Advances in Pure Mathematics, 2011, 1, 250-263 doi:10.4236/apm.2011.15045 Published Online September 2011 (http://www.SciRP.org/journal/apm) Localization of Ringed Spaces William D. Gillam Department of Mathematics, Brown University, Providence, USA E-mail: [email protected] Received March 28, 2011; revised April 12, 2011; accepted April 25, 2011 Abstract Let X be a ringed space together with the data M of a set M x of prime ideals of Xx, for each point x X . We introduce the localization of X ,M , which is a locally ringed space Y and a map of ringed spaces YX enjoying a universal property similar to the localization of a ring at a prime ideal. We use this to prove that the category of locally ringed spaces has all inverse limits, to compare them to the inverse limit in ringed spaces, and to construct a very general Spec functor. We conclude with a discussion of relative schemes. Keywords: Localization, Fibered Product, Spec, Relative Scheme 1. Introduction Then ( Spec of) the natural surjection kx kx Let Top , LRS , RS , and Sch denote the categories Xx11,,1Yy ,X 2 x 2 ky()2 of topological spaces, locally ringed spaces, ringed identifies Speckx 12 ky kx with a closed subspace spaces, and schemes, respectively. Consider maps of of Spec and 1 x , x Xx11,,Yy ,X 2 x 2 XX12Y 12 schemes fii: XY ( i =1,2) and their fibered product naturally coincides, under the EGA isomorphism, to the X12Y X as schemes. Let X denote the topological restriction of the structure sheaf of space underlying a scheme X . There is a natural SpecXx,, X x to the closed subspace comparison map 11Yy , 2 2 Spec kx kx Spec . 1 1()2 ky X11, x Yy ,X 2, x 2 : X12YYXXX12 It is perhaps less well-known that this entire discus- which is not generally an isomorphism, even if sion remains true for LRS morphisms ff, . X ,,XY are spectra of fields (e.g. if Y =Spec , 12 12 From the discussion above, we see that it is possible to XX==Spec , the map is two points mapping to 12 describe XX , at least as a set, from the following one point). However, in some sense fails to be an Y 21 data: isomorphism only to the extent to which it failed in the 1) the ringed space fibered product X RS X case of spectra of fields: According to [EGA I.3.4.7] the 12Y 1 (which carries the data of the rings fiber x , x over a point x , xXX Xx11,,Yy , X 2 x 2 12 12 12Y as stalks of its structure sheaf) and (with common image y ==fx fx) is naturally 11 2 2 2) the subsets Sxx ,Spec bijective with the set 12 Xx11, Yy ,X 2, x 2 It turns out that one can actually recover X12Y X as Spec kx12ky kx. a scheme solely from this data, as follows: Given a pair X , M consisting of a ringed space X and a subset In fact, one can show that this bijection is a M x SpecXx, for each x X , one can construct a homeomorphism when 1 x , x is given the topo- loc 12 locally ringed space XM, with a map of ringed loc logy it inherits from X12Y X . One can even describe 1 spaces X , MX . In a special case, this constru- the sheaf of rings x12, x inherits from X12Y X ction coincides with M. Hakim’s spectrum of a ringed as follows: Let topos. Performing this general construction to Sxx,:=Spec z:z 1There is no sense in which this sheaf of rings on 12 Xx11, Yy ,X 2,, x 2 Xxi i Spec kx 12 ky kx is “quasi-coherent”. It isn’t even a module over = for i = 1,2 . the usual structure sheaf of Spec kx kx xi 12ky Copyright © 2011 SciRes. APM W. D. GILLAM 251 RS ficient to upgrade descent theorems for RS to descent XXSxx1212Y ,, theorems for Sch . yields the comparison map , and, in particular, the scheme X12Y X . A similar construction in fact yields 2. Localization all inverse limits in LRS (§ 3.1) and the comparison map to the inverse limit in RS , and allows one to easily We will begin the localization construction after making prove that a finite inverse limits of schemes, taken in a few definitions. LRS , is a scheme (Theorem 8). Using this description Definition 1. A morphism f : AB of sheaves of of the comparison map one can easily describe some rings on a space X is called a localization morphism3 circumstances under which it is an isomorphism (§ 3.2), iff there is a multiplicative subsheaf SA so that f and one can easily see, for example, that it is a loca- is isomorphic to the localization A SA1 of A at lization morphism (Definition 1), hence has zero cotangent S .4 A morphism of ringed spaces f : XY is called #1 complex. a localization morphism iff ff: YX is a lo- The localization construction also allows us construct calization morphism. (§ 3.3), for any X LRS , a very general relative spec A localization morphism A B in Rings X is functor both flat and an epimorphism in RingsX .5 In par- ticular, the cotangent complex (hence also the sheaf of Spec : Algop LRS X XX Kähler differentials) of a localization morphism is zero which coincides with the usual one when X is a [Ill II.2.3.2]. The basic example is: For any affine scheme X =Spec A , AX is a localization mor- scheme and we restrict to quasi-coherent X algebras. X We can also construct (§ 3.5) a “good geometric realiza- phism. tion” functor from M. Hakim’s stack of relative schemes Definition 2. Let A be a ring, SA Spec any sub- 2 over a locally ringed space X to LRS X . It should set. We write SpecAS for the locally ringed space be emphasized at this point that there is essentially only whose underlying topological space is S with the topol- one construction, the localization of a ringed space of ogy it inherits from Spec A and whose sheaf of rings is § 2.2, in this paper, and one (fairly easy) theorem (Theo- the inverse image of the structure sheaf of Spec A . rem 2) about it; everything else follows formally from If A is clear from context, we drop the subscript and general nonsense. simply write Spec S . There is one possible point of Despite all these results about inverse limits, I stum- confusion here: If I A is an ideal, and we think of bled upon this construction while studying direct limits. I Spec A I as a subset of Spec A , then was interested in comparing the quotient of, say, a finite SpecA Spec A IAI Spec étale groupoid in schemes, taken in sheaves on the étale site, with the same quotient taken in LRS . In order to (though they have the same topological space). compare these meaningfully, one must somehow put them in the same category. An appealing way to do this 2.1. Prime Systems is to prove that the (functor of points of the) LRS quo- tient is a sheaf on the étale site. In fact, one can prove Definition 3. Let XX=, X be a ringed space. A that for any X LRS , the presheaf prime system M on X is a map x M x assigning a subset M x SpecXx, to each point x X . For YHomYX , LRS prime systems M , N on X we write M N to is a sheaf on schemes in both the fppf and fpqc topolo- mean M x N x for all x X . Prime systems on X gies. Indeed, one can easily describe a topology on RS , form a category PS X where there is a unique mor- analogous to the fppf and fpqc topologies on schemes, phism from M to N iff M N . The intersection and prove it is subcanonical. To upgrade this to a sub- iiM of prime systems M i PSX is defined by canonical topology on LRS one is naturally confronted iiMM:= i i . with the comparison of fibered products in LRS and x x RS . In particular, one is confronted with the question of A primed ringed space X , M is a ringed space X whether is an epimorphism in the category of ringed equipped with a prime system M . Prime ringed spaces spaces. I do not know whether this is true for arbitrary form a category PRS where a morphism f :, XM YN, is a morphism of ringed spaces f satisfy- LRS morphisms f12, f , but in the case of schemes it is possible to prove a result along these lines which is suf- 3See [Ill II.2.3.2] and the reference therein. 4 2Hakim already constructed such a functor, but ours is different from See [Ill II.2.3.2] and the reference therein. 5 hers. Both of these conditions can be checked at stalks. Copyright © 2011 SciRes. APM 252 W. D. GILLAM ing sheaf on X . The category SecF of local sections of is the category whose objects are pairs Us, Spec fM N F x xfx() where U is an open subset of X and s F U , and for every x X . where there is a unique morphism Us,, Vt if The inverse limit of a functor iM i to PS X is UV and tU s. loc clearly given by iiM .