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arXiv:1104

.4951v2

[math.DG

]16Nov2012

An introduction to C-schemes

and C-algebraic geometry

Dominic Joyce

1 Introduction

IfX is a manifold then the R-algebra C(X) of smooth functions c: X R

is a C

-ring. That is, for each smooth functionf :R

n

R

there is an n-foldoperation f : C(X)n C(X) acting by f : c1, . . . , cn f(c1, . . . , cn),

and these operations fsatisfy many natural identities. Thus,C(X) actuallyhas a far richer structure than the obvious R-algebra structure.

In [7]the author set out the foundations of a version of algebraic geometry inwhich rings or algebras are replaced by C-rings, focussing onC-schemes, acategory of geometric objects which generalize manifolds, and whose morphismsgeneralize smooth maps, quasicoherent and coherent sheaves on C-schemes,and C-stacks, in particular DeligneMumford C-stacks, a 2-category of ge-ometric objects which generalize orbifolds. This paper is a survey of[7].

C-rings andC-schemes were first introduced in synthetic differential ge-ometry, see for instance Dubuc [3], Moerdijk and Reyes [15] and Kock [11].Following Dubucs discussion of models of synthetic differential geometry [2]

and oversimplifying a bit, symplectic differential geometers are interested in C

-schemes as they provide a category CSchof geometric objects which includessmooth manifolds and certain infinitesimal objects, and all fibre products existinCSch, andCSchhas some other nice properties to do with open covers,and exponentials of infinitesimals.

Synthetic differential geometry concerns proving theorems about manifoldsusing synthetic reasoning involving infinitesimals. But one needs to check thesemethods of synthetic reasoning are valid. To do this you need a model, somecategory of geometric spaces including manifolds and infinitesimals, in whichyou can think of your synthetic arguments as happening. Once you know thereexists at least one model with the properties you want, then as far as syntheticdifferential geometry is concerned the job is done. For this reason C-schemeswere not developed very far in synthetic differential geometry.

Recently,C-rings andC-ringed spaces appeared in a very different con-text, as part of Spivaks definition ofderived manifolds [18], which are an ex-tension to differential geometry of Jacob Luries derived algebraic geometryprogramme. The author [810] is developing an alternative theory of deriveddifferential geometry which simplifies, and goes beyond, Spivaks derived man-

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ifolds. Our notion of derived manifolds are called d-manifolds. We also studyd-manifolds with boundary, andd-manifolds with corners, and orbifold versions

of all these, d-orbifolds. To define d-manifolds and d-orbifolds we need theo-ries ofC-schemes,C-stacks, and quasicoherent sheaves upon them, much ofwhich had not been done, so the author set up the foundations of these in [ 7].

D-manifolds and d-orbifolds will have important applications in symplecticgeometry, and elsewhere. Many areas of symplectic geometry involve mod-uli spacesMg,m(J, ) of stable J-holomorphic curves in a symplectic manifold(M, ). The original motivation for [810] was to find a good geometric de-scription for the geometric structure on such moduli spacesMg,m(J, ). In theLagrangian Floer cohomology theory of Fukaya, Oh, Ohta and Ono[5], modulispaces Mg,m(J, ) are given the structure of Kuranishi spaces. The notion ofKuranishi space seemed to the author to be unsatisfactory. In trying improveit, using ideas from Spivak [18], the author arrived at the theory of[810]. Theauthor believes the correct definition of Kuranishi space in the work of Fukayaet al. [5] should be that a Kuranishi space is a d-orbifold with corners.

Section2 explains C-rings and their modules, 3introducesC-schemes,and quasicoherent and coherent sheaves upon them, and 4discussesC-stacks,particularly DeligneMumford C-stacks, their relation to orbifolds, quasi-coherent and coherent sheaves on DeligneMumford C-stacks, and orbifoldstrata of DeligneMumford C-stacks.

Acknowledgements.I would like to thank Eduardo Dubuc and Jacob Lurie forhelpful conversations.

2 C-rings

We begin by explaining the basic objects out of which our theories are built, C

-rings, or smooth rings, following [7, 2, 3 & 5]. Everything in 2.12.2wasalready known in synthetic differential geometry, and can be found in Moerdijkand Reyes [15,Ch. I], Dubuc[24]or Kock [11, III].

2.1 Two definitions ofC-ring

Definition 2.1. A C-ring is a set C together with operations f : Cn C

for alln 0 and smooth mapsf : Rn R, where by convention whenn = 0 wedefineC0 to be the single point {}. These operations must satisfy the followingrelations: suppose m, n 0, and fi: R

n R fori = 1, . . . , m and g : Rm Rare smooth functions. Define a smooth function h : Rn R by

h(x1, . . . , xn) = g

f1(x1, . . . , xn), . . . , f m(x1 . . . , xn)

,

for all (x1, . . . , xn) Rn. Then for all (c1, . . . , cn) C

n we have

h(c1, . . . , cn) = g

f1(c1, . . . , cn), . . . , fm(c1, . . . , cn)

.

We also require that for all 1 j n, defining j : Rn R by j :

(x1, . . . , xn)xj , we have j (c1, . . . , cn) = cj for all (c1, . . . , cn) Cn.

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Usually we refer to C as the C-ring, leaving the operations f implicit.A morphism between C-rings

C, (f)f:RnR C

,D, (f)f:RnR C

is a map : C D such that f(c1), . . . , (cn) = f(c1, . . . , cn) forall smooth f : Rn R and c1, . . . , cn C. We will writeCRings for thecategory ofC-rings.

Here is the motivating example:

Example 2.2. Let Xbe a manifold, and write C(X) for the set of smoothfunctionsc: X R. Forn 0 and f : Rn R smooth, define f :C(X)n C(X) by

f(c1, . . . , cn)

(x) =f

c1(x), . . . , cn(x)

, (1)

for all c1, . . . , cn C(X) and x X. It is easy to see thatC(X) and theoperations f form a C-ring.

Now let f : X Y be a smooth map of manifolds. Then pullback f :

C(Y) C(X) mappingf :c c fis a morphism ofC-rings. Further-more, everyC-ring morphism : C(Y) C(X) is of the form = f fora unique smooth mapf :XY.

Write CRingsop for the opposite category ofCRings, with directionsof morphisms reversed, and Man for the category of manifolds without bound-ary. Then we have a full and faithful functorFC

RingsMan :Man C

Ringsop

acting byFCRings

Man (X) =C(X) on objects and FC

RingsMan (f) =f

on mor-phisms. This embeds Man as a full subcategory ofCRingsop.

Note thatC-rings are far more general than those coming from manifolds.For example, ifXis any topological space we could define a C-ringC0(X) tobe the set of continuous c: X R, with operations fdefined as in (1). ForXa manifold with dim X >0, the C-ringsC(X) and C0(X) are different.

There is a more succinct definition ofC-rings using category theory:

Definition 2.3. Write Euc for the full subcategory ofMan spanned by theEuclidean spaces Rn. That is, the objects of Euc are the manifolds Rn forn = 0, 1, 2, . . ., and the morphisms in Euc are smooth maps f : Rn Rm.Write Sets for the category of sets. In both Euc andSets we have notions of(finite) products of objects (that is, Rn+m = Rn Rm, and products S T ofsets S, T), and products of morphisms. Define a (category-theoretic) C-ringto be a product-preserving functor F :Euc Sets.

Here is how this relates to Definition 2.1. Suppose F : Euc Sets is aproduct-preserving functor. Define C = F(R). Then C is an object in Sets,that is, a set. Supposen 0 andf : Rn Ris smooth. Then fis a morphisminEuc, soF(f) : F(Rn) F(R) = C is a morphism in Sets. SinceFpreservesproductsF(Rn) =F(R) F(R) = Cn, so F(f) maps Cn C . We definef :C

n C by f =F(f). ThenC, (f)f:RnR C

is a C ring.

As in[7, Prop. 2.5],[15,p. 2122] we have:

Proposition 2.4. In the category CRings of C-rings, all small colimitsexist, and so in particular pushouts and all finite colimits exist.

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Definition 2.5. Let C be a C-ring. Then we may giveC the structure ofa commutative R-algebra. Define addition + on C by c+ c = f(c, c

) for

c, c C , wheref : R2 Ris f(x, y) = x + y. Define multiplication on C byc c = g(c, c), where g : R

2 R is f(x, y) = xy. Define scalar multiplicationby R byc= (c), where : R R is(x) = x. Define elements 0 and1 in C by 0 = 0() and 1 = 1(), where 0 : R

0 Rand 1 : R0 Rare themaps 0 : 0 and 1 : 1. One can then show using the relations on the fthat all the axioms of a commutative R-algebra are satisfied. In Example2.2,this yields the obvious R-algebra structure on the smooth functions c : X R.

Anideal I in C is an ideal I C in C regarded as a commutative R-algebra.Then we make the quotient C/I into a C-ring as follows. Iff : Rn R issmooth, define If : (C/I)

n C/Iby

If(c1+ I , . . . , cn+ I)

(x) = f

c1(x), . . . , cn(x)

+ I.

To show this is well-defined, we must show it is independent of the choice ofrepresentatives c1, . . . , cn in C for c1 + I , . . . , cn + I in C/I. By HadamardsLemma there exist smooth functions gi : R

2n R for i = 1, . . . , nwith

f(y1, . . . , yn) f(x1, . . . , xn) =n

i=1(yi xi)gi(x1, . . . , xn, y1, . . . , yn)

for all x1, . . . , xn, y1, . . . , yn R. Ifc1, . . . , cn are alternative choices for c1, . . . ,

cn, so that ci+ I=ci+ I fori = 1, . . . , nand ci ci I, we have

f

c1(x), . . . , cn(x)

f

c1(x), . . . , cn(x)

=n

i=1(ci ci)gi

c1(x), . . . , c

n(x), c1(x), . . . , cn(x)

.

The second line lies in I as ci ci I and Iis an ideal, so If is well-defined,

and clearly C/I, (I

f)f:RnR C is aC

-ring.We will use the notation (fa : a A) to denote the ideal in a C-ring Cgenerated by a collection of elements fa C , a A. That is,

(fa : a A) =n

i=1 fai ci : n 0, a1, . . . , an A, c1, . . . , cn C

.

2.2 Special classes ofC-ring

We definefinitely generated, finitely presented, local, andfair C-rings.

Definition 2.6. AC-ring Cis calledfinitely generatedif there existc1, . . . , cnin C which generate C over all C-operations. That is, for each c C there

exists smooth f : Rn

R with c = f(c1, . . . , cn). Given such C, c1, . . . , cn,define : C(Rn) C by (f) = f(c1, . . . , cn) for smoothf : R

n R, whereC(Rn) is as in Example2.2with X = Rn. Then is a surjective morphismof C-rings, so I = Ker is an ideal in C(Rn), and C = C(Rn)/I as aC-ring. Thus, C is finitely generated if and only ifC = C(Rn)/I for somen 0 and ideal I in C(Rn).

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An idealI inC(Rn) is calledfinitely generated ifI= (f1, . . . , f k) for somef1, . . . , f k C(R

n). AC-ring C is calledfinitely presented ifC=C(Rn)/I

for somen 0, whereIis a finitely generated ideal in C(Rn).A difference with conventional algebraic geometry is that C(Rn) is not

noetherian, so ideals in C(Rn) may not be finitely generated, and C finitelygenerated does not imply C finitely presented.

Definition 2.7. A C-ring C is called a C-local ring if regarded as an R-algebra, as in Definition 2.5, C is a local R-algebra with residue field R. Thatis, C has a unique maximal ideal mC with C/mC = R.

IfC,D are C-local rings with maximal ideals mC ,mD, and : C D isa morphism ofC rings, then using the fact that C/mC = R = D/mD we seethat 1(mD) = mC , that is, is a localmorphism ofC-local rings. Thus,there is no difference between morphisms and local morphisms.

Example 2.8. For n 0 and p Rn

, define C

p (Rn

) to be the set of germsof smooth functions c : Rn R at p Rn. That is, Cp (Rn) is the quotient

of the set of pairs (U, c) with p U Rn open and c : U R smoothby the equivalence relation (U, c) (U, c) if there exists p V U U

open with c|V c|V. Define operations f : (Cp (Rn))m Cp (R

n) forf : Rm R smooth by (1). ThenCp (R

n) is a C-local ring, with maximal

ideal m=

[(U, c)] : c(p) = 0

.

Definition 2.9. An ideal I in C(Rn) is called fair if for each f C(Rn),f lies in I if and only ifp(f) lies in p(I) Cp (R

n) for all p Rn, whereCp (R

n) is as in Example 2.8 and p : C(Rn) Cp (R

n) is the naturalprojection p : c [(R

n, c)]. A C-ring C is called fair if it is isomorphic toC(Rn)/I, whereI is a fair ideal.

Our term fair was introduced in [7] for brevity, but the idea was alreadywell-known. They were introduced by Dubuc[3, Def. 11] under the name C-rings of finite type presented by an ideal of local character, and in more recentwork would be called finitely generated and germ-determined C-rings.

As in [7, 2], if C(Rm)/I = C(Rn)/J then I is finitely generated, orfair, if and only ifJ is. Thus, to decide whether a C-ring C is finitely pre-sented, or fair, it is enough to test one presentation C = C(Rn)/I. Also, Cfinitely presented implies Cfair implies Cfinitely generated. WriteCRingsfp,CRingsfa and CRingsfg for the full subcategories of finitely presented,fair, and finitely generatedC-rings inCRings, respectively. Then

CRingsfp CRingsfa CRingsfg CRings.

From[7, Prop.s 2.23 & 2.25] we have:

Proposition 2.10. The subcategoriesCRingsfg, CRingsfp are closed un-der pushouts and all finite colimits in CRings, but CRingsfa is not.Nonetheless, pushouts and finite colimits exist inCRingsfa,though they maynot coincide with pushouts and finite colimits inCRings.

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Given morphisms : C D and : C E in CRings, the pushoutD ,C, E in CRings should be thought of as a completed tensor product

DCE. The tensor product D C E is an R-algebra, but in general not a C-ring, and to get a C-ring we must take a completion DCE. When C = R,the trivial C-ring, the pushout D R E is the coproduct D E= D RE. Forexample, one can show that C(Rm)RC(R

n)=C(Rm+n).Here is [7,Prop. 3.1].

Proposition 2.11. (a)If X is a manifold without boundary then theC-ringC(X) of Example 2.2 is finitely presented.

(b) If X is a manifold with boundary, or with corners, and X=, then theC-ringC(X) of Example 2.2 is fair, but is not finitely presented.

To save space we will say no more about manifolds with boundary or cornersandC-geometry in this paper. More information can be found in [ 710].

Example 2.12. AWeil algebra[2, Def. 1.4] is a finite-dimensional commutativeR-algebraWwhich has a maximal ideal m withW/m= Rand mn = 0 for somen > 0. Then by Dubuc [2, Prop. 1.5] or Kock [11, Th. III.5.3], there is aunique way to make W into a C-ring compatible with the given underlyingcommutative R-algebra. This C-ring is finitely presented[11, Prop. III.5.11].C-rings from Weil algebras are important in synthetic differential geometry,in arguments involving infinitesimals.

2.3 Modules over C-rings, and cotangent modules

In [7, 5] we discuss modules over C-rings.

Definition 2.13. Let C be a C-ring. A C-module M is a module over Cregarded as a commutative R-algebra as in Definition2.5. C-modules form anabelian category, which we write as C-mod. For example, C is a C -module, andmore generally C RV is a C-module for any real vector space V.

A C-module M is called finitely presentedif there exists an exact sequenceCRR

m C RRn M0 in C-mod for somem, n 0. We write C-modfp

for the full subcategory of finitely presented C-modules in C-mod. Then C-modfp

is closed under cokernels and extensions in C-mod. But it may not be closedunder kernels, so C-modfp may not be an abelian category.

Let : C D be a morphism of C-rings. If M is a C-module then(M) =MCDis a D-module. This induces a functor : C-mod D-mod,which maps C-modfp D-modfp.

Example 2.14. Let Xbe a manifold, and EXbe a vector bundle. WriteC(E) for the vector space of smooth sections e of E. Then C(X) actson C(E) by (c, e) c e for c C(X) and e C(E), so C(E) is aC(X)-module, which is finitely presented.

Now let X, Y be manifolds and f : X Y a smooth map. Then f :C(Y) C(X) is a morphism ofC-rings. IfEis a vector bundle over Y,

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thenf(E) is a vector bundle overX. Under the functor (f) : C(Y)-mod C(X)-mod of Definition2.13, we see that (f)

C(E)

= C(E) C(Y)

C(X) is isomorphic as a C(X)-module toCf(E).Every commutative algebraA has a natural module A called the module of

Kahler differentials, which is a kind of analogue for A of the cotangent bundleTX of a manifold X. In [7, 5.3] we define the cotangent module C of aC-ring C, which is the C-version of the module of Kahler differentials.

Definition 2.15. Let C be a C-ring, and M a C-module. A C-derivationis an R-linear map d : C Msuch that whenever f : Rn Ris a smooth mapandc1, . . . , cn C , we have

df(c1, . . . , cn) =n

i=1 fxi

(c1, . . . , cn) dci.

Note that d is not a morphism of C-modules. We call such a pair M, d acotangent module for C if it has the universal property that for any C-moduleM and C-derivation d : C M, there exists a unique morphism of C-modules : MM with d = d.

Define C to be the quotient of the free C-module with basis of symbolsdc for c C by the C-submodule spanned by all expressions of the formdf(c1, . . . , cn)

ni=1 f

xi

(c1, . . . , cn) dci for f : Rn R smooth and

c1, . . . , cn C, and define dC : C C by dC : c dc. Then C , dC is acotangent module for C. Thus cotangent modules always exist, and are uniqueup to unique isomorphism.

Let C ,Dbe C-rings with cotangent modules C , dC , D, dD, and : C Dbe a morphism ofC-rings. Then makes D into a C-module, and there isa unique morphism : C Din C-mod with dD= dC. This induces

a morphism () : C C D D in D-mod with () (dC idD) = dD.If : C D, : D E are morphisms ofC-rings then = .

Example 2.16. LetXbe a manifold. Then the cotangent bundle TXis a vec-tor bundle overX, so as in Example2.14it yields aC(X)-moduleC(TX).The exterior derivative d : C(X) C(TX) is a C-derivation. TheseC(TX), d have the universal property in Definition2.15, and so form a cotan-gent module for C(X).

Now let X, Ybe manifolds, and f :XY be smooth. Then f(T Y), T Xare vector bundles over X, and the derivative of f is a vector bundle mor-phism df : T X f(T Y). The dual of this morphism is (df) : f(TY)TX. This induces a morphism ofC(X)-modules ((df)) : C

f(TY)

C(TX). This ((df)) is identified with (f) in Definition2.15under the

natural isomorphismC

f

(T

Y)=C

(T

Y) C(Y)C

(X).Definition2.15abstracts the notion of cotangent bundle of a manifold in a

way that makes sense for any C-ring. From [7,Th.s 5.13 & 5.16] we have:

Theorem 2.17. (a) SupposeC is a finitely presented C-ring. ThenC is afinitely presented C-module.

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(b) Suppose we are given a pushout diagram of finitely generated C-rings:

C

E

D

F,

so that F= D C E. Then the following sequence of F-modules is exact:

CC ,C, F()()DD,D,F

E E,E, F()() F 0.

Here () : C C ,C, F D D,D,Fis induced by : C D,and so on.

3 C

-schemesWe now summarize material in[7, 4] onC-schemes, and in [7, 6] on coherentand quasicoherent sheaves on C-schemes. Much of3.1goes back to Dubuc[3].

3.1 The definition ofC-schemes

The basic definitions are modelled on the definitions of schemes in Hartshorne[6,II.2], but replacing rings by C-rings throughout.

Definition 3.1. A C-ringed space X = (X, OX) is a topological space Xwith a sheafOX ofC-rings on X. That is, for each open setU Xwe aregiven aC ringOX(U), and for each inclusion of open setsV UXwe aregiven a morphism ofC-rings UV : OX(U) OX(V), called the restrictionmaps, and all this data satisfies the usual sheaf axioms [6, II.1].

A morphism f = (f, f) : (X, OX) (Y, OY) of C ringed spaces is a

continuous mapf :XYand a morphism f :f1(OY) OX of sheaves ofC-rings on X. There is another way to write the data f: since direct imageof sheaves f is right adjoint to inverse image f

1, there is a natural bijection

HomX

f1(OY), OX

=HomY

OY , f(OX )

. (2)

Write f : OY f(OX) for the morphism of sheaves ofC-rings on Y corre-

sponding to f under (2), so that

f :f1(OY) OX f: OY f(OX). (3)

Depending on the application, either f

or f may be more useful. We chooseto regardf as primary and write morphisms as f= (f, f) rather than (f, f),because in [8] we find it convenient to work uniformly using pullbacks, ratherthan mixing pullbacks and pushforwards.

A local C-ringed space X= (X, OX) is a C-ringed space for which thestalks OX,x of OX at x are C-local rings for all x X. Since morphisms

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ofC-local rings are automatically local morphisms, morphisms of local C-ringed spaces (X, OX), (Y, OY) are just morphisms ofC-ringed spaces, with-

out any additional locality condition. Write CRS for the category ofC-ringed spaces, and LCRSfor the full subcategory of local C-ringed spaces.

For brevity, we will use the notation that underlined upper case lettersX , Y , Z , . . . representC-ringed spaces (X, OX), (Y, OY), (Z, OZ), . . ., and un-derlined lower case letters f , g, . . . represent morphisms of C-ringed spaces

(f, f), (g, g), . . . . When we write x X we mean that X = (X, OX ) andx X. When we write U is open in X we mean that U = (U, OU) andX= (X, OX) withUXan open set and OU=OX |U.

Definition 3.2. Write CRingsop for the opposite category of CRings.Theglobal sections functor : LCRS CRingsop acts on objects (X, OX)in LCRS by : (X, OX) OX(X) and on morphisms (f, f) : (X, OX)(Y, OY) by : (f, f

)f(Y), forf: OX f(OY) as in (3). As in[3, Th. 8]

there is a spectrum functorSpec : CRingsop LCRS, defined explicitlyin[7, Def. 4.12], which is a right adjoint to , that is, for all C CRingsandX LCRS there are functorial isomorphisms

HomCRings(C, (X))=HomLCRS(X, SpecC). (4)

For anyC-ring Cthere is a natural morphism ofC-rings C :C (SpecC)corresponding to idX in (4) withX= Spec C. By[3, Th. 13], the restriction of

Spec to (CRingsfa)op is full and faithful.A localC-ringed spaceXis called anaffineC-schemeif it is isomorphic

in LCRS to SpecC for some C-ring C. We callX a finitely presented, orfair, affineC-scheme ifX=SpecC for C that kind ofC-ring.

Let X = (X, OX) be a local C-ringed space. We call X a C-scheme if

X can be covered by open sets U X such that (U, OX |U) is an affine C-scheme. We call a C-schemeX locally fair, or locally finitely presented, ifXcan be covered by open U X with (U, OX |U) a fair, or finitely presented,affine C-scheme, respectively.

WriteCSchlf, CSchlfp, CSchfor the full subcategories of locally fair,and locally finitely presented, and all, C-schemes in LCRS, respectively.We call aC-schemeX separated, orparacompact, if the underlying topologicalspaceXis Hausdorff, or paracompact.

Example 3.3. LetXbe a manifold. Define a C-ringed spaceX= (X, OX)to have topological spaceXandOX(U) =C(U) for each openUX, whereC(U) is theC-ring of smooth maps c : U R, and ifV UXare opendefine UV : C(U) C(V) by UV : c c|V. Then X = (X, OX) is a

localC-ringed space. It is canonically isomorphic to Spec C(X), and so isan affineC-scheme. It is locally finitely presented.

Define a functor FCSch

Man : Man CSchlfp CSch by FC

SchMan =

Spec FCRings

Man . ThenFCSchMan is full and faithful, and embeds Man as a full

subcategory ofCSch.

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By [7,Prop.s 4.10, 4.11, 4.25, 4.32, Cor.s 4.18, 4.21 & Th. 4.33] we have:

Theorem 3.4. (a)All finite limits exist in the category C

RS.(b) The full subcategories CSchlfp, CSchlf, CSch, LCRS in CRSare closed under all finite limits in CRS. Hence, fibre products and all finitelimits exist in each of these subcategories.

(c)If C is a finitely generatedC-ring thenSpecCis a fair affineC-scheme.

(d)Let (X, OX)be a finitely presented, or fair, affineC-scheme, andUXbe an open subset. Then (U, OX |U) is also a finitely presented, or fair, affineC-scheme, respectively. However, this does not hold for general affine C-schemes.

(e) Let (X, OX) be a locally finitely presented, locally fair, or general, C-scheme, andUXbe open. Then (U, OX |U)is also a locally finitely presented,or locally fair, or general, C-scheme, respectively.

(f)The functorFC

SchMan takes transverse fibre products inManto fibre products

in CSch.

In[7, Def. 4.34 & Prop. 4.35] we discuss partitions of unityonC-schemes,building on ideas of Dubuc[4].

Definition 3.5. Let X = (X, OX) be a C-scheme. Consider a formal sumaA ca, where A is an indexing set and ca OX(X) for a A. We sayaA ca is a locally finite sum on X ifXcan be covered by open UX such

that for all but finitely many a A we have XU(ca) = 0 in OX(U).By the sheaf axioms for OX , if

aA ca is a locally finite sum there exists

a unique c OX(X) such that for all open U X such that XU(ca) = 0in OX(U) for all but finitely many a A, we have XU(c) =

aA XU(ca)

in OX(U), where the sum makes sense as there are only finitely many nonzeroterms. We call c the limit ofaA ca, writtenaA ca = c.Let c OX(X). SupposeVi X is open and XVi(c) = 0 OX(Vi) for

i I, and let V =

iIVi. ThenV X is open, andXV(c) = 0 OX(V) asOX is a sheaf. Thus taking the union of all open V XwithXV(c) = 0 givesa unique maximal open set Vc Xsuch that XVc(c) = 0 OX(Vc). Definethesupport supp c ofc to be X\ Vc, so that supp c is closed in X. IfUX isopen, we say that c is supported inU if supp c U.

Let {Ua : a A} be an open cover of X. A partition of unity on Xsubordinate to {Ua : a A} is{a : a A} with a OX(X) supported on Uafora A, such that

aA a is a locally finite sum on Xwith

aA a = 1.

Proposition 3.6. Suppose X is a separated, paracompact, locally fair C-

scheme, and {Ua : a A}an open cover of X. Then there exists a partition ofunity{a : a A} onXsubordinate to {Ua: a A}.

Here are some differences between ordinary schemes andC-schemes:

Remark 3.7. (i) IfA is a ring or algebra, then points of the correspondingscheme Spec A are prime ideals in A. However, if C is a C-ring then (by

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definition) points of SpecC are maximal ideals in C with residue field R, orequivalently, R-algebra morphismsx : C R. This has the effect that ifX is a

manifold then points of Spec C(X) are just points ofX.

(ii) In conventional algebraic geometry, affine schemes are a restrictive class.Central examples such as CPn are not affine, and affine schemes are not closedunder open subsets, so that C2 is affine but C2 \ {0} is not. In contrast, affineC-schemes are already general enough for many purposes. For example:

All manifolds are affine C-schemes.

Open C-subschemes of fair affine C-schemes are fair and affine.

Separated, second countable, locally fair C-schemes are affine.

Affine C-schemes are always separated (Hausdorff), so we need general C-schemes to include non-Hausdorff behaviour.

(iii)In conventional algebraic geometry the Zariski topology is too coarse formany purposes, so one has to introduce the etale topology. In C-algebraicgeometry there is no need for this, as affine C-schemes are Hausdorff.

(iv)Even very basicC-rings such as C(Rn) forn >0 are not noetherian asR-algebras. So C-schemes should be compared to non-noetherian schemes inconventional algebraic geometry.

3.2 Quasicoherent and coherent sheaves on C-schemes

In [7, 6] we discuss sheaves of modules on C-schemes.

Definition 3.8. Let X = (X, OX) be a C-scheme. An OX -module E on Xassigns a module E(U) over OX(U) for each open set U X, with OX(U)-

actionU : OX(U) E(U) E(U), and a linear map EUV :E(U) E(V) foreach inclusion of open sets V UX, such that the following commutes:

OX(U) E(U)UVEUV

U E(U)EUV

OX(V) E(V) V E(V),

and all this data E(U), EUVsatisfies the usual sheaf axioms [6, II.1] .A morphism of OX-modules : E F assigns a morphism of OX(U)-

modules(U) :E(U) F(U) for each open set UX, such that(V) EUV =FUV (U) for each inclusion of open sets V U X. Then OX -modulesform anabelian category, which we write as OX-mod.

As in [7, 6.2], the spectrum functor Spec : CRingsop CSch has

a counterpart for modules: if C is a C

-ring and (X, OX) = SpecC we candefine a functor MSpec : C-mod OX -mod. Let X = (X, OX) be a C-scheme, and E an OX -module. We call E quasicoherent if X can be coveredby open Uwith U=SpecC for some C-ring C, and under this identificationE|U =MSpec M for some C-module M. We callEcoherent if furthermore wecan take these C -modulesMto be finitely presented. We callEavector bundle

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where: G H is a group morphism and f : XY is a morphism ofC-schemes withf= ()ffor all G. Iff= (f , ) andg = (g, )

are two such 1-morphisms, then a 2-morphism : f g is roughly anelement Hsuch that () = ()1 for all G, and g = f.

There is a good notion offibre productin a 2-category. All fibre productsof DeligneMumfordC-stacks exist, as DeligneMumford C-stacks.

4.1 The definition ofC-stacks

The next few definitions assume a lot of standard material from stack theory,which is summarized in [7,7].

Definition 4.1. Define a Grothendieck topology Jon the category CSchofC-schemes to have coverings{ia : Ua U}aA whereVa = ia(Ua) is open inU with ia: Ua (Va, OU|Va) an isomorphism for all a A, and U=aA

Va.Up to isomorphisms of theUa, the coverings {ia: Ua U}aA ofUcorrespondexactly to open covers{Va: a A}ofU. Then (CSch, J) is a site.

The stacks on (CSch, J) form a 2-category Sta(CSch,J). The site(CSch, J) is subcanonical. Thus, if X is any C-scheme we have an as-sociated stack on (CSch, J) which we write as X. A C-stack is a stack Xon (CSch, J) such that the diagonal 1-morphism X :X X X is repre-sentable, and there exists a surjective 1-morphism : U Xcalled anatlas forsome C-schemeU. C-stacks form a 2-category CSta. All 2-morphismsin CSta are invertible, that is, they are 2-isomorphisms.

Remark 4.2. So far as the author knows, [7] is the first paper to considerstacks on the site (CSch, J). Note that Behrend and Xu [1, Def. 2.15] usethe term C-stack to mean something different, a geometric stack over the

site (Man, JMan) of manifolds without boundary with Grothendieck topologyJMan given by open covers. These are called smooth stacks by Metzler [13].

Write ManSta for the 2-category of geometric stacks on (Man, JMan), asin[1,12,13]. The functorFC

SchMan of Example3.3embeds the site (Man, JMan)

into (CSch, J). Thus, restricting from (CSch, J) to (Man, JMan) definesa natural truncation 2-functor FManStaCSta :C

StaManSta.A C-stack X encodes all morphisms F : U X for C-schemes U,

whereas its image FManStaCSta (X) remembers only morphisms F : U X formanifolds U. Thus the truncation functor FManStaCSta loses information, as itforgets morphisms from C-schemes which are not manifolds. This includesany information aboutnonreduced C-schemes. For our applications in [810]this nonreduced information will be essential, so we must consider stacks on(CSch, J) rather than on (Man, JMan).

Definition 4.3. A groupoid object (U , V , s , t , u , i , m) in CSch, or simplygroupoid in CSch, consists of objects U, V in CSch and morphisms s, t :

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Definition 4.5. Let X be a C-stack. We say that X is separated if thediagonal 1-morphism X :X X Xis universally closed. IfX = Xfor some

C-scheme X= (X, OX) thenXis separated if and only if X :XX Xis closed, that is, if and only ifXis Hausdorff, so X is separated.

Definition 4.6. LetX be aC-stack. AC-substackY Xis a substack ofXwhich is also aC-stack. It has a natural inclusion 1-morphism iY :Y X.We call Y an open C-substack ofX ifiY is a representable open embedding.An open cover {Ya : a A} ofX is a family of open C-substacksYa Xwith

aA iYa :

aA Ya X surjective.

DeligneMumford stacks in algebraic geometry are locally modelled on quo-tient stacks [X/G] forXan affine scheme andG a finite group. This motivates:

Definition 4.7. A DeligneMumford C-stack is a C-stack X which ad-mits an open cover {Ya : a A} with each Ya equivalent to a quotient stack[Ua/Ga] for Ua an affine C

-scheme and Ga a finite group. We call X a lo-cally fair, orlocally finitely presented, DeligneMumfordC-stack if it has suchan open cover with each Ua a fair, or finitely presented, affine C

-scheme, re-spectively. WriteDMCStalf, DMCStalfp andDMCStafor the full 2-subcategories of locally fair, locally finitely presented, and all, DeligneMumfordC-stacks inCSta.

From[7, Th.s 8.5, 9.10, 9.16 & Prop. 9.6] we have:

Theorem 4.8. (a)All fibre products exist in the 2-category CSta.

(b) DMCSta, DMCStalf andDMCStalfp are closed under fibre prod-ucts in CSta.

(c) DMCSta,DMCStalf andDMCStalfp are closed under taking openC-substacks in CSta.

(d)A C-stack X is separated and DeligneMumford if and only if it is equiv-alent to a groupoid stack [V U] where U, V are separated C-schemes,s: V U is etale, ands t: V U U is universally closed.

(e) A C-stack X is separated, DeligneMumford and locally fair (or locallyfinitely presented) if and only if it is equivalent to some [V U] with U, Vseparated, locally fair (or locally finitely presented) C-schemes, s : V Uetale, and s t: V U U proper.

A C-stack Xhas an underlying topological spaceXtop.

Definition 4.9. Let Xbe aC-stack. Write for the point Spec R inCSch,

and for the associated point in C

Sta. Define Xtop to be the set of 2-isomorphism classes [x] of 1-morphisms x : X. If U X is an openC-substack inX, thenUtop Xtop. DefineTXtop =

Utop : U Xis an open

C-substack in X

. Then (Xtop, TXtop) is a topological space, which we callthe underlying topological space ofX, and usually write asXtop.

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Iff :X Yis a 1-morphism ofC-stacks then there is a natural continuousmap ftop : Xtop Ytop defined by ftop([x]) = [f x]. Iff, g : X Yare 1-

morphisms and : f g is a 2-isomorphism then ftop = gtop. MappingX Xtop,f ftop and 2-morphisms to identities defines a 2-functorF

TopCSta :

CSta Top, where the category of topological spaces Topis regarded as a2-category with only identity 2-morphisms.

If X = (X, OX) is a C-scheme, so that X is a C-stack, then Xtop isnaturally homeomorphic toX, and we will identify Xtop withX. Iff= (f, f

) :

X= (X, OX) Y = (Y, OY) is a morphism ofC-schemes, so that f : X Y

is a 1-morphism ofC-stacks, then ftop : Xtop Ytop is f :XY.We call a DeligneMumfordC-stack Xparacompactif the underlying topo-

logical spaceXtop is paracompact.

Definition 4.10. LetX be a C-stack, and [x] Xtop. Pick a representativex for [x], so that x : X is a 1-morphism. Let G be the group of 2-

morphisms : x x. There is a natural C-scheme G = (G, OG) withG = x,X,x, which makes G into a C-group (a group object in CSch,

just as a Lie group is a group object in Man). With [x] fixed, thisC-group G isindependent of choices up to noncanonical isomorphism; roughly, G is canonicalup to conjugation in G. We define the isotropy group (or stabilizer group, ororbifold group) IsoX([x]) of [x] to be thisC-groupG, regarded as aC-groupup to noncanonical isomorphism.

Iff :X Yis a 1-morphism ofC-stacks and [x] Xtop with ftop([x]) =[y] Ytop, for y = f x, then we define f : IsoX([x]) IsoY([y]) byf() = idf . Then f is a group morphism, and extends to a C-groupmorphism. It is independent of choices ofx [x], y [y] up to conjugationin IsoX([x]), IsoY([y]).

IfX is a DeligneMumford C

-stack then IsoX([x]) is a finite group for all[x] inXtop, which is discrete as aC-group. Here are[7, Th.s 9.17(a) & 9.19].

Proposition 4.11. Let Xbe a DeligneMumfordC-stack and[x] Xtop, sothat IsoX([x]) = H for some finite group H. Then there exists an open C-substack U inX with [x] Utop Xtop and an equivalence U [Y /H], whereY = (Y, OY) is an affineC-scheme with an action of H, and [x] Utop =Y/Hcorresponds to a fixed point y of H in Y.

Theorem 4.12. SupposeX is a DeligneMumford C-stack with IsoX([x])={1} for all [x] Xtop. Then X is equivalent to X for some C-scheme X.

In conventional algebraic geometry, a stack with all stabilizer groups trivialis (equivalent to) an algebraic space, but may not be a scheme, so the categoryof algebraic spaces is larger than the category of schemes. Here algebraic spaces

are spaces which are locally isomorphic to schemes in the etale topology, butnot necessarily locally isomorphic to schemes in the Zariski topology.

In contrast, as Theorem4.12shows, in C-algebraic geometry there is nodifference between C-schemes and C-algebraic spaces. This is because inC-geometry the Zariski topology is already fine enough, as in Remark3.7(iii),so we gain no extra generality by passing to the etale topology.

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4.2 Orbifolds as DeligneMumford C-stacks

Example3.3defined a functorFCSch

Man :Man C

Schembedding manifoldsas a full subcategory ofC-schemes. Similarly, one might expect to define a(2)-functor FDMC

StaOrb : Orb DMC

Sta embedding the (2-)category oforbifolds as a full (2-)subcategory of DeligneMumford C-stacks. In fact, in[7,9.6] we took a slightly different approach: we defined a full 2-subcategoryOrbin DMCSta, and then showed this is equivalent to other definitions of the(2-)category of orbifolds. The reason for this is that there is not one definitionof orbifolds, but several, and our new definition of orbifolds as examples ofC-stacks may be as useful as some of the other definitions.

Orbifolds (without boundary) are spaces locally modelled on Rn/G for G afinite group acting linearly on Rn, just as manifolds are spaces locally modelledon Rn. They were introduced by Satake [17], who called them V-manifolds.Moerdijk [14] defines orbifolds as proper etale Lie groupoids in Man. Both

[14, 17] regard orbifolds as an ordinary category, if the question arises at all.However, for issues such as fibre products or pullbacks of vector bundles thiscategory is badly behaved, and it becomes clear that orbifolds should really bea 2-category, as for stacks in algebraic geometry.

There are two main routes in the literature to defining a 2-category of orb-ifolds Orb. The first, as in Pronk [16] and Lerman [12, 3.3], is to define a2-categoryGpoid of proper etale Lie groupoids, and then to define Orb as a(weak) 2-category localization ofGpoidat a suitable class of 1-morphisms. Thesecond, as in Behrend and Xu[1, 2], Lerman [12, 4] and Metzler[13, 3.5], is todefine orbifolds as a class of DeligneMumford stacks on the site (Man, JMan)of manifolds with Grothendieck topology JMan coming from open covers. Ourapproach is similar to the second route, but defines orbifolds as a class ofC-stacks, that is, as stacks on the site (CSch, J) rather than on (Man, JMan).

Definition 4.13. A C-stack X is called an orbifold if it is equivalent to agroupoid stack [V U] for some groupoid (U , V , s , t , u , i , m) in CSchwhichis the image under FC

SchMan of a groupoid (U,V,s ,t ,u,i,m) in Man, where

s: V Uis an etale smooth map, and st: V UUis a proper smooth map.That is, X is the C-stack associated to a proper etale Lie groupoid in Man.As a C-stack, every orbifold X is a separable, paracompact, locally finitelypresented DeligneMumfordC-stack. WriteOrb for the full 2-subcategory oforbifolds inCSta.

Here is [7, Th. 9.24 & Cor. 9.25]. Since equivalent (2-)categories are con-sidered to be the same, the moral of Theorem 4.14 is that our orbifolds areessentially the same objects as those considered by other recent authors.

Theorem 4.14. The 2-category Orb of orbifolds defined above is equivalentto the2-categories of orbifolds considered as stacks onMan defined in Metzler[13, 3.4] and Lerman [12, 4], and also equivalent as a weak 2-category to theweak 2-categories of orbifolds regarded as proper etale Lie groupoids defined inPronk [16] and Lerman [12, 3.3].

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(d) Suppose W, X, Y, Z are locally fair DeligneMumford C-stacks with a2-Cartesian square

Wf

e

Yh

X g Z

inDMCStalf,so thatW= X ZY. Then the following is exact inqcoh(W) :

(g e)(TZ)

e(g)Ie,g(TZ)

f(h)If,h(TZ)(TZ)

e(TX)f(TY)

efTW 0.

4.4 Orbifold strata of DeligneMumford C-stacks

Let X be a DeligneMumford C-stack, with topological space Xtop. Theneach point [x] Xtop has an orbifold group IsoX([x]), a finite group defined

up to isomorphism. For each finite group we write X,top =

[x] Xtop :

IsoX([x]) =

. This is a locally closed subset ofXtop, coming from a locally

closed C-substack X ofXwith inclusionO(X) :

X X, with

Xtop =

isomorphism classesof finite groups

X,top. (9)

One can show that for each , the closure X ,top ofX,top in Xtop satisfies

X,top

isomorphism classes of finite groups : is isomorphic to a subgroup of

X,top.

Thus (9) is a stratification ofXtop, and the X are called orbifold strataofX.In[7, 11.1] we define six variations of this idea, DeligneMumford C-stacks

writtenX, X, X, and open C-substacksX X, X X

, X X.

The geometric points and orbifold groups ofX, . . . , X are given by:

(i) Points ofX are isomorphism classes [x, ], where [x] Xtop and : IsoX([x]) is an injective morphism, and IsoX([x, ]) is the centralizer of() in IsoX([x]). Points ofX

X

are [x, ] with an isomorphism,and IsoX ([x, ])

=C(), the centre of .

(ii) Points ofX are pairs [x, ], where [x] Xtop and IsoX([x]) isisomorphic to , and IsoX([x, ]) is the normalizer of in IsoX([x]).

Points ofX X are [x, ] with = IsoX([x]), and Iso X ([x, ])=.(iii) Points [x, ] ofX, X are the same as for X

, X , but with orbifoldgroups Iso X([x, ])=Iso X([x, ])/ and IsoX ([x, ])

={1}.

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There are 1-morphismsO(X), . . . , (X) forming a strictly commutative dia-gram, where the columns are inclusions of open C-substacks:

X (X)

O(X)

Aut() X

(X)

O (X)

X X

X

X(X)

O(X)

Aut() X(X)

O(X)

X.

(10)

HereO(X), O(X),(X) are proper and representable, and (X) is proper.Also Aut() acts onX, X , with X

[X/ Aut()] and X [X/ Aut()].

Note that there are in general no natural 1-morphisms from X, X to anyof X, X, X, X

, X . Although X or

X correspond most closely to theusual idea of orbifold stratum, we will find that X and X are most useful in

applications to orbifold and d-orbifold (co)bordism in [8, Ch. 13], in which it isvital thatO(X) : X X and O(X) : X Xare proper.

As in [7, 11.2], representable 1-morphisms and their 2-morphisms lift toorbifold strata X, X, X. That is, if f : X Y is representable then wehave natural representable 1-morphisms f : X Y, f : X X, f :X X, and if : f g is a 2-morphism of representable f, g : X Ythen we have natural 2-morphisms :f g , : f g, : f g.These behave in a strongly functorial way for orbifold strata X, X, so that(g f) =g f, and so on, and in a weakly functorial way for orbifold strata

X, so that (g f) is 2-isomorphic to g f, and so on.We can describe orbifold strata of quotient C-stacks,[7, Th. 11.9]:

Theorem 4.24. Let Xbe a separated C-scheme and G a finite group acting

onXby isomorphisms, and writeX= [X/G] for the quotientC-stack, whichis a DeligneMumford C-stack. Let be a finite group. Then there areequivalences of C-stacks

X

injective group morphisms : G X()

/G

, (11)

X

injective group morphisms : G X()

/G

, (12)

X

subgroups G: =X

/G

, (13)

X

subgroups G: =X

/G

, (14)

X

conjugacy classes [] of subgroups G with =

X

{g G : = gg1}/

, (15)

X

conjugacy classes [] of subgroups G with =X {g G : = gg1}/. (16)

Here for each subgroup G, we write X for the closed C-subscheme inXfixed by in G, and X for the openC

-subscheme inX of points inXwhose stabilizer group inG is exactly .

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Under the equivalences (11)(16), the 1-morphisms in (10) are identifiedup to2-isomorphism with 1-morphisms between quotient C-stacks induced by

naturalC-scheme morphisms between X(), X , . . . .For example, the dis-joint union over [] of the inclusion X() X is a G-equivariant morphism

X() X , inducing a1-morphism [

X

()/G] [X/G]. This is iden-

tified with O(X) :X X by (11).

In[7,11.411.6] we study pullbacks of sheaves to orbifold strata. We findthat ifElies in qcoh(X) then the pullback E :=O(X)(E) to X has a naturalaction of , so we have a splitting E =Etr E

ntinto subsheaves with trivial tr

and nontrivial nt -representations. For cotangent sheaves we have a naturalisomorphismT(X)=(TX)tr. A similar picture holds for sheaves on orbifoldstrata X, X.

References

[1] K. Behrend and P. Xu, Differentiable stacks and gerbes, J. SymplecticGeom. 9 (2011), 285341.math.DG/0605694.

[2] E.J. Dubuc,Sur les modeles de la geometrie differentielle synthetique, Cah.Topol. Geom. Differ. Categ. 20 (1979), 231279.

[3] E.J. Dubuc, C-schemes, Amer. J. Math. 103 (1981), 683690.

[4] E.J. Dubuc, Open covers and infinitary operations in C-rings, Cah.Topol. Geom. Differ. Categ. 22 (1981), 287300.

[5] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono,Lagrangian intersection Floer

theory anomaly and obstruction, Parts I & II. AMS/IP Studies in Ad-vanced Mathematics, 46.1 & 46.2, A.M.S./International Press, 2009.

[6] R. Hartshorne,Algebraic Geometry, Graduate Texts in Math. 52, Springer,New York, 1977.

[7] D. Joyce, Algebraic Geometry over C-rings,arXiv:1001.0023,2010.

[8] D. Joyce, D-manifolds and d-orbifolds: a theory of derived differentialgeometry, book in preparation, 2012. Preliminary version available athttp://people.maths.ox.ac.uk/joyce/.

[9] D. Joyce,An introduction to d-manifolds and derived differential geometry,arXiv:1206.4207,2012.

[10] D. Joyce,D-manifolds, d-orbifolds and derived differential geometry: a de-tailed summary,arXiv:1208.4948, 2012.

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The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, U.K.

E-mail: joyce@maths.ox.ac.uk

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