From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6312 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Proving enough injectives for modules over a Grothendieck topos Date: Mon, 11 Oct 2010 10:00:09 +0100 (BST) Message-ID: References: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1286827988 17742 80.91.229.12 (11 Oct 2010 20:13:08 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 11 Oct 2010 20:13:08 +0000 (UTC) Cc: categories@mta.ca To: Colin McLarty Original-X-From: majordomo@mlist.mta.ca Mon Oct 11 22:13:05 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P5Ok1-00065Y-2G for gsmc-categories@m.gmane.org; Mon, 11 Oct 2010 22:13:01 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35782) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P5Oj1-0006zI-1h; Mon, 11 Oct 2010 17:11:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P5Oiy-0000Zr-2M for categories-list@mlist.mta.ca; Mon, 11 Oct 2010 17:11:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6312 Archived-At: Dear Colin, Yes, the argument is correct, and it'll be in volume 3 of the Elephant. I don't know why it hasn't been published elsewhere. Peter On Sun, 10 Oct 2010, Colin McLarty wrote: > It seems to me there is a nearly elementary proof that the category of > sheaves of modules over any sheaf of rings in a Grothendieck topos has > enough injectives. I want to check it here because I do not see it > published anywhere. People normally cite Grothendieck's Tohoku to > prove the result. > > To give the outline: Johnstone TOPOS THEORY (p. 261) proves the > category of Abelian groups in any Grothendieck topos has enough > injectives. Eisenbud COMMUTATIVE ALGEBRA (p. 621) uses adjunctions > between module categories (over sets) to turn injectives in Abelian > groups into enough injectives in any module category (over sets). > These adjunctions lift to the categories of Modules in ringed toposes > (E,R) over a fixed base topos E -- and so the category of R-Modules in > E has enough injectives. > > In more detail, Eisenbud does not put it in terms of adjunctions. But > in effect he shows the underlying group functor R-Mod-->AbGrp from > R-modules to Abelian groups (over Set) has a right adjoint taking each > Abelian group A to the R-module of additive functions from R to A > (i.e. the R-module of Abelian group morphisms from the underlying > group of R to A). > > Of course the underlying group functor is also left exact (indeed has > a left adjoint, change of base from the integers Z to R). And a right > adjoint to any exact functor preserves injectives. So take any > R-module M, embed it additively into an injective Abelian group M>-->Q > and map this back to R-modules by the right adjoint. It is easy to > see that M has a monic to the domain of the morphism and so embeds in > an injective R-module. > > Grothendieck in SGA 4 casts this in terms of ringed toposes. So far > as I can see he does not talk about the right adjoint to the > underlying group functor, and does not connect this to injectives, but > it lifts routinely unless I am missing something. > > Every ring R in a topos E is also an additive Abelian group in E. On > the other hand, for any Abelian group A in E we can form the group > HOM(R,A) where I write HOM to indicate this is an E object and not the > set of arrows. But HOM(R,A) is also an R-module by the action of R > on arguments: any r \in R times a function f \in HOM(R,A) is the > function that multiplies by r first and then applies f. > > So HOM(R,_) is a functor from Abelian groups in E to R-modules in E, > and the same straightforward calculation as we use in Set shows it is > right adjoint to the underlying group functor. So it preserves > injectives, and the same step as in Eisenbud shows we can use it to > embed every R-module into an injective R-module. > > Is that reasonably clear? Is it right? > > best, Colin > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]