From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9639 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Point-free affine real line? Date: Thu, 31 May 2018 10:40:41 +0100 Message-ID: Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1527811218 8458 195.159.176.226 (1 Jun 2018 00:00:18 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 1 Jun 2018 00:00:18 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Jun 01 02:00:14 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1fOXUL-00026V-Ut for gsmc-categories@m.gmane.org; Fri, 01 Jun 2018 02:00:14 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52209) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1fOXVq-0005d4-RT; Thu, 31 May 2018 21:01:46 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1fOXVB-0006KP-5I for categories-list@mlist.mta.ca; Thu, 31 May 2018 21:01:05 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9639 Archived-At: Algebraic geometry defines the affine line over a field k as an affine schem= e, the spectrum of k[X]. It includes a copy of k, each element a being prese= nt as the irreducible polynomial X-a, with local ring the ring of fractions g= ot by inverting polynomials f(X) such that f(a) is non-zero. You can carry this out for the real line R, but it is very much R as a set, a= nd the copy of R in the underlying space of the spectrum has the discrete to= pology. Does algebraic geometry provide an analogous construction that could lead to= the point-free R? Can the locally ringed space be topologized (point-free) s= o that the copy of R has its usual topology? I've run into various problems. 1. It is not obvious to me that R[X] exists point-free. By that I mean that,= without presupposing a set R[X], or using non-geometric constructions, I ca= n't see how to define a geometric theory whose models are the polynomials. T= he problem comes with trying to pin down the requirement that all but finite= ly many of the coefficients of a polynomial must be zero. You cannot continu= ously define the degree of a polynomial, because the function R -> N, a |-> d= egree(aX + 1), is not continuous. That suggests the construction as Spec(R[X]) might have to be adjusted. Is t= here still some locally ringed space that does the trick? 2. The "structure sheaf" cannot be a sheaf. We hope its fibres are point-fre= e local rings, but, whatever they are, they must be R-algebras and so cannot= have the discrete topology. The space is locally ringed by some bundle othe= r than a sheaf (local homeomorphism). 3. The usual local rings, got as rings of fractions as described above, may b= e problematic point-free in the same way as R[X] is. I don't know what would= do instead. The power series ring R[[X]]? (At least as fibre over 0.) It do= es have the property of inverting those polynomials f for which f(0) is non-= zero. And it can be defined point-free, as R^N. (However, the finitely prese= nted approximations R[X|X^n =3D 0] happily exist point-free.) Thanks for any references you can provide, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]