From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1764 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Picard group of a ringoid Date: Mon, 18 Dec 2000 08:33:46 -0500 (EST) Message-ID: References: <200012170121.eBH1LIP10578@transbay.net> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018083 340 80.91.229.2 (29 Apr 2009 15:14:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:43 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Dec 18 17:24:19 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eBIKcsF06455 for categories-list; Mon, 18 Dec 2000 16:38:54 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Authentication-Warning: triples.math.mcgill.ca: barr owned process doing -bs X-Sender: barr@triples.math.mcgill.ca In-Reply-To: <200012170121.eBH1LIP10578@transbay.net> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 37 Original-Lines: 31 Xref: news.gmane.org gmane.science.mathematics.categories:1764 Archived-At: I have never seen a name for this. I think, if it hasn't been defined before, I would be inclined to call it the Morita group. There is a large groupoid, let me call it the Morita groupoid, whose objects are rings and for which a morphism R --> S is a left S, right R bimodule M such that tensoring with M gives an equivalence between the category of left R modules and left S modules. This is locally small since M must be a finitely generated projective left S module and the group you are dealing with is simply the group of endomorphisms of R in that groupoid. the whole theory is due to Morita (and the main theorem, the Morita theorem). This is for rings, of course. I assume that a ringoid is a small preadditive category. A preadditive category with finitely many objects is Morita equivalent to a ring so it will be true for them. Beyond that, it would have to be examined because I am not sure what corresponds to finitely generated. On Sat, 16 Dec 2000, Bill Rowan wrote: > Tensor product gives a monoid structure on the class of isomorphism types > of R,R-bimodules, for a ring or ringoid R. Restricting to those elements > for which there is a two-sided inverse yields a group. I am inclined to call > this the nonabelian Picard group and denote it by NPic(R). If we start with > a commutative ring R, then the usual Picard group of R, Pic(R), can be viewed > as an abelian subgroup of NPic(R). > > Has anyone seen this before? Does anyone have some other idea about what this > should be called? > > Bill Rowan >