Localization of Ringed Spaces

Home , Morphism, Ringed space

Advances in Pure Mathematics, 2011, 1, 250-263 doi:10.4236/apm.2011.15045 Published Online September 2011 (http://www.SciRP.org/journal/apm)

William D. Gillam Department of Mathematics, Brown University, Providence, USA E-mail: [email protected] Received March 28, 2011; revised April 12, 2011; accepted April 25, 2011

Abstract

Let X be a ringed space together with the data M of a set M x of prime ideals of Xx, for each point x  X . We introduce the localization of  X ,M  , which is a locally ringed space Y and a map of ringed spaces YX enjoying a universal property similar to the localization of a ring at a prime ideal. We use this to prove that the category of locally ringed spaces has all inverse limits, to compare them to the inverse limit in ringed spaces, and to construct a very general Spec functor. We conclude with a discussion of relative schemes.

Keywords: Localization, Fibered Product, Spec, Relative Scheme

1. Introduction Then ( Spec of) the natural surjection kx kx  Let Top , LRS , RS , and Sch denote the categories Xx11,,1Yy ,X 2 x 2 ky()2 of topological spaces, locally ringed spaces, ringed identifies Speckx  12 ky kx with a closed subspace spaces, and schemes, respectively. Consider maps of of Spec  and   1 x , x Xx11,,Yy ,X 2 x 2 XX12Y 12 schemes fii: XY ( i =1,2) and their fibered product naturally coincides, under the EGA isomorphism, to the X12Y X as schemes. Let X denote the topological restriction of the structure sheaf of space underlying a scheme X . There is a natural SpecXx,, X x to the closed subspace comparison map 11Yy , 2 2 Spec kx kx Spec  . 1  1()2 ky   X11, x Yy ,X 2, x 2  : X12YYXXX12 It is perhaps less well-known that this entire discus- which is not generally an isomorphism, even if sion remains true for LRS morphisms ff, . X ,,XY are spectra of fields (e.g. if Y =Spec , 12 12 From the discussion above, we see that it is possible to XX==Spec , the map  is two points mapping to 12 describe  XX , at least as a set, from the following one point). However, in some sense  fails to be an Y 21 data: isomorphism only to the extent to which it failed in the 1) the ringed space fibered product X RS X case of spectra of fields: According to [EGA I.3.4.7] the 12Y 1 (which carries the data of the rings  fiber  x , x over a point x , xXX Xx11,,Yy , X 2 x 2 12 12 12Y as stalks of its structure sheaf) and (with common image y ==fx  fx) is naturally 11 2 2 2) the subsets Sxx ,Spec  bijective with the set 12 Xx11, Yy ,X 2, x 2 It turns out that one can actually recover X12Y X as Spec kx12ky kx. a scheme solely from this data, as follows: Given a pair  X , M  consisting of a ringed space X and a subset In fact, one can show that this bijection is a M x  SpecXx, for each x  X , one can construct a homeomorphism when  1 x , x is given the topo- loc 12 locally ringed space XM, with a map of ringed loc logy it inherits from X12Y X . One can even describe 1 spaces X , MX . In a special case, this constru- the sheaf of rings  x12, x inherits from X12Y X ction coincides with M. Hakim’s spectrum of a ringed as follows: Let topos. Performing this general construction to Sxx,:=Spec z:z 1There is no sense in which this sheaf of rings on 12  Xx11, Yy ,X 2,, x 2 Xxi i Spec kx 12 ky kx  is “quasi-coherent”. It isn’t even a module over = for i = 1,2 . the usual structure sheaf of Spec kx kx xi  12ky 

RS ficient to upgrade descent theorems for RS to descent XXSxx1212Y ,, theorems for Sch . yields the comparison map  , and, in particular, the scheme X12Y X . A similar construction in fact yields 2. Localization all inverse limits in LRS (§ 3.1) and the comparison map to the inverse limit in RS , and allows one to easily We will begin the localization construction after making prove that a finite inverse limits of schemes, taken in a few definitions. LRS , is a scheme (Theorem 8). Using this description Definition 1. A morphism f : AB of sheaves of of the comparison map  one can easily describe some rings on a space X is called a localization morphism3 circumstances under which it is an isomorphism (§ 3.2), iff there is a multiplicative subsheaf SA so that f and one can easily see, for example, that it is a loca- is isomorphic to the localization A  SA1 of A at lization morphism (Definition 1), hence has zero cotangent S .4 A morphism of ringed spaces f : XY is called #1 complex. a localization morphism iff ff: YX is a lo- The localization construction also allows us construct calization morphism. (§ 3.3), for any X  LRS , a very general relative spec A localization morphism A  B in Rings X  is functor both flat and an epimorphism in RingsX .5 In particular, the cotangent complex (hence also the sheaf of Spec :  Algop LRS X XX Kähler differentials) of a localization morphism is zero which coincides with the usual one when X is a [Ill II.2.3.2]. The basic example is: For any affine scheme X =Spec A , AX   is a localization mor- scheme and we restrict to quasi-coherent X algebras. X We can also construct (§ 3.5) a “good geometric realiza- phism. tion” functor from M. Hakim’s stack of relative schemes Definition 2. Let A be a ring, SA Spec any sub- 2 over a locally ringed space X to LRS X . It should set. We write SpecAS for the locally ringed space be emphasized at this point that there is essentially only whose underlying topological space is S with the topol- one construction, the localization of a ringed space of ogy it inherits from Spec A and whose sheaf of rings is § 2.2, in this paper, and one (fairly easy) theorem (Theo- the inverse image of the structure sheaf of Spec A . rem 2) about it; everything else follows formally from If A is clear from context, we drop the subscript and general nonsense. simply write Spec S . There is one possible point of Despite all these results about inverse limits, I stum- confusion here: If I  A is an ideal, and we think of bled upon this construction while studying direct limits. I Spec A I as a subset of Spec A , then was interested in comparing the quotient of, say, a finite SpecA  Spec A IAI  Spec étale groupoid in schemes, taken in sheaves on the étale site, with the same quotient taken in LRS . In order to (though they have the same topological space). compare these meaningfully, one must somehow put them in the same category. An appealing way to do this 2.1. Prime Systems is to prove that the (functor of points of the) LRS quotient is a sheaf on the étale site. In fact, one can prove Definition 3. Let XX=, X  be a ringed space. A that for any X  LRS , the presheaf prime system M on X is a map x  M x assigning a subset M  Spec to each point x  X . For YHomYX , x Xx, LRS  prime systems M , N on X we write M  N to is a sheaf on schemes in both the fppf and fpqc topolo- mean M x  N x for all x  X . Prime systems on X gies. Indeed, one can easily describe a topology on RS , form a category PS X  where there is a unique mor- analogous to the fppf and fpqc topologies on schemes, phism from M to N iff M  N . The intersection and prove it is subcanonical. To upgrade this to a sub- iiM of prime systems M i  PSX is defined by canonical topology on LRS one is naturally confronted iiMM:= i i . with the comparison of fibered products in LRS and x x RS . In particular, one is confronted with the question of A primed ringed space  X , M  is a ringed space X whether  is an epimorphism in the category of ringed equipped with a prime system M . Prime ringed spaces spaces. I do not know whether this is true for arbitrary form a category PRS where a morphism f :, XM  YN, is a morphism of ringed spaces f satisfy- LRS morphisms f12, f , but in the case of schemes it is   possible to prove a result along these lines which is suf- 3See [Ill II.2.3.2] and the reference therein. 4 2Hakim already constructed such a functor, but ours is different from See [Ill II.2.3.2] and the reference therein. 5 hers. Both of these conditions can be checked at stalks.

Copyright © 2011 SciRes. APM 252 W. D. GILLAM ing sheaf on X . The category SecF of local sections of is the category whose objects are pairs Us, Spec fM N F   x xfx() where U is an open subset of X and s F U  , and for every x  X . where there is a unique morphism Us,,  Vt if

The inverse limit of a functor iM i to PS X  is UV and tU s. . loc clearly given by iiM . As a set, the topological space X will be the set of loc Remark 1. Suppose YN,  PRS and f : XY pairs  x, z , where x  X and zM x . Let   X  is an RS morphism. The inverse image f *N is the denote the category of subsets of X loc whose mor- prime system on X defined by phisms are inclusions. For Us, SecX , set * 1 UUs ,:=,  xz Xloc : x Us , z . fN:= Spec fx  Nfx  x  x =Spec:fN1  . This defines a functor Xx, x  fx loc UX: SecX   Formation of inverse image prime systems enjoys the expected naturality in f : g**fM=  fgM* . We satisfying: can alternatively define a PRS morphism f :, XM UUs ,,=,||  UVt  UU VsUV t UV

 YN, to be an RS morphism f : XY such n * UUs,= UUs , n . that c M  fN (i.e. together with a PSX mor-  >0 * phism M  fN). loc The first formula implies that UXSecX     X loc For X LRS , the local prime system X on is a basis for a topology on X where a basic open is defined by  := m . If Y is another locally Xx,  x neighborhood of  x, z is a set UUs, where x U , ringed space, then a morphism f : XY in RS loc sx  z . We always consider X with this topology. defines a morphism of primed ringed spaces The map fX:, Y , iff f is a morphism in LRS , XY loc so we have a fully faithful functor π : X  X

: LRS PRS x, zx (1) 1 XX ,,X is continuous because π UUU=,1. We construct a sheaf of rings  on X loc as X loc and we may regard LRS as a full subcategory of follows. For an open subset VX loc , we let  V X loc   PRS . be the set of At the “opposite extreme” we also have, for any

ssxz=,   Xx, X  RS , the terminal prime system  X defined by    z xz,  V Xx,,:= Spec Xx (i.e. the terminal object in PS X  ). For YM,  PRS , we clearly have satisfying the local consistency condition: For every  x, zV , there is a basic open neighborhood UUt ,  Hom Y,,, M X = Hom Y ,, X PRS X RS  of  x, z contained in V and a section a so the functor  U t n X t  : RS PRS (2) such that, for every  x,,zUUt , we have XX , X a  x is right adjoint to the forgetful functor PRS RS sxz,=n Xx,  . t z given by X , MX . x (Of course, one can always take n =1 since 2.2. Localization UUt,= UUt ,n .) The set  V becomes a ring     X loc  under coordinatewise addition and multiplication, and Now we begin the main construction of this section. Let the obvious restriction maps make  a sheaf of rings X loc X , M be a primed ringed space. We now construct a on X loc . There is a natural isomorphism loc loc locally ringed space XM, (written X if M is loc = X, x clear from context), and a PRS morphism Xxz,, z loc  π :,X loc  XM , called the localization of taking the germ of s =,sxz    U in the stalk X   X loc X at M .  to sxz ,   . This map is injective be- Xxzloc ,,  Xx, z Definition 4. Let X be a topological space, F a cause of the local consistency condition and surjective

Copyright © 2011 SciRes. APM W. D. GILLAM 253 because, given any ab  , we can lift ab, to at zM is identified with the localization of  at Xx, z x Xx, ab, X  U on some neighborhood U of x and define z , hence π is a localization morphism (Definition 1). s  UUb, by letting Proof. With the exception of the fiber description, X loc  everything in the proposition was noted during the con- sxz,:= axX bx  ,x. This s manifestly satisfies z struction of the localization. Clearly there is a natural the local consistency condition and has s xz,= ab. In 1 loc bijection of sets M x = π  x taking zM x to particular, X , with this sheaf of rings, is a locally 1  x, zx π   . We first show that the topology inher- ringed space. loc loc ited from X coincides with the one inherited from To lift π to a map of ringed spaces π : X  X we Spec . By definition of the topology on X loc , a ba- use the tautological map Xx, sic open neighborhood of zM x is a set of the form # π :Xl π* oc X UUs ,=  Mxx z M :, sx z of sheaves of rings on X defined on an open set where U is a neighborhood of x in X and UX by s X U  satisfies sx  z . Clearly this set depends only on the stalk of s  of s at x , and any π# UU: π U=,1 UU x Xx, Xl  * XXocloc   element t Xx, lifts to a section tUX () on some ss . neighborhood of X , so the basic neighborhoods of x z zM x are the sets of the form It is clear that the induced map on stalks zMtzx :  π :=  xz,, X xXxz loc,( , ) X, x z where tzx  . But for the same set of t , the sets is the natural localization map, so π1  = zM x,zz x Dt  :=pp SpecXx, : t  and hence π defines a PRS morphism form a basis for neighborhoods of z in Spec so π :,X loc   XM ,. Xx, X loc the result is clear.

Remark 2. It would have been enough to construct the We next show that the sheaf of rings on M x inher- loc loc localization X , X at the terminal prime system. ited from X is the same as the one inherited from loc Then to construct the localization XM, at any SpecXx, . Given f Xx, , a section of  locM x loc X other prime system, we just note that XM, is over the basic open set M x  Df is an element clearly a subset of X , loc , and we give it the topol- X ssz=       Xx, z ogy and sheaf of rings it inherits from this inclusion. The zM Df loc x  construction of X , X is “classical.” Indeed, M. loc satisfying the local consistency condition: For all Hakim [Hak] describes a construction of X , X zMx Df  , there is a basic open neighborhood that makes sense for any ringed topos X (she calls it loc the spectrum of the ringed topos [Hak IV.1]), and attrib- UUt ,  of  x, z in X and an element atn  U such that, for all utes the analogous construction for ringed spaces to C.   X  t   n Chevalley [Hak IV.2]. Perhaps the main idea of this zMx DfUUt   ,  , we have szat z  z  . work is to define “prime systems,” and to demonstate Note that their ubiquity. The additional flexibility afforded by M x Df  UUt ,= Mxx  Dft non-terminal prime systems is indispensible in the appli- and atn  Dft. The sets Dft M cations of § 3. It is not clear to me whether this setup x  xx SpecXx,     x  x generalizes to ringed topoi. cover MDfx   SpecXx, , and we have a “global We sum up some basic properties of the localization formula” s showing that the stalks of the various n map π below. atx  x  agree at any zMx Df, so they glue to yield an element g sMDf   with Proposition 1. Let X , M be a primed ringed space SpecXx, x loc g ssz= . We can define a morphism of sheaves on with localization π : X  X . For x  X , the fiber  z   1 M x by defining it on basic opens, so this defines a π x is naturally isomorphic in LRS to Spec M x 6 morphism of sheaves g :MM (Definition 2). Under this identification, the stalk of π X loc x SpecXx, x which is easily seen to be an isomorphism on stalks. 6 1Rloc S By “fiber” here we mean π xX:=X   x ,Xx, , which is just Remark 3. Suppose  XM,  PRS and UX is 1 loc an open subspace of X . Then it is clear from the con- the set theoretic preimage π x  X with the topology and sheaf loc loc struction of π :,X MX that of rings it inherits from X . This differs from another common usage  loc loc RS 1 . of “fiber” to mean XxkxX ,   . π UU=,X UMU ,

The following theorem describes the universal prop- commute in RingsY loc  . The stalk of this diagram at erty of localization. (,)x zX loc is a diagram Theorem 2. Let f :,XM YN , be a morphism  fxz, in PRS . Then there is a unique morphism   fz1 loc loc Xx, Yfx,  x fXM:, YN , in LRS making the diagram z π 1 πxz,  fx, fx  z XM,,loc f  YNloc  ππ   (3) Xx, fx Yfx,  f XY in Rings where the vertical arrows are the natural lo- commute in RS . Localization defines a functor calization maps; these are epimorphisms, and the universal property of localization ensures that there is a PRS LRS unique local morphism of local rings fx,z completing XM,, XMloc this diagram. We now want to show that there is actually a (necessarily unique) map ff#1:   of X loc Y fXM:, YN , fXM :,loc YN ,loc sheaves of rings on X loc whose stalk at x, z is the map fx,z . By the universal property of sheafification, retracting the inclusion functor : LRS PRS and we can work with the presheaf inverse image f 1 preX loc right adjoint to it: For any Y LRS , there is a natural instead. A section Vs,  of this presheaf over an open bijection subset WX loc is represented by a pair Vs, where loc VY loc is an open subset of Y loc containing f W  HomLRS Y,, X M = HomPRS  Y ,Y ,, X M .  and Proof. We first establish the existence of such a mor- ssyz=, V .    Yloc   Y, y z phism f . The fact that f is a morphism of primed (,)yz V ringed spaces means that we have a function I claim that we can define a section # MNx  f x f Vs,  W by the formula pre   X loc   #1 1 fpre Vs,,:=, xz s f x fx z . zfz x    for each x  X , so we can complete the diagram of It is clear that this element is independent of replacing topological spaces V with a smaller neighborhood of f W  and restricting s , but we still must check that XYloc f  , Nloc ππ fVs,   p,re    X x z  (,)xz W f XY satisfies the local consistency condition. Suppose (at least on the level of sets) by setting a  U 1 loc t n X t fxz,:= fx  , fx z Y . witnesses local consistency for s  V on a basic To see that f is continuous it is enough to check Yloc open subset UUt,  V. Then it is straightforward to that the preimage f 1 UUs, is open in X loc for    check that the restriction of each basic open subset UUs, of Y loc . But it is clear from the definitions that ffa#1    1 Y f UUt, f 11UUs,= U f U , f# f1 s fft#1 n (note ff#1  s= f s ). to f 1UUt ,   W witnesses local consistency of x x fx Now we want to define a map ff#1:   of f # Vs, on X loc Y pre   sheaves of rings on Y (with “local stalks”) making the f 11UUt,= W U f U , f# f1 t W . diagram     # # f It is clear that our formula for fpre Vs,  respects   f 1 XYloc loc restrictions and has the desired stalks and commutativity,  so its sheafification provides the desired map of sheaves

-1 # of rings. 111π f loc loc ππXX  f This completes the construction of f : XY in

LRS making (3) commute in RS . We now establish Define a prime system N on X , AX by the uniqueness of f . Suppose f : XYloc loc is a Nx:= Spec A = Spec AX= . morphism in LRS that also makes (3) commute in x   Xx, RS . We first prove that f = f  on the level of Let aX:, X    XA ,X  be the natural RS topological spaces. For x  X the commutativity of (3) morphism. Then  = aN* and the natural PRS  X ,X ensures that f xz,= f  x , z for some morphisms zNf ()xYSpec ,f ()x , so it remains only to show that (X , , )XA , , N *, A ,Spec A 1 X X ,  X   zfz =.x  The commutativity of (3) on the level of X loc stalks at x, zX gives a commutative diagram of =*,,A  rings *, A

f  yield natural isomorphisms xz,  Xx, z Yfx,  loc z loc πxz, πfxz,  XX,=,,XX =,, XANX   X ,X  loc π-1f # =*,,SpecAA Xx,   Yfx,  where the vertical arrows are the natural localization in LRS . Proof. Note that the stalk aA:  of at maps. From the commutativity of this diagram and the x Xx, Xx, a 1 x  X is the localization map A  Ax , and, by defi- fact that f  ()=mmz z (because f x,z is local) * xz,  nition, aN is the set prime ideals z of A pulling we find  x x back to x  A under aAx :  Ax . The only such prime z = π1 ()m fxz(), z ideal is the maximal ideal m x  Ax , so 1 1 aN* ={m }= . = π fxz(), f  m z   x Xx,, xz, x X 11 = fxxπ m z Next, it is clear from the description of the localization of a PRS morphism that the localizations of the mor- =()fz1 x phisms in question are bijective on the level of sets. In- as desired. This proves that f = f  on topological deed, the bijections are given by # spaces, and we already argued the uniqueness of f x  xxxx,,*,,m   (which can be checked on stalks) during its construction. x The last statements of the theorem follow easily once so to prove that they are continuous, we just need to we prove that the localization morphism prove that they have the same topology. Indeed, we will loc π :,X X  X is an isomorphism for any show that they all have the usual (Zariski) topology on X LRS . On the level of topological spaces, it is clear X =Spec A . This is clear for X ,,X be-  X ,X  that π is a continuous bijection, so to prove it is an cause localization retracts  (Theorem 2), so isomorphism we just need to prove it is open. To prove loc XX,,XX =, , and it is clear for this, it is enough to prove that for any Us,Sec  ,  X ,X   X the image of the basic open set UUs, under π is *,A ,Spec A because of the description of the fibers of open in X . Indeed, localization in Proposition 1. For X ,,ANX , we note that the sets UUs ,  , as U ranges over connected πUUs,= x U : sxx open subsets of X (or any other family of basic opens =:xUs* for that matter), form a basis for the topology on x Xx, loc XA,,X N . Since U is connected, s  AUX   = A, is open in U , hence in X , because invertibility at the and UUs ,  is identified with the usual basic open stalk implies invertibility on a neighborhood. To prove subset Ds() X under the bijections above. This that π is an isomorpism of locally ringed spaces, it proves that the LRS morphisms in question are iso- remains only to prove that π# : is an isomorphisms on the level of spaces, so it remains only to XlocX morphism of sheaves of rings on X = X loc . Indeed, prove that they are isomorphisms on the level of sheaves # loc Proposition 1 says the stalk of π at x, m x  X is of rings, which we can check on stalks using the descrip- the localization of the local ring Xx, at its unique tion of the stalks of a localization in Proposition 1. maximal ideal, which is an isomorphism in LAn . Remark 4. If X LRS , and M is a prime system Lemma 3. Let A Rings be a ring, on X , the map π : X loc  X is not generally a mor- loc X ,:=SpecX A , and let * be the punctual space. phism in LRS , even though XX, LRS . For ex-

Copyright © 2011 SciRes. APM 256 W. D. GILLAM ample, if X is a point whose “sheaf” of rings is a local limits follows formally from the adjointness in Theorem ring A, m , and M ={p } for some pm , then 2. loc X is a point with the “sheaf” of rings Ap , and the Corollary 5. The category LRS has all inverse limits. # “stalk” of π is the localization map lA :  Ap . Even Proof. Suppose iX i is an inverse limit system in though A, Ap are local, this is not a local morphism LRS . Composing with  yields an inverse limit sys- 1 because lAppmp =  . tem iX iX, in PRS . By the theorem, the lo-   i loc calization XM, of the inverse limit  X , M  locof 3. Applications iX , is the inverse limit of iX ,  iXi  iXi in LRS . But loclocalization retracts  (Theorem 2) so In this section we give some applications of localization iX , is our original inverse limit system  iXi  of ringed spaces. iX i . We can also obtain the following result of C. 3.1. Inverse Limits Chevalley mentioned in [Hak IV.2.4]. Corollary 6 The functor We first prove that LRS has all inverse limits. RS LRS Theorem 4. The category PRS has all inverse limits, loc and both the localization functor PRS LRS and the XX , X forgetful functor PRS RS preserve them. is right adjoint to the inclusion LRS RS . Proof. Suppose iXM , is an inverse limit sys- ii Proof. This is immediate from the adjointness tem in PRS . Let X be the inverse limit of iX in i property of localization in Theorem 2 and the adjointness Top and let π : X  X be the projection. Let  ii X property of the functor  : For Y LRS we have be the direct limit of i  π1 in Rings X and let iXi  #1 loc πiiX: π  X be the structure map to the direct limit,  i HomLRS  Y,, X X  so we may regard XX=,X as a ringed space and πi as a morphism of ringed spaces X  X i . It imme- =,,,HomPRS  YYX X diate from the definition of a morphism in RS that X =,.HomRS  Y X is the inverse limit of iX i in RS . Let M i be the prime system on X given by the inverse limit (inter- Our next task is to compare inverse limits in Sch to * section) of the πiiM . Then it is clear from the definition those in LRS . Let *Top be “the” punctual space of a morphism in PRS that X , M is the inverse (terminal object), so RingsRin*= gs . The functor limit of iXM , , but we will spell out the details ii Rings RS for the sake of concreteness and future use. A  *, A Given a point x = xXi  , we have defined M x to  be the set of z Spec such that Xx, is clearly left adjoint to π1 zMSpec for every i , so that π ix,, xiiX xi i defines a PRS morphism π :,X MXM  , . To : RSop Rings iii see that X , M is the direct limit of iXM  ii,  XX  ,X . suppose fii:,YN  X , Mi are morphisms defining a natural transformation from the constant functor By Lemma 3 (or Proposition 1) we have iYN , to iXM , . We want to show that loc loc  ii  *,A :=  *,AA ,Spec  there is a unique PRS morphism f :,YN  XM , = Spec A. with πiif = f for all i . Since X is the inverse limit of iX in RS , we know that there is a unique map Theorem 2 yields an easy proof of the following result, of ringed spaces f :YX with πiif = f for all i , which can be found in the Errata for [EGA I.1.8] printed so it suffices to show that this f is a PRS morphism. at the end of [EGA II]. 1 Let y Y , zN y . We must show fzMy  f x . By Proposition 7. For A Rings , X LRS , the definition of M , we must show natural map π 1 fz1  M= M for every i . But iyfx() π fx f y i  i HomLRS  X,Spec A HomRings  A,, X X π f = f implies f π = f , so ii yif ()xy i is bijective, so Spec : Rings LRS is left adjoint to 11 π f 1 zfz= is in M because f : LRSop Rings . iyfx()  i y fi y i is a PRS morphism. Proof. This is a completely formal consequence of The fact that the localization functor preserves inverse various adjunctions:

HomXA ,Spec = Hom XA ,  *, loc affine by showing that it is isomorphic to Spec of the LRS LRS coequalizer

=HomPRS X ,X , *,A ## CB=()(): fa gaaA =HomRS XA , *, of f ##,:gA B in Rings . =HomRings AX , ,X . Remark 5. The basic results concerning the existence Theorem 8. The category Sch has all finite inverse of inverse limits in LRS and their coincidence with limits, and the inclusion Sch LRS preserves them. inverse limits in Sch are, at least to some extent, “folk Proof. It is equivalent to show that, for a finite inverse theorems”. I do not claim originality here. The construction of fibered products in LRS can perhaps be attrib- limit system iX i in Sch , the inverse limit X in LRS is a scheme. It suffices to treat the case of (finite) uted to Hanno Becker [HB], and the fact that a cartesian products and equalizers. For products, suppose X  is diagram in Sch is also cartesian in LRS is implicit in i the [EGA] Erratum mentioned above. a finite set of schemes and X = iX i is their product in LRS . We want to show X is a scheme. Let x be Remark 6. It is unclear to me whether the 2-category of locally ringed topoi has 2-fibered products, though a point of X , and let x = xX RS be its image in iii Hakim seems to want such a fibered product in [Hak the ringed space product. Let UAii=Spec be an open V.3.2.3]. affine neighborhood of xi in X i . As we saw above, the map X  RSX is a localization and, as men- 3.2. Fibered Products i i tioned in Remark 3, it follows that the product In this section, we will more closely examine the con- UU:= of the U in LRS is an open neighbor- i i i struction of fibered products in LRS and explain the hood of x in X ,7 so it remains only to prove that relationship between fibered products in LRS and there is an isomorphism UA Spec  , hence U is ii those in RS . By Theorem 8, the inclusion affine.8 Indeed, we can see immediately from Proposition Sch LRS preserves inverse limits, so these results 7 that U and Spec  A represent the same functor ii will generalize the basic results comparing fibered prod- on LRS : ucts in Sch to those in RS (the proofs will become HomLRS (YU , ) = HomLRS YU , i much more transparent as well). i Definition 5. Suppose =HomAY , ,  Rings iY  AkB,,mm , 111 , , kB , 2 , m 22 , kLAn and i 1 fii: AB are LAn morphisms, so fiimm = =HomRings iiAY , , Y for i =1,2 . Let iBjj:  B12 A B j =1,2 be the = HomLRS YA ,Specii . natural maps. Set 1 The case of equalizers is similar: Suppose X is the SABB ,,12 :=SpecppB 1A B 2 : i 1  LRS equalizer of morphisms f ,:gY Z of schemes, (4) mpm1 and x  X . Let y Y be the image of x in Y , so =,12i  = 2 . f ()=()=:ygyz. Since YZ, are schemes, we can find Note that the kernel K of the natural surjection affine neighborhoods VB=Spec of y in Y and BBkk WA=Spec of z in Z so that f , g take V into 1212Ak W . As before, it is clear that the equalizer U of bb12  b 1  b 2 f |,|VgVV :  W in LRS is an open neighborhood of x  X , and we prove exactly as above that U is is generated by the expressions m1 1 and 1 m2 , where mii m , so 7This is the only place we need “finite”. If X were infinite, the  i Speckk12kA  Spec BB 1 2 topological space product of the Ui might not be open in the topology is an isomorphism onto SABB ,,12 . In particular, on the topological space product of the X i because the product topology only allows “restriction in finitely many coordinates”. SABB ,,12 =pp SpecBBK1A 2 :  8 There would not be a problem here even if X i were infinite:

Rings has all direct and inverse limits, so the (possibly infinite) ten- is closed in SpecBB12A  . The subset SABB ,,  enjoys the following impor- sor product iiA over  (coproduct in Rings ) makes sense. Our 12 proof therefore shows that any inverse limit (not necessarily finite) of tant property: Suppose gBiii:,mn  C ,, i =1,2 , affine schemes, taken in LRS , is a scheme. are LAn morphisms with g11fgf= 2 2 and

hffB=,12 : 1A B 2 C is the induced map. Then Corollary 10. Suppose x11 X is rational over Y 1 hSABBn  ,,12 . Conversely, every (i.e. with respect to f11: XY ). Then for any RS 1 p SABB,,12 arises in this manner: take x =, xx12 X 1Y X 2, the fiber  x of the

CB= 12A Bp . comparison map  is punctual. In particular, if every Setup: We will work with the following setup point of X1 is rational over Y , then  is bijective. throughout this section. Let f11: XY , f22: XY Proof. Suppose x11 X is rational over Y . Suppose RS be morphisms in LRS . From the universal property of x =, xx12 X 1 X 2. Set y :=fx11 = fx 2 2 . Since fiber products we get a natural “comparison” map x1 is rational, ky   kx 1  , so Spec kx kxSpec kx has a single ele-  :.X LRS XXRS X  1()2 ky  2 1212YY ment. On the other hand, we saw in Definition 5 that this RS Let πiY: X12XX i (=1,2i ) denote the projec- set is in bijective correspondence with the set tions and let g := ff11π = 2π 2. Recall that the structure RS 11 Sxx,Spec sheaf of X  X is ππ . In par- 12  Xx11, Yy ,X 2, x 2 12Y 112XX12g  ticular, the stalk of this structure Ysheaf at a point RS appearing in Theorem 9, so that same theorem says that x =,xx X X is  , where 1 12 1Y 2 Xx11,,Yy , X 2 x 2   x consists of a single point.

y :=gx = f11  x  = f 2  x 2  . Remark 9. Even if every x1  X is rational over Y , the comparison map In this situation, we set LRS RS  : X1212YYXX X Sxx12,:=, SYy , X , x ,  X , x 11 2 2 is not generally an isomorphism on topological spaces, to save notation. LRS even though it is bijective. The topology on X12Y X Theorem 9. The comparison map  is surjective on is generally much finer than the product topology. In this topological spaces. More precisely, for any situation, the set Sxx 12,  always consists of a single x =,xx XRS X,  1 x is in bijective corre- 12 1Y 2   element zxx 12,  : namely, the maximal ideal of spondence with the set Sxx 12,  , and in fact, there is an  given by the kernel of the natural Xx11,,Yy , X 2 x 2 LRS isomorphism surjection

1 LRS  xX:=  X x,  Xx,,122 X x kxky kx=. kx  12YXXXRS  11,,xYy ,X 2x 2 11Yy , 2 2 12Y LRS RS =Spec Sxx,. If we identify X12Y X and X12Y X as sets via  Xx,,  X x 12 11Yy , 2 2  , then the “finer” topology has basic open sets In particular,  1()x is isomorphic as a topological UU U,:=, s xx U U : s zxx , 1YY 2 12 1 2 xx12, 12 space to Speckx1()2ky kx  (but not as a ringed space). The stalk of  at zSxx , is identified 12 as UU12, range over open subsets of X12, X and s with the localization map ranges over

 . 11 Xx,,,,1,Yy Xx 2 Xx 1,Yy Xx 2 ππUU. z  11212XXY12g   Y In particular,  is a localization morphism (Definition This set is not generally open in the product topology 1). because the stalks of Proof. We saw in § 3.1 that the comparison map  RS 11 is identified with the localization of X12Y X at the ππ112XX  12g Y prime system x12,,xSxx 12, so these results follow from Proposition 1. are not generally local rings, so not being in zxx 12, 

Remark 7. When XXY12,,Sch , the first statement does not imply invertibility, hence is not generally an of Theorem 9 is [EGA I.3.4.7]. open condition on  x12, x  . Remark 8. The fact that  is a localization Remark 10. On the other hand, sometimes the topolo- morphism is often implicitly used in the theory of the gies on X1 , X 2 are so fine that the sets cotangent complex. UU 12Y U, s are easily seen to be open in the product Definition 6. Let f : XY be an LRS morphism. topology. For example, suppose k is a topological A point x  X is called rational over Y (or “over field.9 Then one often works in the full subcategory C y := fx “ or “with respect to f ”) iff the map on of locally ringed spaces over k consisting of those residue fields fx : ky kx  is an isomorphism 9I require all finite subsets of k to be closed in the definition of (equivalently: is surjective). “topological field”.

X LRS k satisfying the conditions: hence it is clear from surjectivity of f1 and locality x1 1) Every point x  X is a k point: the composition of  f2  that Xx, is local and π12,π  are x2 x x kkxXx,  yields an isomorphism kkx=   LAn morphisms. for every x  X . Corollary 12. Suppose  π 2) The structure sheaf X is continuous for the topo- X XX2  logy on k in the sense that, for every Us, Sec , 12Y 2 X π12f the function  f1 s(_):Uk X1  Y

x  sx is a cartesian diagram in LRS . Then: 1) If zX 12Y X is rational over Y , then is a continuous function on U . Here s  xkx   1   zz =   . denotes the image of the stalk s  in the residue RS x Xx, 2) Let  x , xX X, and let field kx()= m , and we make the identification 12 1Y 2 Xx, x y := π  xx = π  . Suppose kx is isomorphic, kkx=   using 1). 11 2 2 2 as a field extesion of ky  , to a subfield of kx 1  . One can show that fiber products in C are the same Then there is a point zXSch X rational over X as those in LRS and that the forgetful functor 12Y 1 with πii()=zx, i =1,2. C  Top preserves fibered products (even though Proof. For 1), set  x ,:=xz , C  RS may not). Indeed, given 12 y := π11 xx = π 2 2 . Then we have a commutative s ππ11UU , the set diagram 11212XXY12g  Y π2,Z * kz   kx2 UU 12Y U, s is the preimage of kk under the π f map s _ , and we can see that s _ is continuous as 1,Z 2,x2 follows: By viewing the sheaf theoretic tensor product as f kx 1,x1  ky the sheafification of the presheaf tensor product we see 1  that, for any point  x12, xUU 1Y 2, we can find a of residue fields. By hypothesis, the compositions neighborhood VV12Y of  x12, x  contained in ky   kx i  kz  are isomorphisms for i =1,2 , UU and sections aa,,  V, so it must be that every map in this diagram is an 12Y 11nX1  bb,,  V such that the stalk s agrees isomorphism, hence the diagram is a pushout. On the 12nX2  x12,x with abii  at each x12, xVV 1Y 2. In other hand, according to the first statement of Theorem 9, i x12x particular, the function s _ agrees with the function 1   z is in bijective correspondence with

x12,()xaxbxk ii1  2   Spec kx 12 ky kx= Spec kz , i    on VV12Y . Since this latter function is continuous in which is punctual. the product topology on VV12Y (because each ai (_) , For 2), let ikx:  21 kx be the hypothesized bi (_) is continuous) and continuity is local, s_ is morphism of field extensions of ky(). By the universal continuous. property of the LRS fibered product X12Y X , the

Corollary 11. Suppose  f1,(), :Yfx X x is maps x1 111 surjective for every x  X . Then the comparison map 11  x212,:SpecikxX  is an isomorphism. In particular,  is an isomor- x :Speckx X phism under either of the following hypotheses: 111

1) f1 is an immersion. give rise to a map 2) f :Spec ky Y is the natural map associated 1 g :Speckx X X. to a point y Y .  112 Y Proof. It is equivalent to show that X := XXRS is 12Y Let zX 12Y X be the point corresponding to this map. in LRS and the structure maps πii: X  X are Then we have a commutative diagram of residue fields

LRS morphisms. Say x =,xx12 X and let kx 1  y :=fx11 = fx 2  2. By construction of X , we have a pushout diagram of rings i

()f1 x 1  kz kx Yy,, X11 x 2

f2 π x2 1 x π1,Z π 2 x  Xx,, Xx kx 1  ky 22

so π1,z :()kx 1  kz must be an isomorphism. according to Theorem 4, the diagram stays cartesian upon localizing, so the left vertical arrows also become 3.3. Spec Functor isomorphisms upon localizing, hence

* loc Suppose X LRS and f :  A is an  SpecfYff := , , # X X YX** X Y algebra. Then f may be viewed as a morphism of = Spec B ringed spaces f :,XA X ,X = X. Give X the local prime system X as usual and X , A the =.X inverse image prime system  (Remark 1), so f Remark 11. Hakim [Hak IV.1] defines a “Spec func- may be viewed as a PRS morphism tor” from ringed topoi to locally ringed topoi, but it is not * the same as ours on the common domain of definition. fXAf:,,XXX  X ,, . There is no meaningful situation in which Hakim’s Spec Explicitly: functor agrees with the “usual” one. When X “is” a *1 locally ringed space, Hakim’s SpecX “is” (up to re- fAfXx=:()=.ppmxxX,x x placing a locally ringed space with the corresponding loc By Theorem 2, there is a unique LRS morphism locally ringed topos) our X , X . As mentioned in Remark 2, Hakim’s theory of localization is only devel- * loc loc f :,,XAfXX  X,,X =X oped for the terminal prime system, which can be a bit awkward at times. For example, if X is a locally ringed lifting f to the localizations. We call space at least one of whose local rings has positive Krull loc SpecAXAf := , , * dimension, Hakim’s sequence of spectra yields an infi- XX nite strictly descending sequence of RS morphisms the spectrum (relative to X ) of A . Spec defines a X  Spec Spec X  Spec XX . functor

op The next results show that SpecX takes direct limits SpecXX : RingsX  LRS X . of X algebras to inverse limits in LRS and that Note that Spec =X , ,loc = X by Theo- SpecX is compatible with changing the base X . XX X X Lemma 14. The functor Spec preserves inverse rem 2. X Our functor Spec agrees with the usual one (c.f. limits. X Proof. Let if :  A be a direct limit sys- [Har II.Ex.5.17]) on their common domain of definition:  iX i tem in  Rings X , with direct limit f :  A , Lemma 13. Let f : XY be an affine morphism X   X and structure maps jA:  A. We claim that of schemes. Then Spec f  (as defined above) is na- ii XX* loc turally isomorphic to X in LRS Y . * SpecXXAXAf = , ,  is the inverse limit of Proof. This is local on Y , so we can assume loc * YA= Spec is affine, and hence X = Spec B is also iAXAf SpecXi = , i , i X . By Theorem 4, it is # affine, and f corresponds to a ring map f : AB . * enough to show that  XAf,, X  is the inverse limit Then * of iXAf  ,,ii X in PRS . Certainly  X , A is  fBB== , the inverse limit of iXA  , i  in RS , so we just * XAYY Y *** need to show that fjfXiiiX=   as prime as Y algebras, and the squares in the diagram systems on  X , A (see the proof of Theorem 4), and

# * this is clear because jfii= f , so, in fact, Yf,,*XY f  Y,,  YY ** *   jfii  X = f X for every i . Lemma 15. Let f : XY be a morphism of locally ringed spaces. Then for any Y algebra g :Y  A , # * YB,, f N  YAN,, the diagram  YYY   *  SpecXYf A  Spec A *,BB ,Spec  *,AA ,Spec  X  Y in PRS are cartesian in PRS , where N is the prime system on YA, Y  given by Nyy =  discussed in is cartesian in LRS . Lemma 3. According to that lemma, the right vertical Proof. Note fA*1:= f A  as usual. One f 1 X arrows become isomorphisms upon localizing, and sees easily that Y

* Every fibered category F admits a morphism of XfA,,*1 f g  YAg,, *  XY 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. XY,,XX ,,  YY pre pre Define a category Sch X as follows. Objects of Sch X is cartesian in PRS so the result follows from Theorem are pairs UX, U  consisting of an open subset UX

4. and a scheme XU over SpecX U . A morphism

Example 1. When X is a scheme, but A is not a UX,,UV   VX is a pair UVXX, UV coherent X module, SpecX A may not be a scheme. consisting of an Ouv X  morphism UV (i.e. For example, let B be a local ring, X := SpecB , and UV ) and a morphism of schemes XUV X mak- let x be the unique closed point of X . Let ing the diagram A:= xBRings X be the skyscraper sheaf B sup- *  XX ported at x . Note  = B and UV Xx,  (5) HomRingsX XX ,x*,BB = HomRings x , , SpecXXUV Spec  so we have a natural map X  A in Rings X  commute in Sch . The forgetful functor whose stalk at x is IdB:  B. Then pre SchX  Ouv X  is clearly a fibered category, where pre SpecX A =xA , is the punctual space with “sheaf” of a cartesian arrow is a Sch morphism rings A , mapping in LRS to X in the obvious X UVXX, UV making (6) cartesian in Sch manner. But x , A is not a scheme unless A is (equivalently in LRS ). Since OuvX has a topology, zero dimensional. we can form the associated stack Sch X . The category of Here is another related pathology example: Proceed as relative schemes over X is, by definition, the fiber above, assuming B is a local domain which is not a category Sch  X  of Sch over the terminal object field and let K be its fraction field. Let A:= xK, and X X * X of Ouv X  . let X  A be the unique map whose stalk at x is (The definition of relative scheme makes sense for a BK . Then SpecX A is empty. ringed topos X with trivial modifications.) Suppose X is a scheme, and A is an X algebra such that SpecX A is a scheme. I do not know whether 3.5. Geometric Realization this implies that the structure morphism SpecX A  X is an affine morphism of schemes. Now let X be a locally ringed space. Following [Hak V.3], we now define a functor 3.4. Relative Schemes FXX:()SchXX LRS We begin by recalling some definitions. called the geometric realization. Although a bit abstract, Definition 7. ([SGA1], [Vis 3.1]) Let F : CD be the fastest way to proceed is as follows: a functor. A C morphism f : cc  is called Definition 9. Let LRS X be the category whose cartesian (relative to F ) iff, for any C morphism objects are pairs UX, U  consisting of an open subset g : cc  and any D morphism hFc :  Fc with UX and a locally ringed space XU over Fgh = Ff there is a unique C morphism hc:  c UU,X  , and where a morphism UX,,UV  VX with Fhh= and f = gh . The functor F is called a is a pair UVXX, UV consisting of an fibered category iff, for any D morphism f : dd  Ouv X  morphism UV (i.e. UV ) and an and any object c of C with Fcd= , there is a LRS morphism XUV X making the diagram cartesian morphism f : cc  with F ff= . A morphism of fibered categories XXUV  (7) F ::CD FCD'  UU,,XX  VV is a functor GC:' C satisfying FGF= and taking cartesian arrows to cartesian arrows. If D has a commute in LRS . The forgetful functor UX, U  U topology (i.e. is a site), then a fibered category makes LRS X a fibered category over Ouv X  F :CD is called a stack iff, for any object dD where a cartesian arrow is a morphism and any cover ddi  of d in D , the category UVXX, UV making (8) cartesian in LRS . 1 F d is equivalent to the category F ddi  of In fact the fibered category LRSX  Ouv X  is a descent data (see [Vis 4.1]). stack: one can define locally ringed spaces and mor-

Copyright © 2011 SciRes. APM 262 W. D. GILLAM phisms thereof over open subsets of X locally. Using X  X  f the universal property of stackification, we define FX  to be the morphism of stacks (really, the corresponding YY  morphism on fiber categories over the terminal object X OuvX ) associated to the morphism of fibered in LRS . categories The above two conditions are equivalent by definition of an affine morphism of schemes, and one sees the F pre: Sch pre LRS XX X equivalence of (RA1) and (RA1.1) using Proposition 7,

LRS which ensures that the map YA Spec in (RA1) UX,,UUUX UXSpec ( )  U ,. U X factors through YY SpecY , hence XY=Spec B The map UU,SpecXX  U is the adjunc- Spec A tion morphism for the adjoint functors of Proposition 7. =SpecSpecYYBSpec Y YASpec This functor clearly takes cartesian arrows to cartesian Y =Spec.YY  B arrows. SpecY Y YA Remark 12. Although we loosely follow [Hak V.3.2] in our construction of the geometric realization, our geo- Each of these conditions has some claim to be the metric realization functor differs from Hakim’s on their definition of a relatively affine morphism in LRS . With common domain of definition. the exception of (2), all of the conditions are equivalent, when Y is a scheme, to f being an affine morphism 3.6. Relatively Affine Morphisms of schemes in the usual sense. With the exception of (4), each condition is closed under base change. For each pos- Let f : XY be an LRS morphism. Consider the sible definition of a relatively affine morphism in LRS , one has a corresponding definition of relatively schema- following conditions: tic morphism, namely: f : XY in LRS is rela- RA1. Locally on Y there is an  Y algebra A Y  tively schematic iff, locally on X , f is relatively af- and a cartesian diagram fine. X  SpecA The notion of “relatively schematic morphism” ob- f  tained from (1) is equivalent to: f : XY is in the essential image of the geometric realization functor FY . YY SpecY  in LRS . 3.7. Monoidal Spaces RA2. There is an  algebra A so that f is Y The setup of localization of ringed spaces works equally isomorphic to SpecY A in LRS Y . RA3. Same condition as above, but A is required to well in other settings; for example in the category of be quasi-coherent. monoidal spaces. We will sketch the relevant definitions RA4. For any g : ZY in LRS Y , the map and results. For our purposes, a monoid is a set P equipped with a commutative, associative binary opera- HomZX , Hom f, g tion + such that there is an element 0 P with LRS YXY RingsY **Z 0= pp for all pP . A morphism of monoids is a hgh # * map of sets that respects + and takes 0 to 0. An ideal of a is bijective. monoid P is a subset I  P such that I PI. An Remark 13. The condition (RA1) is equivalent to both ideal I is prime iff its complement is a submonoid (in of the following conditions: [label = RA1.., ref = RA1] particular, its complement must be non-empty). A sub- RA1.1 Locally on Y there is a ring homomorphism monoid whose complement is an ideal, necessarily prime, 2 A  B and a cartesian diagram is called a face. For example, the faces of  are 0,0 ,   0 , and 0   ; the diagonal  :  2 X  SpecB is a submonoid, but not a face. f  If SP is a submonoid, the localization of P at 1 YA Spec S is the monoid SP whose elements are equivalence classes  ps,  , pP , s  S where  ps,=  p , s in LRS . iff there is some tS with tpstps=  , and RA1.2. Locally on Y there is an affine morphism of where  ps,,:=,  p s  p p s s . The natural map schemes X  Y and a cartesian diagram PSP 1 given by pp  ,0 is initial among mon-

Copyright © 2011 SciRes. APM W. D. GILLAM 263 oid homomorphisms hP:  Q with hS Q* . The op Spec :XXMonXX LMS , localization of a monoid at a prime ideal is, by definition, the localization at the complementary face. for any  X ,X LMS which preserves inverse limits, A monoidal space X ,X is a topological space and to prove that the natural map X equipped with a sheaf of monoids  . Monoidal X HomX , ,SpecPPX Hom , spaces form a category MS where a morphism LMS  XX  Mon   † fffX=, :,XY Y , consists of a con- is bijective. tinuous map f : XY together with a map  of sheaves of monoids on X . A monoidal space 4. Acknowledgements X ,X is called local iff each stalk monoid X has a unique maximal ideal m x . Local monoidal spaces This research was partially supported by an NSF Post- form a category LMS where a morphism is a map of doctoral Fellowship. the underlying monoidal spaces such that each stalk map † fx : Yfx,() Xx , is local in the sense 5. References † 1 f mmfx = x . A primed monoidal space is a mo-  [1] M. Hakim, “Topos Annelés et Schémas Relatifs. Ergeb- noidal space equipped with a set of primes M x in each nisse der Mathematik und ihrer Grenzgebiete,” Springer- stalk monoid Xx, . The localization of a primed mo- Verlag, Berlin, 1972. noidal space is a map of monoidal spaces [2] R. Hartshorne, “Algebraic Geometry,” Springer-Verlag loc XM,,XX X , from a local monoidal space Berlin, 1977. constructed in an obvious manner analogous to the con- [3] H. Becker, Faserprodukt in LRS. struction of § 2.2 and enjoying a similar universal pro- http://www.uni-bonn.de/~habecker/Faserprodukt―in―L perty. In particular, we let SpecP denote the locali- RS.pdf. zation of the punctual space with “sheaf” of monoids P [4] L. Illusie, “Complexe Cotangent et Deformations I. L.N.M. 239,” Springer-Verlag, Berlin, 1971. at the terminal prime system. A scheme over 1 is a locally monoidal space locally isomorphic to SpecP for [5] A. Grothendieck and J. Dieudonné, “Éléments de various monoids P . (This is not my terminology.) Géométrie Algébrique,” Springer, Berlin, 1960. The same “general nonsense” arguments of this paper [6] J. Giraud, “Cohomologie non Abélienne,” Springer, Ber- allow us to construct inverse limits of local monoidal lin, 1971. spaces, to prove that a finite inverse limit of schemes [7] A. Vistoli, “Notes on Grothendieck Topologies, Fibered over 1 , taken in local monoidal spaces, is again a Categories, and Descent Theory,” Citeseer, Princeton 2004. scheme over 1 , to construct a relative Spec functor