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 Advances in Pure Mathematics, 2011, 1, 250-263 doi:10.4236/apm.2011.15045 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Localization of Ringed Spaces William D. Gillam Department of Mat hematics, Brow n University, Providence, USA E-mail: wgillam@math.brown.edu 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 xM of prime ideals of ,Xx for each point xX. We introduce the localization of ,XM, 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 Let Top , LRS, RS, and Sch denote the categories of topological spaces, locally ringed spaces, ringed spaces, and schemes, respectively. Consider maps of schemes :iifXY(=1,2i) and their fibered product 12YXX as schemes. Let X denote the topological space underlying a scheme X. There is a natural comparison map 1212:YYXXXX which is not generally an isomorphism, even if 12,,XXY are spectra of fields (e.g. if =SpecY, 12==SpecXX , the map  is two points mapping to one point). However, in some sense  fails to be an isomorphism only to the extent to which it failed in the case of spectra of fields: According to [EGA I.3.4.7] the fiber 112,xx over a point 1212,YxxXX (with common image 1122==yfx fx) is naturally bijective with the set 12Spec .kykx kx In fact, one can show that this bijection is a homeomorphism when 112,xx is given the topo- logy it inherits from 12YXX. One can even describe the sheaf of rings 112,xx inherits from 12YXX as follows: Let 12,, ,11,22,:= Spec: =for =1,2.XxX xXxYyi ixiSxx zzi Then (Spec of) the natural surjection  ,,1()211,22XxX xkyYy kx kx identifies 12 kySpeckx kx with a closed subspace of ,,11,2 2Spec XxX xYy and 11212 ,XXYxx naturally coincides, under the EGA isomorphism, to the restriction of the structure sheaf of ,,11,22Spec XxX xYy to the closed subspace 1()2, ,11,2 2Spec Spec.kyX xXxYykx kx 1 It is perhaps less well-known that this entire discus- sion remains true for LRS morphisms 12,ff. From the discussion above, we see that it is possible to describe 21 XX Y, at least as a set, from the following data: 1) the ringed space fibered product 12YXXRS (which carries the data of the rings ,,11,22XxX xYy as stalks of its structure sheaf) and 2) the subsets 12, ,11,22,SpecXxX xYySxx  It turns out that one can actually recover 12YXX as a scheme solely from this data, as follows: Given a pair ,XM consisting of a ringed space X and a subset ,SpecxXxM for each xX, one can construct a locally ringed space ,locXM with a map of ringed spaces ,locXMX. In a special case, this constru- ction coincides with M. Hakim’s spectrum of a ringed topos. Performing this general construction to 1There is no sense in which this sheaf of rings on 12Spec kykx kx is “quasi-coherent”. It isn’t even a module over the usual structure sheaf of 12Spec kykx kx W. D. GILLAM Copyright © 2011 SciRes. APM 2511212,,YXXSxxRS yields the comparison map , and, in particular, the scheme 12YXX. A similar construction in fact yields all inverse limits in LRS (§3.1) and the comparison map to the inverse limit in RS , and allows one to easily prove that a finite inverse limits of schemes, taken in LRS, is a scheme (Theorem 8). Using this description of the comparison map  one can easily describe some circumstances under which it is an isomorphism (§3.2), and one can easily see, for example, that it is a loca- lization morphism (Definition 1), hence has zero cotangent complex. The localization construction also allows us construct (§3.3), for any XLRS , a very general relative spec functor opSpec :XXXAlg LRS which coincides with the usual one when X is a scheme and we restrict to quasi-coherent X algebras. We can also construct (§3.5) a “good geometric realiza-tion” functor from M. Hakim’s stack of relative schemes over a locally ringed space X to XLRS .2 It should be emphasized at this point that there is essentially only one construction, the localization of a ringed space of §2.2, in this paper, and one (fairly easy) theorem (Theo-rem 2) about it; everything else follows formally from general nonsense. Despite all these results about inverse limits, I stum-bled upon this construction while studying direct limits. I was interested in comparing the quotient of, say, a finite étale groupoid in schemes, taken in sheaves on the étale site, with the same quotient taken in LRS . In order to compare these meaningfully, one must somehow put them in the same category. An appealing way to do this 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 that for any XLRS, the presheaf LRS ,YHomYX is a sheaf on schemes in both the fppf and fpqc topolo-gies. Indeed, one can easily describe a topology on RS , analogous to the fppf and fpqc topologies on schemes, and prove it is subcanonical. To upgrade this to a sub-canonical topology on LRS one is naturally confronted with the comparison of fibered products in LRS and RS . In particular, one is confronted with the question of whether  is an epimorphism in the category of ringed spaces. I do not know whether this is true for arbitrary LRS morphisms 12,ff, but in the case of schemes it is possible to prove a result along these lines which is suf-ficient to upgrade descent theorems for RS to descent theorems for Sch . 2. Localization We will begin the localization construction after making a few definitions. Definition 1. A morphism :fAB of sheaves of rings on a space X is called a localization morphism3 iff there is a multiplicative subsheaf SA so that f is isomorphic to the localization 1ASA of A at S.4 A morphism of ringed spaces :fXY is called a localization morphism iff #1:YXff is a lo-calization morphism. A localization morphism AB in XRings is both flat and an epimorphism in XRings .5 In par-ticular, the cotangent complex (hence also the sheaf of Kähler differentials) of a localization morphism is zero [Ill II.2.3.2]. The basic example is: For any affine scheme = XSpecA , XXA is a localization mor-phism. Definition 2. Let A be a ring, Spec SA any sub-set. We write SpecAS for the locally ringed space whose underlying topological space is S with the topol-ogy it inherits from Spec A and whose sheaf of rings is the inverse image of the structure sheaf of Spec A. If A is clear from context, we drop the subscript and simply write Spec S. There is one possible point of confusion here: If IA is an ideal, and we think of SpecAI as a subset of Spec A, then SpecSpec Spec AAIAI (though they have the same topological space). 2.1. Prime Systems Definition 3. Let =,XXX be a ringed space. A prime system M on X is a map xxM assigning a subset ,SpecxXxM to each point xX. For prime systems ,MN on X we write MN to mean xxMN for all xX. Prime systems on X form a category XPS where there is a unique mor-phism from M to N iff MN. The intersection iiM of prime systems iMXPS is defined by := .iii ixxMM A primed ringed space ,XM is a ringed space X equipped with a prime system M. Prime ringed spaces form a category PRS where a morphism :,fXM ,YN is a morphism of ringed spaces f satisfy-2Hakim already constructed such a functor, but ours is different from hers. 3See [Ill II.2.3.2] and the reference therein. 4See [Ill II.2.3.2] and the reference therein. 5Both of these conditions can be checked at stalks. W. D. GILLAM Copyright © 2011 SciRes. APM 252 ing ()Spec xxfxfM N for every xX. The inverse limit of a functor iiM to PS X is clearly given by iiM. Remark 1. Suppose ,YNPRS and :fXY is an RS morphism. The inverse image *fN is the prime system on X defined by 1*1,:=Spec =Spec: .xfxxXx xfxfNf NfN  Formation of inverse image prime systems enjoys the expected naturality in f: *** =gfMfgM. We can alternatively define a PRS morphism :,fXM ,YN to be an RS morphism :fXY such that c*MfN (i.e. together with a XPS mor-phism *MfN). For XLRS , the local prime system X on X is defined by ,:=Xx xm. If Y is another locally ringed space, then a morphism :fXY in RS defines a morphism of primed ringed spaces :, ,XYfX Y iff f is a morphism in LRS , so we have a fully faithful functor :,,XLRS PRSXX (1) and we may regard LRS as a full subcategory of PRS. At the “opposite extreme” we also have, for any XRS , the terminal prime system X defined by ,,:= SpecXx Xx (i.e. the terminal object in XPS ). For ,YMPRS , we clearly have  ,,, =,,XHomY MXHomY XPRS RS so the functor :,XXXRS PRS (2) is right adjoint to the forgetful functor PRSRS given by ,XMX. 2.2. Localization Now we begin the main construction of this section. Let ,XM be a primed ringed space. We now construct a locally ringed space ,locXM (written locX if M is clear from context), and a PRS morphism π:, ,loc locXXXM called the localization of X at M. Definition 4. Let X be a topological space, F a sheaf on X. The category SecF of local sections of F is the category whose objects are pairs ,Us where U is an open subset of X and sUF, and where there is a unique morphism ,,UsVt if UV and .tU s. As a set, the topological space locX will be the set of pairs ,xz, where xX and xzM. Let locX denote the category of subsets of locX whose mor-phisms are inclusions. For ,XUsSec, set ,:= ,:,.loc xUUsxzXxUsz This defines a functor :locXUXSec satisfying: >0,,=,||,= ,.UV UVnUUsUVtUU VstUUs UUsn The first formula implies that locXUXSec is a basis for a topology on locX where a basic open neighborhood of ,xz is a set ,UUs where xU, xsz. We always consider locX with this topology. The map π:,locXXxzx is continuous because 1π=,1UUU. We construct a sheaf of rings locX on locX as follows. For an open subset locVX, we let locXV be the set of ,,=, Xxzxz Vssxz satisfying the local consistency condition: For every ,xzV, there is a basic open neighborhood ,UUt of ,xz contained in V and a section XntaUt such that, for every ,,xzUUt, we have ,,= .xXxnzxasxzt  (Of course, one can always take =1n since ,= ,nUUtUUt .) The set locXV becomes a ring under coordinatewise addition and multiplication, and the obvious restriction maps make locX a sheaf of rings on locX. There is a natural isomorphism ,,, =locX xzXxz taking the germ of =, locXssxzU in the stalk ,,locXxz to ,,Xxzsxz. This map is injective be- cause of the local consistency condition and surjective W. D. GILLAM Copyright © 2011 SciRes. APM 253because, given any ,Xxzab, we can lift ,ab to ,Xab U on some neighborhood U of x and define ,locXsUUb by letting ,,:= .xXxxzsxza b  This s manifestly satisfies the local consistency condition and has ,=sxz ab. In particular, locX, with this sheaf of rings, is a locally ringed space. To lift π to a map of ringed spaces π:locXX we use the tautological map #*π:πXlocX of sheaves of rings on X defined on an open set UX by   #*π:π=,1.XloclocXXxzUUU UUss It is clear that the induced map on stalks ,, ,,(,)π:=xz XxlocXxzXxz is the natural localization map, so 1,π=xzz xzM and hence π defines a PRS morphism π:, ,loc locXXXM. Remark 2. It would have been enough to construct the localization ,locXX at the terminal prime system. Then to construct the localization ,locXM at any other prime system, we just note that ,locXM is clearly a subset of ,locXX, and we give it the topol-ogy and sheaf of rings it inherits from this inclusion. The construction of ,locXX is “classical.” Indeed, M. Hakim [Hak] describes a construction of ,locXX that makes sense for any ringed topos X (she calls it the spectrum of the ringed topos [Hak IV.1]), and attrib-utes the analogous construction for ringed spaces to C. Chevalley [Hak IV.2]. Perhaps the main idea of this work is to define “prime systems,” and to demonstate their ubiquity. The additional flexibility afforded by non-terminal prime systems is indispensible in the appli-cations of §3. It is not clear to me whether this setup generalizes to ringed topoi. We sum up some basic properties of the localization map π below. Proposition 1. Let ,XM be a primed ringed space with localization π:locXX. For xX, the fiber 1πx is naturally isomorphic in LRS to Spec xM (Definition 2).6 Under this identification, the stalk of π at xzM is identified with the localization of ,Xx at z, hence π is a localization morphism (Definition 1). Proof. With the exception of the fiber description, everything in the proposition was noted during the con-struction of the localization. Clearly there is a natural bijection of sets 1=πxMx taking xzM to 1,πxzx. We first show that the topology inher-ited from locX coincides with the one inherited from ,Spec Xx. By definition of the topology on locX, a ba-sic open neighborhood of xzM is a set of the form ,= :,xxxUUsMzM sz where U is a neighborhood of x in X and XsU satisfies xsz. Clearly this set depends only on the stalk of ,xXxs of s at x, and any element ,Xxt lifts to a section ()XtU on some neighborhood of X, so the basic neighborhoods of xzM are the sets of the form :xzMtz where xtz. But for the same set of t, the sets ,:= Spec :XxDt tpp form a basis for neighborhoods of z in ,Spec Xx so the result is clear. We next show that the sheaf of rings on xM inher-ited from locX is the same as the one inherited from ,Spec Xx. Given ,Xxf, a section of locxXM over the basic open set xMDf is an element ,=XxzzM Dfxssz satisfying the local consistency condition: For all xzM Df , there is a basic open neighborhood ,UUt of ,xz in locX and an element nXtat U such that, for all ,xzMDf UUt , we have nzzszat. Note that ,=xxxMDfUUt MDft and Spec ,nxxxXxat Dft. The sets xxDft M cover ,SpecxXxMDf, and we have a “global formula” s showing that the stalks of the various nxxat agree at any xzM Df , so they glue to yield an element Spec ,xXxgsMDf with =zgssz. We can define a morphism of sheaves on xM by defining it on basic opens, so this defines a morphism of sheaves Spec ,:locxxXxXgMM which is easily seen to be an isomorphism on stalks. Remark 3. Suppose ,XMPRS and UX is an open subspace of X. Then it is clear from the con-struction of π:,locXMX that 1π=, ,locXUUUMU. 6By “fiber” here we mean  1RS,π:= ,locXXxxXx, which is justthe set theoretic preimage 1πlocxX with the topology and sheafof rings it inherits from locX. This differs from another common usage of “fiber” to mean RS ,locXXxkx. W. D. GILLAM Copyright © 2011 SciRes. APM 254 The following theorem describes the universal prop-erty of localization. Theorem 2. Let :, ,fXM YN be a morphism in PRS . Then there is a unique morphism :, ,loc locfXMYN in LRS making the diagram  ππ,,loc locffXM YNXY (3) commute in RS . Localization defines a functor PRSLRS ,,locXM XM :,, :,,loc locfXMYNfXMYN retracting the inclusion functor :LRSPRS and right adjoint to it: For any YLRS , there is a natural bijection ,,=,,, .locYHomYX MHomYX MLRSPRS  Proof. We first establish the existence of such a mor-phism f. The fact that f is a morphism of primed ringed spaces means that we have a function xfxMN 1xzfz for each xX, so we can complete the diagram of topological spaces ππ,locflocfXYNXY (at least on the level of sets) by setting  1,:= ,.locxfxzfx fzY To see that f is continuous it is enough to check that the preimage 1,fUUs is open in locX for each basic open subset ,UUs of locY. But it is clear from the definitions that 11#1,= ,fUUsU fUffs (note #1 =xfxxffsf s). Now we want to define a map #1:loc YXff of sheaves of rings on Y (with “local stalks”) making the diagram #-1 #1π111ππloc locfXYfXXff  commute in locYRings . The stalk of this diagram at (,) locxzX is a diagram   ,1,,1,,ππ,,xzfxfzxxzxfXx xYf xzXx Yfxffz in Rings where the vertical arrows are the natural lo-calization maps; these are epimorphisms, and the uni-versal property of localization ensures that there is a unique local morphism of local rings ,xzf completing this diagram. We now want to show that there is actually a (necessarily unique) map #1:loc YXff of sheaves of rings on locX whose stalk at ,xz is the map ,xzf. By the universal property of sheafification, we can work with the presheaf inverse image 1pre locXf instead. A section ,Vs of this presheaf over an open subset locWX is represented by a pair ,Vs where locVY is an open subset of locY containing fW and ,(,)=, .locY yzYyz Vssyz V I claim that we can define a section #p,re locXfVs W by the formula #1p,,:=,.re xfVsxzs fxfz It is clear that this element is independent of replacing V with a smaller neighborhood of fW and restricting s, but we still must check that p,(,),reX xzxz WfVs satisfies the local consistency condition. Suppose XntaUt witnesses local consistency for locYsV on a basic open subset ,UUt V. Then it is straightforward to check that the restriction of #11#1 ,YnffafUUtfft to 1,fUUt W witnesses local consistency of #p,refVs on 11#1,= ,.fUUtWUfUfftW It is clear that our formula for #p,refVs respects restrictions and has the desired stalks and commutativity, so its sheafification provides the desired map of sheaves of rings. This completes the construction of :loc locfXY in W. D. GILLAM Copyright © 2011 SciRes. APM 255LRS making (3) commute in RS . We now establish the uniqueness of f. Suppose :loc locfXY is a morphism in LRS that also makes (3) commute in RS . We first prove that =ff on the level of topological spaces. For xX the commutativity of (3) ensures that  ,= ,fxzfxz for some (), ()Spec ,fxYfxzN so it remains only to show that 1=.xzfz The commutativity of (3) on the level of stalks at ,locxzX gives a commutative diagram of rings  ,-1 #,,π,π,π,,xzfXx Yf xzzfxzxzfXx Yf x   where the vertical arrows are the natural localization maps. From the commutativity of this diagram and the fact that 1,()=zzxzfmm (because ,xzf is local) we find 1(),11(), ,111=π()=π=π=()zfxzzfxz xzxx zxzfffzmmm as desired. This proves that =ff on topological spaces, and we already argued the uniqueness of #f (which can be checked on stalks) during its construction. The last statements of the theorem follow easily once we prove that the localization morphism π:, locXXX is an isomorphism for any XLRS . On the level of topological spaces, it is clear that π is a continuous bijection, so to prove it is an isomorphism we just need to prove it is open. To prove this, it is enough to prove that for any ,SecXUs, the image of the basic open set ,UUs under π is open in X. Indeed, *,π,=:=:xxxXxUUsxU sxUs is open in U, hence in X, because invertibility at the stalk implies invertibility on a neighborhood. To prove that π is an isomorpism of locally ringed spaces, it remains only to prove that #π:XlocX is an iso- morphism of sheaves of rings on =locXX. Indeed, Proposition 1 says the stalk of #π at ,locxxXm is the localization of the local ring ,Xx at its unique maximal ideal, which is an isomorphism in LAn . Lemma 3. Let ARings be a ring, ,:=Spec XXA, and let*be the punctual space. Define a prime system N on ,XXA by ,:=Spec=Spec = .XxxNx AAX Let :,,XXaX XA be the natural RS morphism. Then *,=XXaN and the natural PRS morphisms  ,*,(,,),,*,,Spec =*,,XXXXAXXA NAAA yield natural isomorphisms ,,=,,=,,=*, ,Specloc locXXXXXlocXX XANAA in LRS . Proof. Note that the stalk ,,:XxxXxaA  of a at xX is the localization map xAA, and, by defi- nition, *xaN is the set prime ideals z of xA pulling back to xA under :xxaA A. The only such prime ideal is the maximal ideal xxAm, so *,,={ }=xXxxXaN m. Next, it is clear from the description of the localization of a PRS morphism that the localizations of the mor-phisms in question are bijective on the level of sets. In-deed, the bijections are given by ,,*,,xxxxxxm so to prove that they are continuous, we just need to prove that they have the same topology. Indeed, we will show that they all have the usual (Zariski) topology on =Spec XA. This is clear for ,,,XXXX be-cause localization retracts  (Theorem 2), so ,,, =,locXXXXXX , and it is clear for *,,Spec AA because of the description of the fibers of localization in Proposition 1. For ,,XXAN, we note that the sets ,UUs, as U ranges over connected open subsets of X (or any other family of basic opens for that matter), form a basis for the topology on ,,locXXA N. Since U is connected, =XsAU A, and ,UUs is identified with the usual basic open subset ()Ds X under the bijections above. This proves that the LRS morphisms in question are iso-morphisms on the level of spaces, so it remains only to prove that they are isomorphisms on the level of sheaves of rings, which we can check on stalks using the descrip-tion of the stalks of a localization in Proposition 1. Remark 4. If XLRS , and M is a prime system on X, the map π:locXX is not generally a mor-phism in LRS , even though ,locXX LRS . For ex- W. D. GILLAM Copyright © 2011 SciRes. APM 256 ample, if X is a point whose “sheaf” of rings is a local ring ,Am, and ={}Mp for some pm, then locX is a point with the “sheaf” of rings Ap, and the “stalk” of #π is the localization map :lA Ap. Even though ,AAp are local, this is not a local morphism because 1=lApppm. 3. Applications In this section we give some applications of localization of ringed spaces. 3.1. Inverse Limits We first prove that LRS has all inverse limits. Theorem 4. The category PRS has all inverse limits, and both the localization functor PRSLRS and the forgetful functor PRSRS preserve them. Proof. Suppose ,iiiXM is an inverse limit sys-tem in PRS . Let X be the inverse limit of iiX in Top and let π:iiXX be the projection. Let X be the direct limit of 1πiXii in XRings and let #1π:πiiX Xi be the structure map to the direct limit, so we may regard =,XXX as a ringed space and πi as a morphism of ringed spaces iXX. It imme-diate from the definition of a morphism in RS that X is the inverse limit of iiX in RS. Let iM be the prime system on X given by the inverse limit (inter-section) of the *πiiM. Then it is clear from the definition of a morphism in PRS that ,XM is the inverse limit of ,iiiXM, but we will spell out the details for the sake of concreteness and future use. Given a point =ixxX, we have defined xM to be the set of ,SpecXxz such that 1,,πSpecixxX xiiizM  for every i, so that πi defines a PRS morphism π:, ,iiiXMXM. To see that ,XM is the direct limit of ,iiiXM suppose :, ,iiifYNXM are morphisms defining a natural transformation from the constant functor ,iYN to ,iiiXM. We want to show that there is a unique PRS morphism :, ,fYN XM with π=iiff for all i. Since X is the inverse limit of iX in RS , we know that there is a unique map of ringed spaces :fYX with π=iiff for all i, so it suffices to show that this f is a PRS morphism. Let yY, yzN. We must show 1yfxfzM. By definition of M, we must show 11π()π=iyfyfxfx iifz MM for every i. But π=iiff implies  ()π=yiifxyff, so 111()π=iy ifx yfzfz is in ifyM because if is a PRS morphism. The fact that the localization functor preserves inverse limits follows formally from the adjointness in Theorem 2. Corollary 5. The category LRS has all inverse limits. Proof. Suppose iiX is an inverse limit system in LRS . Composing with  yields an inverse limit sys-tem ,iXiiX in PRS . By the theorem, the lo-calization ,locXM of the inverse limit ,XM of ,iXiiX is the inverse limit of ,lociXiiX in LRS . But localization retracts  (Theorem 2) so ,lociXiiX is our original inverse limit system iiX. We can also obtain the following result of C. Chevalley mentioned in [Hak IV.2.4]. Corollary 6 The functor  ,locXXXRS LRS is right adjoint to the inclusion LRSRS . Proof. This is immediate from the adjointness property of localization in Theorem 2 and the adjointness property of the functor : For YLRS we have ,,=,,,=,.locXYXHomY XHom YXHomY XLRSPRSRS Our next task is to compare inverse limits in Sch to those in LRS . Let *Top be “the” punctual space (terminal object), so *=RingsRings. The functor *,AARings RS is clearly left adjoint to op: ,.XXXRS Rings By Lemma 3 (or Proposition 1) we have  *,:=*,,Spec =Spec .loc locAAAA Theorem 2 yields an easy proof of the following result, which can be found in the Errata for [EGA I.1.8] printed at the end of [EGA II]. Proposition 7. For ARings , XLRS , the natural map ,Spec,, XHomXAHomA XLRSRings  is bijective, so Spec :RingsLRS is left adjoint to op:LRSRings . Proof. This is a completely formal consequence of various adjunctions: W. D. GILLAM Copyright © 2011 SciRes. APM 257 Hom,Spec =Hom,*,=Hom,, *,=Hom, *,=Hom, ,.locXXXA XAXAXAAXLRSLRSPRSRSRings Theorem 8. The category Sch has all finite inverse limits, and the inclusion SchLRS preserves them. Proof. It is equivalent to show that, for a finite inverse limit system iiX in Sch, the inverse limit X in LRS is a scheme. It suffices to treat the case of (finite) products and equalizers. For products, suppose iX is a finite set of schemes and =iiXX is their product in LRS . We want to show X is a scheme. Let x be a point of X, and let =iiixxXRS be its image in the ringed space product. Let =SpeciiUA be an open affine neighborhood of ix in iX. As we saw above, the map iiXXRS is a localization and, as men- tioned in Remark 3, it follows that the product := iiUU of the iU in LRS is an open neighbor-hood of x in X,7 so it remains only to prove that there is an isomorphism Spec iiUA, hence U is affine.8 Indeed, we can see immediately from Proposition 7 that U and SpeciiA represent the same functor on LRS : Hom(,) =Hom,=Hom, ,=Hom,,= Hom,Spec.iiiYiii YiiYU YUAYAYYALRS LRSRingsRingsLRS The case of equalizers is similar: Suppose X is the LRS equalizer of morphisms ,:fgY Z of schemes, and xX. Let yY be the image of x in Y, so ()= ()=:fygy z. Since ,YZ are schemes, we can find affine neighborhoods =SpecVB of y in Y and =SpecWA of z in Z so that ,fg take V into W. As before, it is clear that the equalizer U of |,|:fVgVV W in LRS is an open neighborhood of xX, and we prove exactly as above that U is affine by showing that it is isomorphic to Spec of the coequalizer ##=()():CBfa gaaA of ##,:fgA B in Rings . Remark 5. The basic results concerning the existence of inverse limits in LRS and their coincidence with inverse limits in Sch are, at least to some extent, “folk theorems”. I do not claim originality here. The construc-tion of fibered products in LRS can perhaps be attrib-uted to Hanno Becker [HB], and the fact that a cartesian diagram in Sch is also cartesian in LRS is implicit in the [EGA] Erratum mentioned above. Remark 6. It is unclear to me whether the 2-category of locally ringed topoi has 2-fibered products, though Hakim seems to want such a fibered product in [Hak V.3.2.3]. 3.2. Fibered Products In this section, we will more closely examine the con-struction of fibered products in LRS and explain the relationship between fibered products in LRS and those in RS. By Theorem 8, the inclusion Sch LRS preserves inverse limits, so these results will generalize the basic results comparing fibered prod-ucts in Sch to those in RS (the proofs will become much more transparent as well). Definition 5. Suppose 1112 22,, ,,,,,,AkBkBkLAnmm m and :iifAB are LAn morphisms, so 1=iifmm for =1,2i. Let 12:jj AiB BB =1,2j be the natural maps. Set 112121112 2,, :=Spec:=,=.ASABBBBiippmpm (4) Note that the kernel K of the natural surjection  1212121 2AkBBkkbbb b is generated by the expressions 11m and 21m, where iimm, so 1212Spec SpeckAkk BB is an isomorphism onto 12,,SABB . In particular, 121 2,, =Spec:ASABBBBKpp is closed in 12SpecABB. The subset 12,,SABB enjoys the following impor-tant property: Suppose :, ,iiigB Cmn, =1,2i, are LAn morphisms with 112 2=gfgf and 7This is the only place we need “finite”. If iX were infinite, the topological space product of the iU might not be open in the topology on the topological space product of the iX because the product to-pology only allows “restriction in finitely many coordinates”. 8There would not be a problem here even if iX were infinite: Rings has all direct and inverse limits, so the (possibly infinite) ten-sor product iiA over  (coproduct in Rings) makes sense. Our proof therefore shows that any inverse limit (not necessarily finite) ofaffine schemes, taken in LRS, is a scheme. W. D. GILLAM Copyright © 2011 SciRes. APM 258 12 12=, :AhffB BC is the induced map. Then  112,,hSABBn. Conversely, every 12,,SABBp arises in this manner: take 12=ACB Bp. Setup: We will work with the following setup throughout this section. Let 11:fXY, 22:fXY be morphisms in LRS . From the universal property of fiber products we get a natural “comparison” map 1212:.YYXXX XLRSRS Let 12π:iY iXXXRS (=1,2i) denote the projec-tions and let 112 2:= π=πgff. Recall that the structure sheaf of 12YXXRS is 1111212ππXXgY. In par-ticular, the stalk of this structure sheaf at a point 12 12=, Yxxx XXRS is ,,11,22XxX xYy, where 1122:= ==.ygxf xfx In this situation, we set 12, ,,112 2,:=, ,YyX xXxSxx S  to save notation. Theorem 9. The comparison map  is surjective on topological spaces. More precisely, for any 12 12=, Yxxx XXRS , 1x is in bijective corre-spondence with the set 12,Sxx , and in fact, there is an LRS isomorphism 112,,11,2 21212,,11,2 2:= ,=Spec, .YXxXxYyXXYXxX xYyxX XxSxx LRS RS In particular, 1()x is isomorphic as a topological space to  1()2Spec kykx kx (but not as a ringed space). The stalk of  at 12,zSxx is identified with the localization map ,,,,1, 21, 2.XxXxXxXxYy Yyz In particular,  is a localization morphism (Definition 1). Proof. We saw in §3.1 that the comparison map  is identified with the localization of 12YXXRS at the prime system 12 12,,xxSxx, so these results follow from Proposition 1. Remark 7. When 12,,XXYSch, the first statement of Theorem 9 is [EGA I.3.4.7]. Remark 8. The fact that  is a localization morphism is often implicitly used in the theory of the cotangent complex. Definition 6. Let :fXY be an LRS morphism. A point xX is called rational over Y (or “over :=yfx “ or “with respect to f”) iff the map on residue fields :xfky kx is an isomorphism (equivalently: is surjective). Corollary 10. Suppose 11xX is rational over Y (i.e. with respect to 11:fXY). Then for any 12 12=, Yxxx XXRS , the fiber 1x of the comparison map  is punctual. In particular, if every point of 1X is rational over Y, then  is bijective. Proof. Suppose 11xX is rational over Y. Suppose 12 12=,xxx XXRS . Set 1122:= =yfx fx. Since 1x is rational, 1ky kx, so 1()2 2Spec Speckykx kxkx has a single ele- ment. On the other hand, we saw in Definition 5 that this set is in bijective correspondence with the set 12, ,11,22,SpecXxX xYySxx  appearing in Theorem 9, so that same theorem says that 1x consists of a single point. Remark 9. Even if every 1xX is rational over Y, the comparison map 1212:YYXXX XLRSRS is not generally an isomorphism on topological spaces, even though it is bijective. The topology on 12YXXLRS is generally much finer than the product topology. In this situation, the set 12,Sxx always consists of a single element 12,zxx : namely, the maximal ideal of ,,11,22XxX xYy given by the kernel of the natural surjection ,,12211,22=.XxX xkyYy kxkx kx If we identify 12YXXLRS and 12YXXRS as sets via , then the “finer” topology has basic open sets 121 212 1212,,:= ,:,YYxxUU UsxxU Uszxx as 12,UU range over open subsets of 12,XX and s ranges over 111121212ππ .XXYgYUU This set is not generally open in the product topology because the stalks of 1111212ππXXgY are not generally local rings, so not being in 12,zxx does not imply invertibility, hence is not generally an open condition on 12,xx. Remark 10. On the other hand, sometimes the topolo-gies on 1X, 2X are so fine that the sets 12,YUUU s are easily seen to be open in the product topology. For example, suppose k is a topological field.9 Then one often works in the full subcategory C of locally ringed spaces over k consisting of those 9I require all finite subsets of k to be closed in the definition of “topological field”. W. D. GILLAM Copyright © 2011 SciRes. APM 259XkLRS satisfying the conditions: 1) Every point xX is a k point: the composition ,Xxkkx yields an isomorphism =kkx for every xX. 2) The structure sheaf X is continuous for the topo- logy on k in the sense that, for every ,XUsSec, the function (_):sUkxsx is a continuous function on U. Here sxkx denotes the image of the stalk ,xXxs in the residue field ,()= Xxxkx m, and we make the identification =kkx using 1). One can show that fiber products in C are the same as those in LRS and that the forgetful functor CTop preserves fibered products (even though CRS may not). Indeed, given 111121212ππXXYgYsUU  , the set 12,YUUU s is the preimage of *kk under the map _s, and we can see that _s is continuous as follows: By viewing the sheaf theoretic tensor product as the sheafification of the presheaf tensor product we see that, for any point 12 1 2,YxxUU , we can find a neighborhood 12YVV of 12,xx contained in 12YUU and sections 111,,nXaa V, 122,,nXbb V such that the stalk ,12xxs agrees with  12iiixxab at each 121 2,YxxVV . In particular, the function _s agrees with the function  121 2,()iiixxaxbxk  on 12YVV. Since this latter function is continuous in the product topology on 12YVV (because each (_)ia, (_)ib is continuous) and continuity is local, _s is continuous. Corollary 11. Suppose 1,() ,1111:YfxX xxf is surjective for every 11xX. Then the comparison map  is an isomorphism. In particular,  is an isomor-phism under either of the following hypotheses: 1) 1f is an immersion. 2) 1:Spec fky Y is the natural map associated to a point yY. Proof. It is equivalent to show that 12:= YXXXRS is in LRS and the structure maps π:iiXX are LRS morphisms. Say 12=,xxx X and let  1122:= =yfx fx. By construction of X, we have a pushout diagram of rings  112122(),,11ππ,,22xxxxfYyX xfXx Xx hence it is clear from surjectivity of 11xf and locality of 22xf that ,Xx is local and 12π,πxx are LAn morphisms. Corollary 12. Suppose 2121π12 2π1YffXXXXY is a cartesian diagram in LRS. Then: 1) If 12YzX X is rational over Y, then 1=zz. 2) Let 12 12,YxxX XRS , and let 1122:= π=π.yxx Suppose 2kx is isomorphic, as a field extesion of ky, to a subfield of 1kx . Then there is a point 12YzX XSch rational over 1X with π()=iizx, =1,2i. Proof. For 1), set 12,:=xxz, 1122:= π=πyxx. Then we have a commutative diagram  2,2,1, 21, 1π2π1ZxZxffkz kxkx ky of residue fields. By hypothesis, the compositions iky kxkz are isomorphisms for =1,2i, so it must be that every map in this diagram is an isomorphism, hence the diagram is a pushout. On the other hand, according to the first statement of Theorem 9, 1z is in bijective correspondence with 12Spec= Spec,kykx kxkz which is punctual. For 2), let 21:ikx kx be the hypothesized morphism of field extensions of ()ky. By the universal property of the LRS fibered product 12YXX, the maps 212,:SpecxikxX 111:Specxkx X give rise to a map 112:Spec .YgkxXX Let 12YzX X be the point corresponding to this map. Then we have a commutative diagram of residue fields 1kx1kx 2kx ky kz 1,πZ i W. D. GILLAM Copyright © 2011 SciRes. APM 260 so 1, 1π:()zkx kz must be an isomorphism. 3.3. Spec Functor Suppose XLRS and :XfA is an X algebra. Then f may be viewed as a morphism of ringed spaces :,,=XfXA XX. Give X the local prime system X as usual and ,XA the inverse image prime system  (Remark 1), so f may be viewed as a PRS morphism *:,,,, .XXXfXAfX Explicitly:  *1,=:()=.XxxxXxxfAfppm By Theorem 2, there is a unique LRS morphism loc loc*:,,,, =XXXfXAf XX lifting f to the localizations. We call loc*Spec:=, ,XXAXAf the spectrum (relative to X) of A. SpecX defines a functor opSpec :.XXXXRings LRS Note that locSpec=, ,=XXXXXX by Theo- rem 2. Our functor SpecX agrees with the usual one (c.f. [Har II.Ex.5.17]) on their common domain of definition: Lemma 13. Let :fXY be an affine morphism of schemes. Then *SpecXXf (as defined above) is na- turally isomorphic to X in YLRS . Proof. This is local on Y, so we can assume = Spec YA is affine, and hence =Spec XB is also affine, and f corresponds to a ring map #:fAB. Then *==,YXAYYfBB as Y algebras, and the squares in the diagram  *#**#,, ,,,, ,,*, ,Spec*, ,SpecXY YYYYYYf fYYBfNYA NBB AA  in PRS are cartesian in PRS , where N is the prime system on ,YYA given by =yNy discussed in Lemma 3. According to that lemma, the right vertical arrows become isomorphisms upon localizing, and according to Theorem 4, the diagram stays cartesian upon localizing, so the left vertical arrows also become isomorphisms upon localizing, hence *#**Spec:= ,,= Spec =.locYXX YfYffBX Remark 11. Hakim [Hak IV.1] defines a “Spec func-tor” from ringed topoi to locally ringed topoi, but it is not the same as ours on the common domain of definition. There is no meaningful situation in which Hakim’s Spec functor agrees with the “usual” one. When X “is” a locally ringed space, Hakim’s SpecX “is” (up to re-placing a locally ringed space with the corresponding locally ringed topos) our ,locXX. As mentioned in Remark 2, Hakim’s theory of localization is only devel-oped for the terminal prime system, which can be a bit awkward at times. For example, if X is a locally ringed space at least one of whose local rings has positive Krull dimension, Hakim’s sequence of spectra yields an infi-nite strictly descending sequence of RS morphisms SpecSpec Spec .XXX The next results show that SpecX takes direct limits of X algebras to inverse limits in LRS and that SpecX is compatible with changing the base X. Lemma 14. The functor SpecX preserves inverse limits. Proof. Let :iX iif A be a direct limit sys-tem in XXRings, with direct limit :XfA, and structure maps :iijA A. We claim that *Spec=, ,locXXAXAf is the inverse limit of *Spec=, ,locXii iXiAXAf. By Theorem 4, it is enough to show that *,, XXAf is the inverse limit of *,,ii XiXAf in PRS . Certainly ,XA is the inverse limit of ,iiXA in RS , so we just need to show that ***=XiiiXfjf as prime systems on ,XA (see the proof of Theorem 4), and this is clear because =iijff , so, in fact, ***=ii XXjf f for every i. Lemma 15. Let :fXY be a morphism of locally ringed spaces. Then for any Y algebra :YgA, the diagram *Spec SpecXYXfYAA is cartesian in LRS . Proof. Note *11:= XfYfAf A as usual. One sees easily that W. D. GILLAM Copyright © 2011 SciRes. APM 261 **1 *,, ,,,, ,,XYXX YYXfA f gYAgXY  is cartesian in PRS so the result follows from Theorem 4. Example 1. When X is a scheme, but A is not a coherent X module, SpecXA may not be a scheme. For example, let B be a local ring, := SpecXB, and let x be the unique closed point of X. Let *:=AxB XRings be the skyscraper sheaf B sup-ported at x. Note ,=Xx B and *,Hom,= Hom,,XXxXxBBRingsRings  so we have a natural map XA in XRings whose stalk at x is :IdB B. Then Spec =,XAxA is the punctual space with “sheaf” of rings A, mapping in LRS to X in the obvious manner. But ,xA is not a scheme unless A is zero dimensional. Here is another related pathology example: Proceed as above, assuming B is a local domain which is not a field and let K be its fraction field. Let *:=AxK, and let XA be the unique map whose stalk at x is BK. Then SpecXA is empty. Suppose X is a scheme, and A is an X algebra such that SpecXA is a scheme. I do not know whether this implies that the structure morphism SpecXAX is an affine morphism of schemes. 3.4. Relative Schemes We begin by recalling some definitions. Definition 7. ([SGA1], [Vis 3.1]) Let :FCD be a functor. A C morphism :fcc is called cartesian (relative to F) iff, for any C morphism :gcc  and any D morphism :hFc Fc with =Fgh Ff there is a unique C morphism :hc c with =Fhh and =fgh. The functor F is called a fibered category iff, for any D morphism :fdd and any object c of C with =Fcd, there is a cartesian morphism :fcc with =Fff. A morphism of fibered categories ::'FCD FCD is a functor :'GC C satisfying =FGF and taking cartesian arrows to cartesian arrows. If D has a topology (i.e. is a site), then a fibered category :FCD is called a stack iff, for any object dD and any cover idd of d in D, the category 1Fd is equivalent to the category iFdd of descent data (see [Vis 4.1]). Every fibered category F admits a morphism of fibered categories, called the associated stack, to a stack universal among such morphisms [Gir I.4.1.2]. Definition 8. ([Hak V.1]) Let X be a ringed space. Define a category preXSch as follows. Objects of preXSch are pairs ,UUX consisting of an open subset UX and a scheme UX over Spec XU. A morphism ,,UVUX VX is a pair ,UVUVX X consisting of an Ouv X morphism UV (i.e. UV) and a morphism of schemes UVXX mak- ing the diagram  Spec SpecUVXXXXUV (5) commute in Sch . The forgetful functor preXXSchOuv is clearly a fibered category, where a cartesian arrow is a preXSch morphism ,UVUVX X making (6) cartesian in Sch (equivalently in LRS ). Since XOuv has a topology, we can form the associated stack XSch . The category of relative schemes over X is, by definition, the fiber category XXSch of XSch over the terminal object X of XOuv . (The definition of relative scheme makes sense for a ringed topos X with trivial modifications.) 3.5. Geometric Realization Now let X be a locally ringed space. Following [Hak V.3], we now define a functor :()XXFXXSchLRS called the geometric realization. Although a bit abstract, the fastest way to proceed is as follows: Definition 9. Let XLRS be the category whose objects are pairs ,UUX consisting of an open subset UX and a locally ringed space UX over ,XUU, and where a morphism ,,UVUX VX is a pair ,UVUVX X consisting of an XOuv morphism UV (i.e. UV) and an LRS morphism UVXX making the diagram  ,,UVXXXXUU VV (7) commute in LRS. The forgetful functor ,UUX U makes XLRS a fibered category over XOuv where a cartesian arrow is a morphism ,UVUVX X making (8) cartesian in LRS. In fact the fibered category XXLRS Ouv is a stack: one can define locally ringed spaces and mor- W. D. GILLAM Copyright © 2011 SciRes. APM 262 phisms thereof over open subsets of X locally. Using the universal property of stackification, we define XF to be the morphism of stacks (really, the corresponding morphism on fiber categories over the terminal object XXOuv ) associated to the morphism of fibered categories :pre preXXXFSchLRS Spec(),, ,.UUUXXUXUXU ULRS The map ,SpecXXUU U is the adjunc-tion morphism for the adjoint functors of Proposition 7. This functor clearly takes cartesian arrows to cartesian arrows. Remark 12. Although we loosely follow [Hak V.3.2] in our construction of the geometric realization, our geo-metric realization functor differs from Hakim’s on their common domain of definition. 3.6. Relatively Affine Morphisms Let :fXY be an LRS morphism. Consider the following conditions: RA1. Locally on Y there is an YY algebra A and a cartesian diagram SpecSpecfYXAYY in LRS . RA2. There is an Y algebra A so that f is isomorphic to SpecYA in YLRS . RA3. Same condition as above, but A is required to be quasi-coherent. RA4. For any :gZY in YLRS , the map **#*Hom, Hom,YXZYYZXf ghghLRS Rings is bijective. Remark 13. The condition (RA1) is equivalent to both of the following conditions: [label = RA1.., ref = RA1] RA1.1 Locally on Y there is a ring homomorphism AB and a cartesian diagram SpecSpecfXBYA in LRS . RA1.2. Locally on Y there is an affine morphism of schemes XY and a cartesian diagram fXXYY in LRS . The above two conditions are equivalent by definition of an affine morphism of schemes, and one sees the equivalence of (RA1) and (RA1.1) using Proposition 7, which ensures that the map Spec YA in (RA1) factors through Spec YYY, hence Spec Spec SpecSpec=Spec=Spec Spec=Spec .AYAYYYAYYXY BYYBYYB Each of these conditions has some claim to be the definition of a relatively affine morphism in LRS. With the exception of (2), all of the conditions are equivalent, when Y is a scheme, to f being an affine morphism of schemes in the usual sense. With the exception of (4), each condition is closed under base change. For each pos- sible definition of a relatively affine morphism in LRS , one has a corresponding definition of relatively schema- tic morphism, namely: :fXY in LRS is rela- tively schematic iff, locally on X, f is relatively af- fine. The notion of “relatively schematic morphism” ob- tained from (1) is equivalent to: :fXY is in the essential image of the geometric realization functor YF. 3.7. Monoidal Spaces The setup of localization of ringed spaces works equally well in other settings; for example in the category of monoidal spaces. We will sketch the relevant definitions and results. For our purposes, a monoid is a set P equipped with a commutative, associative binary opera- tion + such that there is an element 0P with 0=pp for all pP. A morphism of monoids is a map of sets that respects + and takes 0 to 0. An ideal of a monoid P is a subset IP such that IPI. An ideal I is prime iff its complement is a submonoid (in particular, its complement must be non-empty). A sub- monoid whose complement is an ideal, necessarily prime, is called a face. For example, the faces of 2 are 0,0 , 0, and 0; the diagonal 2: is a submonoid, but not a face. If SP is a submonoid, the localization of P at S is the monoid 1SP whose elements are equivalence classes ,ps, pP, sS where ,= ,psps iff there is some tS with =tps tps , and where ,,:=,psp sppss. The natural map 1PSP given by ,0pp is initial among mon- W. D. GILLAM Copyright © 2011 SciRes. APM 263oid homomorphisms :hP Q with *hS Q. The localization of a monoid at a prime ideal is, by definition, the localization at the complementary face. A monoidal space ,XX is a topological space X equipped with a sheaf of monoids X. Monoidal spaces form a category MS where a morphism †=, :,,XYfff XY consists of a con- tinuous map :fXY together with a map  of sheaves of monoids on X. A monoidal space ,XX is called local iff each stalk monoid X has a unique maximal ideal xm. Local monoidal spaces form a category LMS where a morphism is a map of the underlying monoidal spaces such that each stalk map †,() ,:xYfx Xxf is local in the sense 1†=xfxfmm. A primed monoidal space is a mo- noidal space equipped with a set of primes xM in each stalk monoid ,Xx. The localization of a primed mo- noidal space is a map of monoidal spaces loc,,,XXXM X from a local monoidal space constructed in an obvious manner analogous to the con- struction of §2.2 and enjoying a similar universal pro- perty. In particular, we let SpecP denote the locali- zation of the punctual space with “sheaf” of monoids P at the terminal prime system. A scheme over 1 is a locally monoidal space locally isomorphic to SpecP for various monoids P. (This is not my terminology.) The same “general nonsense” arguments of this paper allow us to construct inverse limits of local monoidal spaces, to prove that a finite inverse limit of schemes over 1, taken in local monoidal spaces, is again a scheme over 1, to construct a relative Spec functor opSpec :,XXXXMon LMS for any ,XXLMS which preserves inverse limits, and to prove that the natural map Hom, ,SpecHom,XXXPPXLMS Mon is bijective. 4. Acknowledgements This research was partially supported by an NSF Post-doctoral Fellowship. 5. References [1] M. Hakim, “Topos Annelés et Schémas Relatifs. Ergeb-nisse der Mathematik und ihrer Grenzgebiete,” Springer- Verlag, Berlin, 1972. [2] R. Hartshorne, “Algebraic Geometry,” Springer-Verlag Berlin, 1977. [3] H. Becker, Faserprodukt in LRS. http://www.uni-bonn.de/~habecker/Faserprodukt―in―LRS.pdf. [4] L. Illusie, “Complexe Cotangent et Deformations I. L.N.M. 239,” Springer-Verlag, Berlin, 1971. [5] A. Grothendieck and J. Dieudonné, “Éléments de Géométrie Algébrique,” Springer, Berlin, 1960. [6] J. Giraud, “Cohomologie non Abélienne,” Springer, Ber-lin, 1971. [7] A. Vistoli, “Notes on Grothendieck Topologies, Fibered Categories, and Descent Theory,” Citeseer, Princeton 2004.