 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 14 Original-Lines: 31 Xref: news.gmane.org gmane.science.mathematics.categories:4479 Archived-At: As some of you know, my wife and I are translating Grothendieck's Tohoku paper. Originally, it was suggested that it be published as a TAC reprint, but Grothendieck refuses permission because he "does not believe in copyright". So I thought to retype it and post it on my own site (so sue). I then realized that translation would be easier than retyping. Which brings me to a translation problem. I am not expert in spectral sequences and what I know is from Cartan-Eilenberg. I cannot related G's definition to theirs. G defines a spectral sequence as a pair E=(E^{p,q}_r,E^n), all indexed objects of an abelian category subject to five conditions. The first three refer only to the E^{p,q}_r, including that for each pair p,q the E^{p,q}_r stabilize vis-vis r to a term he calls E^{p,q}_\infty (and not E^{p,q}). The fourth assumes "isomorphisms $\beta^{p,q}:E^{pq}\to G^p(E^{p+q})$. The family $(E^n)$ without filtrations is called the \emph{l'aboutissement} of the spectral sequence $E$." The E^n are assumed filtered and G^p is the associated grading: G^p(A)=F^p(A)/F^{p-1}(A). Now this makes no sense. The only thing called E^{pq} would be the term E^n for n=pq and this is really unlikely. I strongly suspect the domain of \beta^{p,q} is intended to be E^{p,q}_\infty. Finally does anyone have any idea how "aboutissement" is to be translated. It means something like limit, but the usual term for that is of course "limite". The Cartan-Eilenberg development is different enough that there seems not to be any corresponding word. Michael