Localization of Ringed Spaces

doi:10.4236/apm.2011.15045

Advances in Pure Mathematics, 2011, 1, 250-263

doi:10.4236/apm.2011.15045 Published Online September 2011 (http://www.SciRP.org/journal/apm)

Localization of Ringed Spaces

William D. Gillam

Department of 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

: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



 over a point



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



 is given the topo-

logy it inherits from 12Y

X

X. One can even describe

the sheaf of rings



1

12

,

x



 inherits from 12Y

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









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

XRS

(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



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

251







1212

,,

Y

XXSxxRS

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 XLRS , 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 XLRS, the presheaf



LRS ,YHomYX

is a sheaf on schemes in both the fppf and fpqc topolo-

gies. Indeed, one can easily describe a topology on RS ,

analogous to the fppf and fpqc topologies on schemes,

and prove it is subcanonical. To upgrade this to a sub-

canonical topology on LRS one is naturally confronted

with the comparison of fibered products in LRS and

RS . In particular, one is confronted with the question of

whether



is an epimorphism in the category of ringed

spaces. I do not know whether this is true for arbitrary

LRS morphisms 12

,

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

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

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

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

XPS is defined by





:= .

iii i

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

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



,YNPRS 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 XLRS , 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

XRS , the terminal prime system X

 defined by

,,

:= Spec

Xx Xx

 (i.e. the terminal object in





X

PS ).

For



,YMPRS , 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

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

UF, 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

UsSec, set













,:= ,:,.

loc x

UUsxzXxUsz

This defines a functor



:loc

X

UXSec

satisfying:











>0

,,=,||

,= ,.

UV UV

n

UUsUVtUU Vst

UUs UUsn





The first formula implies that







loc

X

UXSec

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

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

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

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

s

zat



.

Note that











,=

x

xx

M

DfUUt MDft

and













Spec ,

n

x

xx

Xx

at Dft

. The sets





x

Dft M

cover





,

Spec

x

Xx

MDf, and we have a “global

formula”

s

showing that the stalks of the various





n

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





,XMPRS 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

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

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









 





commute in





loc

YRings . The stalk of this diagram at

(,) loc

x

zX is a diagram



  



,

1

,

1

,,

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

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

,,

xz

f

Xx Yf x

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

xz

f



mm (because



,

x

z

f

 is local)

we find



1

(),

1

(), ,

11

1

=π()

=π

=()

z

fxz

z

fxz xz

xx z

x

z

f

fz















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 is an isomorphism for any

XLRS . On the level of topological spaces, it is clear

that π is a continuous bijection, so to prove it is an

isomorphism we just need to prove it is open. To prove

this, it is enough to prove that for any



,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m is

the localization of the local ring ,Xx

 at its unique

maximal ideal, which is an isomorphism in LAn .

Lemma 3. Let ARings be a ring,



,:=Spec

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

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

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

aA A. The only such prime

ideal is the maximal ideal

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

xxxxm

so to prove that they are continuous, we just need to

prove that they have the same topology. Indeed, we will

show that they all have the usual (Zariski) topology on

=Spec

X

A. This is clear for



,

,,

XXX

X

 be-

cause localization retracts  (Theorem 2), so









,

,, =,

loc

XX



 , and it is clear for





*,,Spec

A

A because of the description of the fibers of

localization in Proposition 1. For



,,

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

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 Ap. 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 YLRS 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

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 ARings , XLRS , 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

257

 







Hom,Spec =Hom,*,

=Hom,, *,

=Hom, *,

=Hom, ,.

loc

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

 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

xXRS 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

XRS 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

iY

i

ii Y

ii

YU YU

AY

YA









LRS LRS

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

,, ,,,,,,AkBkBkLAnmm m and

:

ii

f

AB are LAn morphisms, so





1=

ii

fmm

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

SABBBBKpp

is closed in





12

SpecA

BB.

The subset





12

,,SABB enjoys the following impor-

tant property: Suppose



:, ,

iii

gB Cmn

, =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

258



12 12

=, :A

hffB BC is the induced map. Then

 

1

12

,,hSABB

n. Conversely, every



12

,,SABBp arises in this manner: take



12

=A

CB Bp.

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

XRS 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

,,

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

XRS 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

XLRS

is generally much finer than the product topology. In this

situation, the set





12

,Sxx always consists of a single

element





12

,zxx : namely, the maximal ideal of

,,

11,22

XxX x

Yy





 given by the kernel of the natural

surjection







,,122

11,22=.

XxX xky

Yy kxkx kx





If we identify 12Y

X

XLRS and 12Y

X

XRS 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

259

X

kLRS 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

UsSec,

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

CTop preserves fibered products (even though

CRS 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

s

agrees

with

 

12

ii

i

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

XXRS 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

()

,,

11

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

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,

1, 2

1, 1

π

2

π

1

Z

x

Z

x

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



, 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

260

so



1, 1

π:()

zkx kz must be an isomorphism.

3.3. Spec Functor

Suppose XLRS 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

XRings 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

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

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 XRings 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

UV









(5)

commute in Sch . The forgetful functor





pre

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

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