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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 x M 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 Let Top , LRS, RS, and Sch denote the categories of topological spaces, locally ringed spaces, ringed spaces, and schemes, respectively. Consider maps of schemes : ii f XY(=1,2i) and their fibered product 12Y X X as schemes. Let X denote the topological space underlying a scheme X . There is a natural comparison map 12 12 :YY X XXX which is not generally an isomorphism, even if 12 ,, X XY 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 1 12 , x x over a point 12 12 ,Y x xXX (with common image 1122 == y fx fx) is naturally bijective with the set 12 Spec . ky kx kx In fact, one can show that this bijection is a homeomorphism when 1 12 , x x is given the topo- logy it inherits from 12Y X X. One can even describe the sheaf of rings 1 12 , x x inherits from 12Y X X as follows: Let 12,, , 11,22 ,:= Spec: =for =1,2. XxX xXx Yyi i xi Sxx zz i Then (Spec of) the natural surjection ,,1()2 11,22 XxX xky Yy kx kx identifies 12 ky Speckx kx with a closed subspace of ,, 11,2 2 Spec XxX x Yy and 1 12 12 , XX Y x x naturally coincides, under the EGA isomorphism, to the restriction of the structure sheaf of ,, 11,22 Spec XxX x Yy to the closed subspace 1()2, , 11,2 2 Spec Spec. kyX xXx Yy kx 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 12Y X XRS (which carries the data of the rings ,, 11,22 XxX x Yy as stalks of its structure sheaf) and 2) the subsets 12, , 11,22 ,Spec XxX x Yy Sxx It turns out that one can actually recover 12Y X X as a scheme solely from this data, as follows: Given a pair , X M consisting of a ringed space X and a subset , Spec x Xx M for each x X, one can construct a locally ringed space ,loc XM with a map of ringed spaces ,loc X MX. 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 12 Spec ky kx kx is “quasi-coherent”. It isn’t even a module over the usual structure sheaf of 12 Spec ky kx kx W. D. GILLAM Copyright © 2011 SciRes. APM 251 1212 ,, Y XXSxxRS yields the comparison map , and, in particular, the scheme 12Y X X. 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 XLRS , a very general relative spec functor op Spec : XX X 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 X LRS .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 XLRS, 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 , f f, 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 : f AB 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 1 A SA of A at S.4 A morphism of ringed spaces : f XY is called a localization morphism iff #1 :YX ff is a lo- calization morphism. A localization morphism A B in X Rings is both flat and an epimorphism in X Rings .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 = X SpecA , XX A 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 I A is an ideal, and we think of Spec A I as a subset of Spec A , then SpecSpec Spec A A IAI (though they have the same topological space). 2.1. Prime Systems Definition 3. Let =, X XX be a ringed space. A prime system M on X is a map x x M assigning a subset , Spec x Xx M to each point x X . For prime systems , M N on X we write M N to mean x x M N for all x X . Prime systems on X form a category X PS where there is a unique mor- phism from M to N iff M N. The intersection ii M of prime systems i M XPS is defined by := . iii i x x MM A primed ringed space , X M is a ringed space X equipped with a prime system M . Prime ringed spaces form a category PRS where a morphism :, f XM ,YN is a morphism of ringed spaces f satisfy- 2Hakim already constructed such a functor, but ours is different fro m 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 x xfx fM N for every x X. The inverse limit of a functor i iM to PS X is clearly given by ii M . Remark 1. Suppose ,YNPRS and : f XY is an RS morphism. The inverse image * f N is the prime system on X defined by 1 * 1 , :=Spec =Spec: . xfx x Xx xfx fNf N fN Formation of inverse image prime systems enjoys the expected naturality in f : * ** = g fMfgM. We can alternatively define a PRS morphism :, f XM ,YN to be an RS morphism : f XY such that c* M fN (i.e. together with a X PS mor- phism * M fN). For XLRS , the local prime system X on X is defined by ,:= Xx x m. If Y is another locally ringed space, then a morphism : f XY in RS defines a morphism of primed ringed spaces :, , XY fX Y iff f is a morphism in LRS , so we have a fully faithful functor : ,, X LRS PRS XX (1) and we may regard LRS as a full subcategory of PRS. At the “opposite extreme” we also have, for any XRS , the terminal prime system X defined by ,, := Spec Xx Xx (i.e. the terminal object in X PS ). For ,YMPRS , we clearly have ,,, =,, X H omY MXHomY X PRS RS so the functor : ,X XX RS PRS (2) is right adjoint to the forgetful functor PRSRS given by , X MX. 2.2. Localization Now we begin the main construction of this section. Let , X M be a primed ringed space. We now construct a locally ringed space ,loc XM (written loc X if M is clear from context), and a PRS morphism π:, , loc loc X X XM 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 s UF, and where there is a unique morphism ,,UsVt if UV and .tU s . As a set, the topological space loc X will be the set of pairs , x z, where x X and x zM. Let loc X denote the category of subsets of loc X whose mor- phisms are inclusions. For ,X UsSec, set ,:= ,:,. loc x UUsxzXxUsz This defines a functor :loc X UXSec satisfying: >0 ,,=,|| ,= ,. UV UV n UUsUVtUU Vst UUs UUsn The first formula implies that loc X UXSec is a basis for a topology on loc X where a basic open neighborhood of , x z is a set ,UUs where x U , x s z . We always consider loc X with this topology. The map π: , loc X X x zx is continuous because 1 π=,1UUU . We construct a sheaf of rings loc X on loc X as follows. For an open subset loc VX, we let loc XV be the set of , , =, Xx z xz V ssxz satisfying the local consistency condition: For every , x zV , there is a basic open neighborhood ,UUt of , x z contained in V and a section X nt aU t such that, for every ,, x zUUt , we have , ,= . xXx n z x a sxzt (Of course, one can always take =1n since ,= , n UUtUUt .) The set loc XV becomes a ring under coordinatewise addition and multiplication, and the obvious restriction maps make loc X a sheaf of rings on loc X . There is a natural isomorphism , ,, = locX x z Xxz taking the germ of =, loc X s sxzU in the stalk ,, loc Xxz to , ,Xx z sxz. This map is injective be- cause of the local consistency condition and surjective W. D. GILLAM Copyright © 2011 SciRes. APM 253 because, given any ,Xx z ab, we can lift ,ab to ,X ab U on some neighborhood U of x and define , loc X s UUb by letting , ,:= . xXx x z sxza b This s manifestly satisfies the local consistency condition and has ,= s xz ab . In particular, loc X , with this sheaf of rings, is a locally ringed space. To lift π to a map of ringed spaces π:loc X X we use the tautological map # * π:π Xloc X of sheaves of rings on X defined on an open set UX by # * π:π=,1 . Xlocloc XX xz UUU UU ss It is clear that the induced map on stalks ,, , ,(,) π:= xz XxlocXx z Xxz is the natural localization map, so 1 , π= x zz x zM and hence π defines a PRS morphism π:, , loc loc X X XM. Remark 2. It would have been enough to construct the localization ,loc X X at the terminal prime system. Then to construct the localization ,loc XM at any other prime system, we just note that ,loc XM is clearly a subset of ,loc X X, and we give it the topol- ogy and sheaf of rings it inherits from this inclusion. The construction of ,loc X X is “classical.” Indeed, M. Hakim [Hak] describes a construction of ,loc X X 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 , X M be a primed ringed space with localization π:loc X X. For x X, the fiber 1 π x is naturally isomorphic in LRS to Spec x M (Definition 2).6 Under this identification, the stalk of π at x zM 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 =π x M x taking x zM to 1 ,π x zx . We first show that the topology inher- ited from loc X coincides with the one inherited from , Spec Xx . By definition of the topology on loc X , a ba- sic open neighborhood of x zM is a set of the form ,= :, xxx UUsMzM sz where U is a neighborhood of x in X and X s U satisfies x s z . Clearly this set depends only on the stalk of , x Xx s of s at x , and any element ,Xx t lifts to a section () X tU on some neighborhood of X , so the basic neighborhoods of x zM are the sets of the form : x zMtz where x tz . But for the same set of t, the sets , := Spec : Xx Dt 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 x M inher- ited from loc X is the same as the one inherited from , Spec Xx . Given ,Xx f , a section of locx X M over the basic open set x M Df is an element , =Xx z zM Df x ssz satisfying the local consistency condition: For all x zM Df , there is a basic open neighborhood ,UUt of , x z in loc X and an element nXt at U such that, for all , x zMDf UUt , we have n z z s zat . Note that ,= x xx M DfUUt MDft and Spec , n x xx Xx at Dft . The sets x x Dft M cover , Spec x Xx MDf, and we have a “global formula” s showing that the stalks of the various n x x at agree at any x zM Df , so they glue to yield an element Spec ,x Xx g sMDf with = z g ssz. We can define a morphism of sheaves on x M by defining it on basic opens, so this defines a morphism of sheaves Spec , :locxx Xx X g MM which is easily seen to be an isomorphism on stalks. Remark 3. Suppose ,XMPRS and UX is an open subspace of X . Then it is clear from the con- struction of π:,loc X MX that 1 π=, ,loc X UUUMU . 6By “fiber” here we mean 1RS , π:= , loc X Xx xXx , which is jus t the set theoretic preimage 1 πloc x X with the topology and shea f of rings it inherits from loc X. This differs from another common usage of “fiber” to mean RS , loc X Xxkx. W. D. GILLAM Copyright © 2011 SciRes. APM 254 The following theorem describes the universal prop- erty of localization. Theorem 2. Let :, , f XM YN be a morphism in PRS . Then there is a unique morphism :, , loc loc fXMYN in LRS making the diagram ππ ,, loc loc f f XM YN XY (3) commute in RS . Localization defines a functor PRSLRS ,, loc XM XM :,, :,, loc loc fXMYNfXMYN retracting the inclusion functor :LRSPRS and right adjoint to it: For any Y LRS , there is a natural bijection ,,=,,, . loc Y HomYX MHomYX M LRSPRS 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 x f x MN 1 x zfz for each x X, so we can complete the diagram of topological spaces ππ ,loc f loc f XYN XY (at least on the level of sets) by setting 1 ,:= ,. loc x fxzfx fzY To see that f is continuous it is enough to check that the preimage 1, f UUs is open in loc X for each basic open subset ,UUs of loc Y. But it is clear from the definitions that 11#1 ,= , f UUsU fUffs (note #1 =xfx x ffsf s ). Now we want to define a map #1 :loc Y X ff of sheaves of rings on Y (with “local stalks”) making the diagram # -1 # 1 π 111 ππ loc loc f XY f XX f f commute in loc YRings . The stalk of this diagram at (,) loc x zX is a diagram , 1 , , 1 ,, π π ,, xz fxfz x xz x f Xx x Yf x z Xx Yfx f fz 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 , x z f completing this diagram. We now want to show that there is actually a (necessarily unique) map #1 :loc Y X ff of sheaves of rings on loc X whose stalk at , x z is the map , x z f. By the universal property of sheafification, we can work with the presheaf inverse image 1 pre loc X f instead. A section ,Vs of this presheaf over an open subset loc WX is represented by a pair ,Vs where loc VY is an open subset of loc Y containing f W and , (,) =, . locY y z Yyz V ssyz V I claim that we can define a section # p, re loc X f Vs W by the formula #1 p,,:=,. re x f Vsxzs fxfz It is clear that this element is independent of replacing V with a smaller neighborhood of f W and restricting s, but we still must check that p, (,) , reX x z xz W fVs satisfies the local consistency condition. Suppose X nt aU t witnesses local consistency for loc Y s V on a basic open subset ,UUt V. Then it is straightforward to check that the restriction of #1 1 #1 , Y n ffa f UUt fft to 1, f UUt W witnesses local consistency of # p, re f Vs on 11#1 ,= ,. f UUtWUfUfftW It is clear that our formula for # p, re f Vs 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 loc f XY in W. D. GILLAM Copyright © 2011 SciRes. APM 255 LRS making (3) commute in RS . We now establish the uniqueness of f . Suppose :loc loc f XY is a morphism in LRS that also makes (3) commute in RS . We first prove that = f f on the level of topological spaces. For x X the commutativity of (3) ensures that ,= , f xzfxz for some (), () Spec , f xYfx zN so it remains only to show that 1 =. x zfz The commutativity of (3) on the level of stalks at ,loc x zX gives a commutative diagram of rings , -1 # ,, π, π, π ,, xz f Xx Yf x z z fxz xz f Xx Yf x where the vertical arrows are the natural localization maps. From the commutativity of this diagram and the fact that 1 ,()= z z xz f mm (because , x z f is local) we find 1 (), 1 1 (), , 11 1 =π() =π =π =() z fxz z fxz xz xx z x z f f fz m m m as desired. This proves that = f f 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:, loc X X X is an isomorphism for any XLRS . 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 ,Sec X Us, the image of the basic open set ,UUs under π is open in X . Indeed, * , π,=: =: xx x Xx UUsxU s xUs 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:Xloc X is an iso- morphism of sheaves of rings on =loc X X. Indeed, Proposition 1 says the stalk of # π at ,loc x x Xm is the localization of the local ring ,Xx at its unique maximal ideal, which is an isomorphism in LAn . Lemma 3. Let ARings be a ring, ,:=Spec X X A, and let*be the punctual space. Define a prime system N on ,X X A by , :=Spec=Spec = . Xx x Nx AAX Let :,, X X aX XA be the natural RS morphism. Then * ,= XXaN and the natural PRS morphisms , *, (,,),,*,,Spec =*,, X XXX A X XA NAA A yield natural isomorphisms , ,=,,=,, =*, ,Spec loc loc X XX XX loc XX XAN AA in LRS . Proof. Note that the stalk ,, :Xx x Xx aA of a at x X is the localization map x A A, and, by defi- nition, * x aN is the set prime ideals z of x A pulling back to x A under : x x aA A. The only such prime ideal is the maximal ideal x x A m, so * ,, ={ }= xXx xX aN 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 ,,*,, x x xxxxm 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 X A. This is clear for , ,, XXX X be- cause localization retracts (Theorem 2), so , ,, =, loc XX XX XX , and it is clear for *,,Spec A A because of the description of the fibers of localization in Proposition 1. For ,, X X AN, 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 ,,loc X XA N. Since U is connected, = X s AU 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:loc X X is not generally a mor- phism in LRS , even though ,loc XX 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 loc X is a point with the “sheaf” of rings A p, and the “stalk” of # π is the localization map :lA Ap. Even though , A Ap are local, this is not a local morphism because 1=lA p ppm. 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 , ii iXM is an inverse limit sys- tem in PRS . Let X be the inverse limit of i iX in Top and let π: ii X X be the projection. Let X be the direct limit of 1 πiX i i in X Rings and let #1 π:π iiX X i be the structure map to the direct limit, so we may regard =, X XX as a ringed space and πi as a morphism of ringed spaces i X X. It imme- diate from the definition of a morphism in RS that X is the inverse limit of i iX in RS. Let i M be the prime system on X given by the inverse limit (inter- section) of the * πii M . Then it is clear from the definition of a morphism in PRS that , X M is the inverse limit of , ii iXM, but we will spell out the details for the sake of concreteness and future use. Given a point =i x xX, we have defined x M to be the set of , SpecXx z such that 1 ,, πSpec ixxX x iii zM for every i, so that πi defines a PRS morphism π:, , iii X MXM. To see that , X M is the direct limit of , ii iXM suppose :, , iii f YNXM are morphisms defining a natural transformation from the constant functor ,iYN to , ii iXM. We want to show that there is a unique PRS morphism :, , f YN XM with π= ii f f for all i. Since X is the inverse limit of iX in RS , we know that there is a unique map of ringed spaces : f YX with π= ii f f for all i, so it suffices to show that this f is a PRS morphism. Let y Y, y zN. We must show 1 y f x fzM . By definition of M , we must show 11 π () π= iy f y fx fx i i fz MM for every i. But π= ii f f implies () π= yii f xy f f, so 11 1 () π= iy i fx y f zfz is in i f y M because i f 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 i iX is an inverse limit system in LRS . Composing with yields an inverse limit sys- tem , iX i iX in PRS . By the theorem, the lo- calization ,loc XM of the inverse limit , X M of , iX i iX is the inverse limit of ,loc iX i iX in LRS . But localization retracts (Theorem 2) so ,loc iX i iX is our original inverse limit system i iX. We can also obtain the following result of C. Chevalley mentioned in [Hak IV.2.4]. Corollary 6 The functor ,loc X XX 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 YLRS we have ,, =,,, =,. loc X YX HomY X Hom YX HomY X LRS PRS RS Our next task is to compare inverse limits in Sch to those in LRS . Let * Top be “the” punctual space (terminal object), so *=Rin g sRin g s. The functor *, A A Rings RS is clearly left adjoint to op : ,. X XX RS Rings By Lemma 3 (or Proposition 1) we have *,:=*,,Spec =Spec . loc loc A AA A 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 ARings , XLRS , the natural map ,Spec,, X HomXAHomA 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, ,. loc X X XA XA X A XA AX LRSLRS PRS RS Rings 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 i iX 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 i X is a finite set of schemes and =i i X X is their product in LRS . We want to show X is a scheme. Let x be a point of X , and let =ii i x xXRS be its image in the ringed space product. Let =Spec ii UA be an open affine neighborhood of i x in i X . As we saw above, the map i i X XRS is a localization and, as men- tioned in Remark 3, it follows that the product := i i UU of the i U in LRS is an open neighbor- hood of x in X ,7 so it remains only to prove that there is an isomorphism Spec ii UA , hence U is affine.8 Indeed, we can see immediately from Proposition 7 that U and Specii A represent the same functor on LRS : Hom(,) =Hom, =Hom, , =Hom,, = Hom,Spec. i i iY i ii Y ii YU YU AY AY YA LRS LRS Rings Rings LRS The case of equalizers is similar: Suppose X is the LRS equalizer of morphisms ,: f gY Z of schemes, and x X. Let y Y be the image of x in Y, so ()= ()=: f ygy z. Since ,YZ are schemes, we can find affine neighborhoods =SpecVB of y in Y and =SpecWA of z in Z so that , f g take V into W. As before, it is clear that the equalizer U of |,|: f VgVV W in LRS is an open neighborhood of x X, and we prove exactly as above that U is affine by showing that it is isomorphic to Spec of the coequalizer ## =()():CBfa gaaA of ## ,: f gA 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 ,, ,,,,,,AkBkBkLAnmm m and : ii f AB are LAn morphisms, so 1= ii fmm for =1,2i. Let 12 : jj A iB BB =1,2j be the natural maps. Set 1 12121 1 12 2 ,, :=Spec: =,=. A SABBBBi i pp mpm (4) Note that the kernel K of the natural surjection 1212 121 2 Ak BBkk bbb b is generated by the expressions 11m and 2 1m , where ii m m, so 1212 Spec Spec kA kk BB is an isomorphism onto 12 ,,SABB . In particular, 121 2 ,, =Spec: A SABBBBKpp is closed in 12 SpecA BB. The subset 12 ,,SABB enjoys the following impor- tant property: Suppose :, , iii gB Cmn , =1,2i, are LAn morphisms with 112 2 = g fgf and 7This is the only place we need “finite”. If i X were infinite, the topological space product of the i U might not be open in the topology on the topological space product of the i X because the product to- p ology only allows “restriction in finitely many coordinates”. 8There would not be a problem here even if i X were infinite: Rings has all direct and inverse limits, so the (possibly infinite) ten- sor product ii A over (coproduct in Rings) makes sense. Our p roof therefore shows that any inverse limit (not necessarily finite) o f affine schemes, taken in LRS, is a scheme. W. D. GILLAM Copyright © 2011 SciRes. APM 258 12 12 =, :A hffB BC is the induced map. Then 1 12 ,,hSABB n. Conversely, every 12 ,,SABBp arises in this manner: take 12 =A CB Bp. Setup: We will work with the following setup throughout this section. Let 11 : f XY, 22 : f XY be morphisms in LRS . From the universal property of fiber products we get a natural “comparison” map 1212 :. YY X XX X LRSRS Let 12 π: iY i X XX RS (=1,2i) denote the projec- tions and let 112 2 := π=π g ff. Recall that the structure sheaf of 12Y X XRS is 11 112 12 ππ XX gY . In par- ticular, the stalk of this structure sheaf at a point 12 12 =, Y x xx XX RS is ,, 11,22 XxX x Yy , where 1122 := ==. y gxf xfx In this situation, we set 12, ,, 112 2 ,:=, , YyX xXx Sxx S to save notation. Theorem 9. The comparison map is surjective on topological spaces. More precisely, for any 12 12 =, Y x xx XX RS , 1 x is in bijective corre- spondence with the set 12 ,Sxx , and in fact, there is an LRS isomorphism 1 12,, 11,2 2 12 12 ,, 11,2 2 := , =Spec, . YXxXx Yy XX Y XxX x Yy xX Xx Sxx LRS RS In particular, 1() x is isomorphic as a topological space to 1()2 Spec ky kx kx (but not as a ringed space). The stalk of at 12 ,zSxx is identified with the localization map ,,,, 1, 21, 2 . XxXxXxXx Yy Yy z In particular, is a localization morphism (Definition 1). Proof. We saw in §3.1 that the comparison map is identified with the localization of 12Y X XRS at the prime system 12 12 ,, x xSxx, 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 : f XY be an LRS morphism. A point x X is called rational over Y (or “over := y fx “ or “with respect to f ”) iff the map on residue fields : x f ky kx is an isomorphism (equivalently: is surjective). Corollary 10. Suppose 11 x X is rational over Y (i.e. with respect to 11 : f XY). Then for any 12 12 =, Y x xx XX RS , the fiber 1 x of the comparison map is punctual. In particular, if every point of 1 X is rational over Y, then is bijective. Proof. Suppose 11 x X is rational over Y. Suppose 12 12 =, x xx XX RS . Set 1122 := = y fx fx. Since 1 x is rational, 1 ky kx, so 1()2 2 Spec Spec ky kx 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 ,Spec XxX x Yy Sxx appearing in Theorem 9, so that same theorem says that 1 x consists of a single point. Remark 9. Even if every 1 x X is rational over Y, the comparison map 1212 :YY X XX X LRSRS is not generally an isomorphism on topological spaces, even though it is bijective. The topology on 12Y X XLRS 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 ,, 11,22 XxX x Yy given by the kernel of the natural surjection ,,122 11,22=. XxX xky Yy kxkx kx If we identify 12Y X XLRS and 12Y X XRS as sets via , then the “finer” topology has basic open sets 12 1 212 1212 , ,:= ,:, YY xx UU UsxxU Uszxx as 12 ,UU range over open subsets of 12 , X X and s ranges over 11 11212 12 ππ . XXY gYUU This set is not generally open in the product topology because the stalks of 11 112 12 ππ XX gY are not generally local rings, so not being in 12 ,zxx does not imply invertibility, hence is not generally an open condition on 12 , x x. Remark 10. On the other hand, sometimes the topolo- gies on 1 X , 2 X are so fine that the sets 12 , Y UUU 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 o f “topological field”. W. D. GILLAM Copyright © 2011 SciRes. APM 259 X kLRS satisfying the conditions: 1) Every point x X is a k point: the composition ,Xx kkx yields an isomorphism =kkx for every x X. 2) The structure sheaf X is continuous for the topo- logy on k in the sense that, for every ,X UsSec, the function (_): s Uk x sx is a continuous function on U. Here s xkx denotes the image of the stalk , x Xx s in the residue field , ()= Xxx kx 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 CTop preserves fibered products (even though CRS may not). Indeed, given 11 11212 12 ππ XXY gY s UU , the set 12 , Y UUU 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 ,Y x xUU , we can find a neighborhood 12Y VV of 12 , x x contained in 12Y UU and sections 11 1 ,, nX aa V, 12 2 ,, nX bb V such that the stalk , 12 x x s agrees with 12 ii i x x ab at each 121 2 ,Y x xVV . In particular, the function _ s agrees with the function 121 2 ,() ii i x xaxbxk on 12Y VV. Since this latter function is continuous in the product topology on 12Y VV (because each (_) i a, (_) i b is continuous) and continuity is local, _ s is continuous. Corollary 11. Suppose 1,() , 111 1:YfxX x x f is surjective for every 11 x X. Then the comparison map is an isomorphism. In particular, is an isomor- phism under either of the following hypotheses: 1) 1 f is an immersion. 2) 1:Spec f ky Y is the natural map associated to a point y Y. Proof. It is equivalent to show that 12 := Y X XXRS is in LRS and the structure maps π: ii X X are LRS morphisms. Say 12 =, x xx X and let 1122 := = y fx fx. By construction of X , we have a pushout diagram of rings 11 21 2 2 () ,, 11 π π ,, 22 x xx x f YyX x f Xx Xx hence it is clear from surjectivity of 11 x f and locality of 22 x f that ,Xx is local and 12 π,π x x are LAn morphisms. Corollary 12. Suppose 2 12 1 π 12 2 π 1 Y f f X XX X Y is a cartesian diagram in LRS. Then: 1) If 12Y zX X is rational over Y, then 1=zz . 2) Let 12 12 ,Y x xX X RS , and let 1122 := π=π. y xx Suppose 2 kx is isomorphic, as a field extesion of ky, to a subfield of 1 kx . Then there is a point 12Y zX X Sch rational over 1 X with π()= ii zx, =1,2i. Proof. For 1), set 12 ,:= x xz , 1122 := π=π y xx. Then we have a commutative diagram 2, 2, 1, 2 1, 1 π 2 π 1 Z x Z x f f kz kx kx ky of residue fields. By hypothesis, the compositions i ky 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 12 Spec= Spec, ky kx 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 12Y X X , the maps 212 ,:Spec x ikxX 111 :Spec x kx X give rise to a map 112 :Spec . Y g kxXX Let 12Y zX X be the point corresponding to this map. Then we have a commutative diagram of residue fields 1 kx 1 kx 2 kx ky kz 1, π Z i W. D. GILLAM Copyright © 2011 SciRes. APM 260 so 1, 1 π:() zkx kz must be an isomorphism. 3.3. Spec Functor Suppose XLRS and :X f A is an X algebra. Then f may be viewed as a morphism of ringed spaces :,,= X f XA XX. Give X the local prime system X as usual and , X A the inverse image prime system (Remark 1), so f may be viewed as a PRS morphism * :,,,, . XXX fXAfX Explicitly: *1 , =:()=. XxxxXx x fAf ppm By Theorem 2, there is a unique LRS morphism loc loc * :,,,, = XXX f XAf XX lifting f to the localizations. We call loc * Spec:=, , XX AXAf the spectrum (relative to X ) of A . SpecX defines a functor op Spec :. XX X XRings LRS Note that loc Spec=, ,= XXXX X X 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 : f XY be an affine morphism of schemes. Then * SpecXX f (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 X B is also affine, and f corresponds to a ring map #: f AB. Then *==, Y XAY Y fBB as Y algebras, and the squares in the diagram * # * * # ,, ,, ,, ,, *, ,Spec*, ,Spec XY YY YY Y Yf fY YBfNYA N BB AA in PRS are cartesian in PRS , where N is the prime system on ,Y YA given by = y Ny 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 =. loc YXX Y fYff B X 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 Spec X “is” (up to re- placing a locally ringed space with the corresponding locally ringed topos) our ,loc X X. 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 . X XX The next results show that Spec X takes direct limits of X algebras to inverse limits in LRS and that Spec X is compatible with changing the base X . Lemma 14. The functor SpecX preserves inverse limits. Proof. Let : iX i if A be a direct limit sys- tem in X X Rings, with direct limit :X f A, and structure maps : ii jA A. We claim that * Spec=, ,loc XX AXAf is the inverse limit of * Spec=, ,loc Xii iX iAXAf. By Theorem 4, it is enough to show that * ,, X XAf is the inverse limit of * ,, ii X iXAf in PRS . Certainly , X A is the inverse limit of ,i iXA in RS , so we just need to show that *** = XiiiX fjf as prime systems on , X A (see the proof of Theorem 4), and this is clear because = ii jff , so, in fact, *** = ii XX jf f for every i. Lemma 15. Let : f XY be a morphism of locally ringed spaces. Then for any Y algebra :Y g A, the diagram * Spec Spec XY X f Y A A is cartesian in LRS . Proof. Note *1 1 := X fY fAf A as usual. One sees easily that W. D. GILLAM Copyright © 2011 SciRes. APM 261 * *1 * ,, ,, ,, ,, XY XX YY XfA f gYAg XY 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, SpecX A may not be a scheme. For example, let B be a local ring, := Spec X B, and let x be the unique closed point of X . Let * := A xB XRings be the skyscraper sheaf B sup- ported at x . Note ,= Xx B and *, Hom,= Hom,, XXx X x BB Rings Rings so we have a natural map X A in X Rings whose stalk at x is : I dB B. Then Spec =, X A xA is the punctual space with “sheaf” of rings A, mapping in LRS to X in the obvious manner. But , x A 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 * := A xK, and let X A be the unique map whose stalk at x is BK. Then SpecX A is empty. Suppose X is a scheme, and A is an X algebra such that SpecX A is a scheme. I do not know whether this implies that the structure morphism SpecX A X is an affine morphism of schemes. 3.4. Relative Schemes We begin by recalling some definitions. Definition 7. ([SGA1], [Vis 3.1]) Let : F CD be a functor. A C morphism : f cc is called cartesian (relative to F ) iff, for any C morphism : g cc and any D morphism :hFc Fc with = F gh Ff there is a unique C morphism :hc c with = F hh and = f gh. The functor F is called a fibered category iff, for any D morphism : f dd and any object c of C with = F cd , there is a cartesian morphism : f cc with = F ff. A morphism of fibered categories ::' F CD FCD is a functor :'GC C satisfying = F GF and taking cartesian arrows to cartesian arrows. If D has a topology (i.e. is a site), then a fibered category : F CD is called a stack iff, for any object dD and any cover i dd of d in D, the category 1 F d is equivalent to the category i F dd 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 p re X Sch as follows. Objects of p re X Sch are pairs ,U UX consisting of an open subset UX and a scheme U X over Spec XU. A morphism ,, UV UX VX is a pair ,UV UVX X consisting of an Ouv X morphism UV (i.e. UV) and a morphism of schemes UV X X mak- ing the diagram Spec Spec UV XX XX UV (5) commute in Sch . The forgetful functor pre X X SchOuv is clearly a fibered category, where a cartesian arrow is a p re X Sch morphism ,UV UVX X making (6) cartesian in Sch (equivalently in LRS ). Since X Ouv has a topology, we can form the associated stack X Sch . The category of relative schemes over X is, by definition, the fiber category X X Sch of X Sch over the terminal object X of X Ouv . (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 :() XX F XXSchLRS called the geometric realization. Although a bit abstract, the fastest way to proceed is as follows: Definition 9. Let X LRS be the category whose objects are pairs ,U UX consisting of an open subset UX and a locally ringed space U X over ,X UU, and where a morphism ,, UV UX VX is a pair ,UV UVX X consisting of an X Ouv morphism UV (i.e. UV) and an LRS morphism UV X X making the diagram ,, UV XX XX UU VV (7) commute in LRS. The forgetful functor ,U UX U makes X LRS a fibered category over X Ouv where a cartesian arrow is a morphism ,UV UVX X making (8) cartesian in LRS. In fact the fibered category X X 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 X F to be the morphism of stacks (really, the corresponding morphism on fiber categories over the terminal object X XOuv ) associated to the morphism of fibered categories : pre pre XXX FSchLRS Spec() ,, ,. UUUX X UXUXU ULRS The map ,Spec XX UU 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 : f XY be an LRS morphism. Consider the following conditions: RA1. Locally on Y there is an YY algebra A and a cartesian diagram Spec Spec f Y X A YY in LRS . RA2. There is an Y algebra A so that f is isomorphic to SpecY A in YLRS . RA3. Same condition as above, but A is required to be quasi-coherent. RA4. For any : g ZY in YLRS , the map ** # * Hom, Hom, YXZ Y Y ZXf g hgh 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 A B and a cartesian diagram Spec Spec f X B YA in LRS . RA1.2. Locally on Y there is an affine morphism of schemes X Y and a cartesian diagram f X X YY 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 Y YY, hence Spec Spec Spec Spec =Spec =Spec Spec =Spec . A YA Y Y YA Y Y XY B YYB YYB 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: : f XY 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: : f XY is in the essential image of the geometric realization functor Y F . 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 I P such that I PI. 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 1 SP whose elements are equivalence classes ,ps, pP , s S where ,= ,psps iff there is some tS with =tps tps , and where ,,:=,psp sppss . The natural map 1 PSP given by ,0pp is initial among mon- W. D. GILLAM Copyright © 2011 SciRes. APM 263 oid 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 ,X X is a topological space X equipped with a sheaf of monoids X . Monoidal spaces form a category M S where a morphism † =, :,, XY fff XY consists of a con- tinuous map : f XY together with a map of sheaves of monoids on X . A monoidal space ,X X is called local iff each stalk monoid X has a unique maximal ideal x m. Local monoidal spaces form a category LMS where a morphism is a map of the underlying monoidal spaces such that each stalk map † ,() , : x Yfx Xx f is local in the sense 1 †= x fx fmm . A primed monoidal space is a mo- noidal space equipped with a set of primes x M in each stalk monoid ,Xx . The localization of a primed mo- noidal space is a map of monoidal spaces loc ,,, XX XM 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 op Spec :, XX XXMon LMS for any ,X XLMS which preserves inverse limits, and to prove that the natural map Hom, ,SpecHom, XX X PPX 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―L RS.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. |