From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5985 Path: news.gmane.org!not-for-mail From: zoran skoda Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck's relative point of view Date: Fri, 9 Jul 2010 18:03:16 +0200 Message-ID: References: Reply-To: zoran skoda NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1278765759 3478 80.91.229.12 (10 Jul 2010 12:42:39 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 10 Jul 2010 12:42:39 +0000 (UTC) Cc: categories@mta.ca To: Colin McLarty Original-X-From: categories@mta.ca Sat Jul 10 14:42:38 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OXZO7-00017i-Pp for gsmc-categories@m.gmane.org; Sat, 10 Jul 2010 14:42:35 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OXYyK-0003pT-Sd for categories-list@mta.ca; Sat, 10 Jul 2010 09:15:56 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5985 Archived-At: Dear colleagues, the flexibility of definitions with respect to the change of base is in my understanding not the main aspect in Grothendieck's relative point of view. The idea is that the general *study of properties* of objects should be better replaced by the more fundamental study of properties of morphisms. This is different from just looking how the properties of *objects* change under base change. Thus the notions like affine scheme, quasicompact scheme and so on, are replaced by the study of affine, flat, quasicompact, quasi-separated, proper etc. morphisms over an arbitrary base. The definition of many local properties can be extended say from schemes to algebraic stacks by pulling back to a chart by a scheme (or algebraic space= ) and checking the property there. Many properties of morphisms can be expressed in terms of properties of functors among the corresponding categories of quasicoherent sheaves, what enables nowdays further "noncommutative" generalizations in terms of generalizations of such modules. Thus while affine scheme corresponds to (the spectrum of a) ring, affine morphism generalizes a ring morphism in the sense that the direct inverse image functor is faithful and has both left and right adjoint; therefore it corresponds to a monad having a right adjoint. Thus being represented by an algebra on the object level gets generalized to being represented by a sufficiently good monad at the morphism level. Now, if one emphasises the change of base aspect then this means that you look just at properties of morphisms measured with respect to the codomain fibration; if we emphsise on the behaviour with respect to quasicoherent modules, then we measure the properties of morphism with respect to to the stack of quasicoherent sheaves. It is often the case that the properties of morphisms in geometry are measured by action on/behaviour with respect to specific kinds of objects. For example, in algebraic geometry there are two main kinds of global finiteness: quasicompact morphism and quasi-separated morphism; both derived from a notion of quasi-compact object. Zoran =C5=A0koda [For admin and other information see: http://www.mta.ca/~cat-dist/ ]