diff --git "a/1001.0023.txt" "b/1001.0023.txt" new file mode 100644--- /dev/null +++ "b/1001.0023.txt" @@ -0,0 +1,7151 @@ +arXiv:1001.0023v7 [math.AG] 1 Nov 2016Algebraic Geometry over C∞-rings +Dominic Joyce +Abstract +IfXis a manifold then the R-algebra C∞(X) of smooth functions +c:X→Ris aC∞-ring. That is, for each smooth function f:Rn→R +there is an n-fold operation Φ f:C∞(X)n→C∞(X) acting by Φ f: +(c1,...,c n)/mapsto→f(c1,...,c n), and these operations Φ fsatisfy many natural +identities. Thus, C∞(X) actually has a far richer structure than the +obviousR-algebra structure. +We explain the foundations of a version of algebraic geometr y in which +rings or algebras are replaced by C∞-rings. As schemes are the basic +objects in algebraic geometry, the new basic objects are C∞-schemes, a +category of geometric objects which generalize manifolds, and whose mor- +phisms generalize smooth maps. We also study quasicoherent sheaves on +C∞-schemes, and C∞-stacks, in particular Deligne–Mumford C∞-stacks, +a 2-category of geometric objects generalizing orbifolds. +Many of these ideas are not new: C∞-rings and C∞-schemes have long +been part of synthetic differential geometry. But we develop them in new +directions. In [36–38], the author uses these tools to define d-manifolds +andd-orbifolds , ‘derived’ versions of manifolds and orbifolds related to +Spivak’s ‘derived manifolds’ [64]. +Contents +1 Introduction 3 +2C∞-rings 5 +2.1 Two definitions of C∞-ring . . . . . . . . . . . . . . . . . . . . . 6 +2.2C∞-rings as commutative R-algebras, and ideals . . . . . . . . . 7 +2.3 Local C∞-rings, and localization . . . . . . . . . . . . . . . . . . 9 +2.4 FairC∞-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 +2.5 Pushouts of C∞-rings . . . . . . . . . . . . . . . . . . . . . . . . 15 +2.6 Flat ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 +3 TheC∞-ringC∞(X)of a manifold X 17 +4C∞-ringed spaces and C∞-schemes 20 +4.1 Some basic topology . . . . . . . . . . . . . . . . . . . . . . . . . 20 +4.2 Sheaves on topological spaces . . . . . . . . . . . . . . . . . . . . 21 +4.3C∞-ringed spaces and local C∞-ringed spaces . . . . . . . . . . . 24 +14.4 The spectrum functor . . . . . . . . . . . . . . . . . . . . . . . . 26 +4.5 Affine C∞-schemes and C∞-schemes . . . . . . . . . . . . . . . . 31 +4.6 Complete C∞-rings . . . . . . . . . . . . . . . . . . . . . . . . . . 34 +4.7 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . 37 +4.8 A criterion for affine C∞-schemes . . . . . . . . . . . . . . . . . . 39 +4.9 Quotients of C∞-schemes by finite groups . . . . . . . . . . . . . 42 +5 Modules over C∞-rings and C∞-schemes 44 +5.1 Modules over C∞-rings . . . . . . . . . . . . . . . . . . . . . . . 44 +5.2 Cotangent modules of C∞-rings . . . . . . . . . . . . . . . . . . . 45 +5.3 Sheaves ofOX-modules on a C∞-ringed space ( X,OX) . . . . . 50 +5.4 Sheaves on affine C∞-schemes, MSpec and Γ . . . . . . . . . . . 51 +5.5 Complete modules over C∞-rings . . . . . . . . . . . . . . . . . . 56 +5.6 Cotangent sheaves of C∞-schemes . . . . . . . . . . . . . . . . . 58 +6C∞-stacks 61 +6.1C∞-stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 +6.2 Properties of 1-morphisms of C∞-stacks . . . . . . . . . . . . . . 64 +6.3 Open C∞-substacks and open covers . . . . . . . . . . . . . . . . 66 +6.4 The underlying topological space of a C∞-stack . . . . . . . . . . 67 +6.5 Gluing C∞-stacks by equivalences . . . . . . . . . . . . . . . . . 70 +7 Deligne–Mumford C∞-stacks 71 +7.1 Quotient C∞-stacks, 1-morphisms, and 2-morphisms . . . . . . . 71 +7.2 Deligne–Mumford C∞-stacks . . . . . . . . . . . . . . . . . . . . 73 +7.3 Characterizing Deligne–Mumford C∞-stacks . . . . . . . . . . . . 76 +7.4 Quotient C∞-stacks, 1- and 2-morphisms as local models for +objects, 1- and 2-morphisms in DMC∞Sta. . . . . . . . . . . . 80 +7.5 Effective Deligne–Mumford C∞-stacks . . . . . . . . . . . . . . . 86 +7.6 Orbifolds as Deligne–Mumford C∞-stacks . . . . . . . . . . . . . 87 +8 Sheaves on Deligne–Mumford C∞-stacks 89 +8.1 Quasicoherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . 89 +8.2 Writing sheaves in terms of a groupoid presentation . . . . . . . 92 +8.3 Pullback of sheaves as a weak 2-functor . . . . . . . . . . . . . . 93 +8.4 Cotangent sheaves of Deligne–Mumford C∞-stacks . . . . . . . . 96 +9 Orbifold strata of C∞-stacks 99 +9.1 The definition of orbifold strata XΓ,...,ˆXΓ +◦. . . . . . . . . . . . 100 +9.2 Lifting 1- and 2-morphisms to orbifold strata . . . . . . . . . . . 108 +9.3 Orbifold strata of quotient C∞-stacks [X/G] . . . . . . . . . . . 109 +9.4 Sheaves on orbifold strata . . . . . . . . . . . . . . . . . . . . . . 111 +9.5 Sheaves on orbifold strata of quotients [ X/G] . . . . . . . . . . . 114 +9.6 Cotangent sheaves of orbifold strata . . . . . . . . . . . . . . . . 116 +2A Background material on stacks 118 +A.1 Introduction to 2-categories . . . . . . . . . . . . . . . . . . . . . 118 +A.2 Grothendieck topologies, sites, prestacks, and stacks . . . . . . . 122 +A.3 Descent theory on a site . . . . . . . . . . . . . . . . . . . . . . . 125 +A.4 Properties of 1-morphisms . . . . . . . . . . . . . . . . . . . . . . 126 +A.5 Geometric stacks, and stacks associated to groupoids . . . . . . . 128 +References 132 +Glossary of Notation 137 +Index 140 +1 Introduction +LetXbe a smooth manifold, and write C∞(X) for the set of smooth functions +c:X→R. ThenC∞(X) is a commutative R-algebra, with operations of +addition, multiplication, and scalar multiplication defined pointwise. How ever, +C∞(X) has much more structure than this. For example, if c:X→Ris +smooth then exp( c) :X→Ris smooth, and this defines an operation exp : +C∞(X)→C∞(X) which cannot be expressed algebraically in terms of the R- +algebra structure. More generally, if n/greaterorequalslant0 andf:Rn→Ris smooth, define +ann-fold operation Φ f:C∞(X)n→C∞(X) by +/parenleftbig +Φf(c1,...,cn)/parenrightbig +(x) =f/parenleftbig +c1(x),...,cn(x)/parenrightbig +, +for allc1,...,cn∈C∞(X) andx∈X. These operations satisfy many identities: +supposem,n/greaterorequalslant0, andfi:Rn→Rfori= 1,...,mandg:Rm→Rare smooth +functions. Define a smooth function h:Rn→Rby +h(x1,...,xn) =g/parenleftbig +f1(x1,...,xn),...,fm(x1...,xn)/parenrightbig +, +for all (x1,...,xn)∈Rn. Then for all c1,...,cn∈C∞(X) we have +Φh(c1,...,cn) = Φg/parenleftbig +Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig +.(1.1) +AC∞-ring/parenleftbig +C,(Φf)f:Rn→RC∞/parenrightbig +is a setCwith operations Φ f:Cn→Cfor +allf:Rn→Rsmooth satisfying identities (1.1), and one other condition. For +exampleC∞(X) is aC∞-ring for any manifold X, but there are also many C∞- +rings which do not come from manifolds, and can be thought of as rep resenting +geometric objects which generalize manifolds. +The most basic objects in conventional algebraic geometry are com mutative +ringsR, or commutative K-algebrasRfor some field K. The ‘spectrum’ Spec R +ofRis an affine scheme, and Ris interpreted as an algebra of functions on +SpecR. More general kinds of spaces in algebraic geometry — schemes and +stacks — are locally modelled on affine schemes Spec R. This book lays down +the foundations of Algebraic Geometry over C∞-rings, in which we replace +3commutative rings in algebraic geometry by C∞-rings. It includes the study of +C∞-schemes andDeligne–Mumford C∞-stacks, two classes of geometric spaces +generalizing manifolds and orbifolds, respectively. +This is not a new idea, but was studied years ago as part of synthetic dif- +ferential geometry , which grew out of ideas of Lawvere in the 1960s; see for +instance Dubuc [23] on C∞-schemes, and the books by Moerdijk and Reyes [54] +and Kock [44]. However, we have new things to say, as we are motivat ed by +different problems (see below), and so are asking different question s. +Following Dubuc’s discussion of ‘models of synthetic differential geome try’ +[21] and oversimplifying a bit, synthetic differential geometers are in terested +inC∞-schemes as they provide a category C∞Schof geometric objects which +includes smooth manifolds and certain ‘infinitesimal’ objects, and all fi bre prod- +ucts exist in C∞Sch, andC∞Schhas some other nice properties to do with +open covers, and exponentials of infinitesimals. +Synthetic differential geometry concerns proving theorems abou t manifolds +usingsyntheticreasoninginvolving‘infinitesimals’. But oneneedstoch eckthese +methods of synthetic reasoning are valid. To do this you need a ‘mode l’, some +category of geometric spaces including manifolds and infinitesimals, in which +you can think of your synthetic arguments as happening. Once you know there +exists at least one model with the properties you want, then as far as synthetic +differential geometry is concerned the job is done. For this reason C∞-schemes +have not been developed very far in synthetic differential geometr y. +Recently,C∞-rings andC∞-ringed spaces appeared in a very different con- +text, in the theory of derived differential geometry , the differential-geometric +analogue of the derived algebraic geometry of Lurie [48] and To¨ en– Vezzosi +[66,67], which studies derived smooth manifolds andderived smooth orbifolds . +This began with a short section in Lurie [48, §4.5], where he sketched how to +define an∞-category of derivedC∞-schemes , including derived manifolds. +Lurie’s student David Spivak [64] worked out the details of this, defi ning +an∞-category of derived manifolds. Simplifications and extensions of Sp ivak’s +theory were given by Borisov and Noel [9,10] and the author [36–38 ]. An al- +ternative approach to the foundations of derived differential geo metry involving +differential graded C∞-rings is proposed by Carchedi and Roytenberg [12,13]. +The author’s notion of derived manifolds [36–38] are called d-manifolds , and +are built using our theory of C∞-schemes and quasicoherent sheaves upon them +below. They form a 2-category. We also study orbifold versions, d-orbifolds , +which are built using our theory of Deligne–Mumford C∞-stacks and their qua- +sicoherent sheaves below. +Many areas of symplectic geometry involve studying moduli spaces o fJ- +holomorphic curves in a symplectic manifold, which are made into Kuranishi +spacesin the framework of Fukaya, Oh, Ohta and Ono [26,27]. The author +argues that Kuranishi spaces are really derived orbifolds , and has given a new +definition [39,41] of a 2-category of Kuranishi spaces Kurwhich is equivalent +to the 2-category of d-orbifolds dOrbfrom [36–38]. Because of this, derived +differential geometry will have important applications in symplectic ge ometry. +To set up our theory of d-manifolds and d-orbifolds requires a lot of pre- +4liminary work on C∞-schemes and C∞-stacks, and quasicoherent sheaves upon +them. That is the purpose of this book. We have tried to present a c om- +plete, self-contained account which should be understandable to r eaders with +a reasonable background in algebraic geometry, and we assume no f amiliarity +with synthetic differential geometry. We expect this material may h ave other +applications quite different to those the author has in mind in [36–38]. +Section 2 explains C∞-rings. The archetypal examples of C∞-rings,C∞(X) +for manifolds X, are discussed in §3. Section 4 studies C∞-schemes, and§5 +modules over C∞-rings and sheaves of modules over C∞-schemes. +Sections 6–9 discuss C∞-stacks. Section 6 defines the 2-category C∞Sta +ofC∞-stacks, analogues of Artin stacks in algebraic geometry, and §7 the 2- +subcategory DMC∞StaofDeligne–Mumford C∞-stacks, which are C∞-stacks +locallymodelled on[ U/G]forUanaffineC∞-schemeand Gafinitegroupacting +onU, and are analogues of Deligne–Mumford stacks in algebraic geometr y. We +show that orbifolds Orbmay be regarded as a 2-subcategory of DMC∞Sta. +Section 8 studies quasicoherent sheaves on Deligne–Mumford C∞-stacks, gen- +eralizing§5, and§9 orbifold strata of Deligne–Mumford C∞-stacks. +Appendix Asummarizesbackgroundonstacksfrom[3,4,29,46,49 ,55], foruse +in§6–§9. Stacks are a very technical area, and §A is too terse to help a beginner +learn the subject, it is intended only to establish notation and definit ions for +those already familiar with stacks. Readers with no experience of st acks are +advised to first consult an introductory text such as Vistoli [68], G omez [29], +Laumon and Moret-Bailly [46], or the online ‘Stacks Project’ [34]. +Much of§2–§4 is already understood in synthetic differential geometry, such +as in the work of Dubuc [23] and Moerdijk and Reyes [54]. But we believ e it is +worthwhile giving a detailed and self-contained exposition, from our o wn point +of view. Sections 5–9 are new, so far as the author knows, though §5–§8 are +based on well known material in algebraic geometry. +Acknowledgements. I would like to thank Omar Antolin, Eduardo Dubuc, Kelli +Francis-Staite, Jacob Gross, Jacob Lurie, and Ieke Moerdijk for helpful conver- +sations, and a referee for many useful comments. This research was supported +by EPSRC grants EP/H035303/1 and EP/J016950/1. +2C∞-rings +We begin by explaining the basic objects out of which our theories are built, +C∞-rings, orsmooth rings . The archetypal example of a C∞-ring is the vector +spaceC∞(X) of smooth functions c:X→Rfor a manifold X. Everything in +thissectionis knowntoexpertsin syntheticdifferentialgeometry, andmuchofit +canbe found in Moerdijkand Reyes[54, Ch. I], Dubuc [21–24] orKock [44,§III]. +We introducesomenew notationfor brevity, in particular, our fairC∞-ringsare +known in the literature as ‘finitely generated and germ determined C∞-rings’. +52.1 Two definitions of C∞-ring +We first define C∞-rings in the style of classical algebra. +Definition 2.1. AC∞-ringis a setCtogether with operations +Φf:Cn=/rightanglenwncopies/rightanglene +C×···×C−→C +for alln/greaterorequalslant0 and smooth maps f:Rn→R, where by convention when n= 0 we +defineC0to be the single point {∅}. These operations must satisfy the following +relations: suppose m,n/greaterorequalslant0, andfi:Rn→Rfori= 1,...,mandg:Rm→R +are smooth functions. Define a smooth function h:Rn→Rby +h(x1,...,xn) =g/parenleftbig +f1(x1,...,xn),...,fm(x1...,xn)/parenrightbig +, +for all (x1,...,xn)∈Rn. Then for all ( c1,...,cn)∈Cnwe have +Φh(c1,...,cn) = Φg/parenleftbig +Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig +. +We also require that for all 1 /lessorequalslantj/lessorequalslantn, definingπj:Rn→Rbyπj: +(x1,...,xn)/ma√sto→xj, we have Φ πj(c1,...,cn) =cjfor all (c1,...,cn)∈Cn. +Usually we refer to Cas theC∞-ring, leaving the operations Φ fimplicit. +Amorphism betweenC∞-rings/parenleftbig +C,(Φf)f:Rn→RC∞/parenrightbig +,/parenleftbig +D,(Ψf)f:Rn→RC∞/parenrightbig +is a mapφ:C→Dsuch that Ψ f/parenleftbig +φ(c1),...,φ(cn)/parenrightbig +=φ◦Φf(c1,...,cn) for +all smooth f:Rn→Randc1,...,cn∈C. We will write C∞Ringsfor the +category of C∞-rings. +Here is the motivating example, which we will study at greater length in §3: +Example 2.2. LetXbe a manifold, which may be without boundary, or with +boundary, or with corners. Write C∞(X) for the set of smooth functions c: +X→R. Forn/greaterorequalslant0 andf:Rn→Rsmooth, define Φ f:C∞(X)n→C∞(X) by +/parenleftbig +Φf(c1,...,cn)/parenrightbig +(x) =f/parenleftbig +c1(x),...,cn(x)/parenrightbig +, (2.1) +for allc1,...,cn∈C∞(X) andx∈X. It is easy to see that C∞(X) and the +operations Φ fform aC∞-ring. +Example 2.3. TakeXto be the point∗in Example 2.2. Then C∞(∗) =R, +with operations Φ f:Rn→Rgiven by Φ f(x1,...,xn) =f(x1,...,xn). This +makesRinto the simplest nonzero example of a C∞-ring, the initial object +inC∞Rings. +Note thatC∞-rings are far more general than those coming from manifolds. +For example, if Xis any topological space we could define a C∞-ringC0(X) to +be the set of continuous c:X→Rwith operations Φ fdefined as in (2.1). For +Xa manifold with dim X >0, theC∞-ringsC∞(X) andC0(X) are different. +There is a more succinct definition of C∞-rings using category theory: +6Definition 2.4. WriteManfor the category of manifolds, and Eucfor the full +subcategory of Manwith objects the Euclidean spaces Rn. That is, the objects +ofEucareRnforn= 0,1,2,...,and the morphisms in Eucare smooth maps +f:Rm→Rn. Write Setsfor the category of sets. In both EucandSets +we have notions of (finite) products of objects (that is, Rm+n=Rm×Rn, and +productsS×Tof setsS,T), and products of morphisms. +Define a ( category-theoretic )C∞-ringto be a product-preserving functor +F:Euc→Sets. HereFshould also preserve the empty product, that is, it +mapsR0inEucto the terminal object in Sets, the point∗. +C∞-rings in this sense are an example of an algebraic theory in the sense of +Ad´ amek, Rosick´ y and Vitale [1], and many of the basic categorical p roperties +ofC∞-rings follow from this. +Here is how this relates to Definition 2.1. Suppose F:Euc→Setsis a +product-preserving functor. Define C=F(R). Then Cis an object in Sets, +that is, a set. Suppose n/greaterorequalslant0 andf:Rn→Ris smooth. Then fis a morphism +inEuc, soF(f) :F(Rn)→F(R) =Cis a morphism in Sets. SinceFpreserves +productsF(Rn) =F(R)×···×F(R) =Cn, soF(f) mapsCn→C. We define +Φf:Cn→Cby Φf=F(f). The fact that Fis a functor implies that the Φ f +satisfy the relations in Definition 2.1, so/parenleftbig +C,(Φf)f:Rn→RC∞/parenrightbig +is aC∞ring. +Conversely, if/parenleftbig +C,(Φf)f:Rn→RC∞/parenrightbig +is aC∞-ring then we define F:Euc→ +SetsbyF(Rn) =Cn, and iff:Rn→Rmis smooth then f= (f1,...,fm) for +fi:Rn→Rsmooth, and we define F(f) :Cn→CmbyF(f) : (c1,...,cn)/ma√sto→/parenleftbig +Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig +. ThenFis a product-preserving functor. +This defines a 1-1 correspondence between C∞-rings in the sense of Definition +2.1, and category-theoretic C∞-rings in the sense of Definition 2.4. +As in Moerdijk and Reyes [54, p. 21–22] we have: +Proposition 2.5. In the category C∞RingsofC∞-rings, all limits and all +filtered colimits exist, and regarding C∞-rings as functors F:Euc→Sets +as in Definition 2.4,they may be computed objectwise in Eucby taking the +corresponding limits/filtered colimits in Sets. +Also, all small colimits exist, though in general they are no t computed ob- +jectwise in Eucby taking colimits in Sets. In particular, pushouts and all finite +colimits exist in C∞Rings. +We will write D∐φ,C,ψEorD∐CEfor the pushout of morphisms φ:C→D, +ψ:C→EinC∞Rings. WhenC=R, the initial object in C∞Rings, pushouts +D∐REare called coproducts and are usually written D⊗∞E. ForR-algebras +A,Bthe coproduct is the tensor product A⊗B. But the coproduct D⊗∞Eof +C∞-ringsD,Eis generally different from their coproduct D⊗EasR-algebras. +For example we have C∞(Rm)⊗∞C∞(Rn)∼=C∞(Rm+n), which contains but +is much larger than the tensor product C∞(Rm)⊗C∞(Rn) form,n>0. +2.2C∞-rings as commutative R-algebras, and ideals +EveryC∞-ringChas an underlying commutative R-algebra: +7Definition 2.6. LetCbe aC∞-ring. Then we may give Cthe structure of +acommutative R-algebra. Define addition ‘+’ on Cbyc+c′= Φf(c,c′) for +c,c′∈C, wheref:R2→Risf(x,y) =x+y. Define multiplication ‘ ·’ onCby +c·c′= Φg(c,c′), whereg:R2→Risf(x,y) =xy. Define scalar multiplication +byλ∈Rbyλc= Φλ′(c), whereλ′:R→Risλ′(x) =λx. Define elements 0 +and 1 in Cby 0 = Φ 0′(∅) and 1 = Φ 1′(∅), where 0′:R0→Rand 1′:R0→R +are the maps 0′:∅/ma√sto→0 and 1′:∅/ma√sto→1. The relations on the Φ fimply that +all the axioms of a commutative R-algebra are satisfied. In Example 2.2, this +yields the obvious R-algebra structure on the smooth functions c:X→R. +Here is another way to say this. In an R-algebraA, then-fold ‘operations’ +Φ :An→A, that is, all the maps An→Awe can construct using only addition, +multiplication, scalar multiplication, and the elements 0 ,1∈A, correspond ex- +actly to polynomials p:Rn→R. Since polynomials are smooth, the operations +of anR-algebra are a subset of those of a C∞-ring, and we can truncate from +C∞-rings to R-algebras. As there are many more smooth functions f:Rn→R +than there are polynomials, a C∞-ring has far more structure and operations +than a commutative R-algebra. +Definition 2.7. AnidealIinCis an idealI⊂CinCregarded as a commu- +tativeR-algebra. Then we make the quotient C/Iinto aC∞-ring as follows. If +f:Rn→Ris smooth, define ΦI +f: (C/I)n→C/Iby +ΦI +f(c1+I,...,cn+I) = Φf(c1,...,cn)+I. +To show this is well-defined, we must show it is independent of the choic e of +representatives c1,...,cninCforc1+I,...,cn+IinC/I. By Hadamard’s +Lemma there exist smooth functions gi:R2n→Rfori= 1,...,nwith +f(y1,...,yn)−f(x1,...,xn) =/summationtextn +i=1(yi−xi)gi(x1,...,xn,y1,...,yn) +for allx1,...,xn,y1,...,yn∈R. Ifc′ +1,...,c′ +nare alternative choices for c1,..., +cn, so thatc′ +i+I=ci+Ifori= 1,...,nandc′ +i−ci∈I, we have +Φf(c′ +1,...,c′ +n)−Φf(c1,...,cn) =/summationtextn +i=1(c′ +i−ci)Φgi(c′ +1,...,c′ +n,c1,...,cn). +The second line lies in Iasc′ +i−ci∈IandIis an ideal, so ΦI +fis well-defined, +and clearly/parenleftbig +C/I,(ΦI +f)f:Rn→RC∞/parenrightbig +is aC∞-ring. +IfCis aC∞-ring, we will use the notation ( fa:a∈A) to denote the +ideal inCgenerated by a collection of elements fa,a∈AinC, in the sense of +commutative R-algebras. That is, +(fa:a∈A) =/braceleftbig/summationtextn +i=1fai·ci:n/greaterorequalslant0,a1,...,an∈A,c1,...,cn∈C/bracerightbig +. +Definition 2.8. AC∞-ringCis calledfinitely generated if there exist c1,...,cn +inCwhich generate Cover allC∞-operations. That is, for each c∈Cthere +exists a smooth map f:Rn→Rwithc= Φf(c1,...,cn). (This is a much +weaker condition than Cbeing finitely generated as a commutative R-algebra.) +8By Kock [44, Prop. III.5.1], C∞(Rn) is the free C∞-ring withngenerators. +Given such C,c1,...,cn, defineφ:C∞(Rn)→Cbyφ(f) = Φf(c1,...,cn) for +smoothf:Rn→R, whereC∞(Rn) is as in Example 2.2 with X=Rn. Then +φis a surjective morphism of C∞-rings, soI= Kerφis an ideal in C∞(Rn), +andC∼=C∞(Rn)/Ias aC∞-ring. Thus, Cis finitely generated if and only if +C∼=C∞(Rn)/Ifor somen/greaterorequalslant0 and ideal IinC∞(Rn). +AnidealIinaC∞-ringCiscalledfinitelygenerated ifIisafinitelygenerated +ideal of the underlying commutative R-algebra of Cin Definition 2.6, that is, +I= (i1,...,ik)forsomei1,...,ik∈C. AC∞-ringCiscalledfinitelypresented if +C∼=C∞(Rn)/Ifor somen/greaterorequalslant0, 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 Cfinitely +generated does not imply Cfinitely presented. +WriteC∞RingsfgandC∞Ringsfpfor the full subcategories of finitely +generated and finitely presented C∞-rings in C∞Rings. +Example 2.9. AWeil algebra [21, Def. 1.4]is afinite-dimensional commutative +R-algebraWwhich has a maximal ideal mwithW/m∼=Randmn= 0 for some +n >0. Then by Dubuc [21, Prop. 1.5] or Kock [44, Th. III.5.3], there is a +unique way to make Winto aC∞-ring compatible with the given underlying +commutative R-algebra. This C∞-ring is finitely presented [44, Prop. III.5.11]. +C∞-rings from Weil algebras are important in synthetic differential geo metry, +in arguments involving infinitesimals. See [11, §2] for a detailed study of this. +2.3 Local C∞-rings, and localization +Definition 2.10. AC∞-ringCis called localif regarded as an R-algebra, as +in Definition 2.6, Cis a local R-algebra with residue field R. That is, Chas a +unique maximal ideal mCwithC/mC∼=R. +IfC,Dare localC∞-rings with maximal ideals mC,mD, andφ:C→Dis +a morphism of C∞rings, then using the fact that C/mC∼=R∼=D/mDwe see +thatφ−1(mD) =mC, that is,φis alocalmorphism of local C∞-rings. Thus, +there is no difference between morphisms and local morphisms. +Remark 2.11. We use the term ‘local C∞-ring’ following Dubuc [23, Def. 4]. +They are also called C∞-local rings in Dubuc [22, Def. 2.13], pointed local C∞- +ringsin [54,§I.3] andArchimedean local C∞-ringsin [52,§3]. +Moerdijk and Reyes [52–54] use the term ‘local C∞-ring’ to mean a C∞-ring +which is a local R-algebra, but which need not have residue field R. +The next example is taken from Moerdijk and Reyes [54, §I.3]. +Example 2.12. WriteC∞(N) for theR-algebraofallfunctions f:N→R. It is +a finitely generated C∞-ring isomorphic to C∞(R)/{f∈C∞(R) :f|N= 0}. Let +Fbe anon-principal ultrafilter onN, in the sense of Comfort and Negrepontis +[16], and let I⊂Cbe the prime ideal of f:N→Rsuch that{n∈N:f(n) = 0} +lies inF. ThenC=C∞(N)/Iis a finitely generated C∞-ring which is a local +9R-algebra by [54, Ex. I.3.2], that is, it has a unique maximal ideal mC, but its +residue field is not Rby [54, Cor. I.3.4]. Hence Cis a localC∞-ring in the sense +of [52–54], but not in our sense. +Localizations ofC∞-rings are studied in [22,23,52,53], see [54, p. 23]. +Definition 2.13. LetCbe aC∞-ring andSa subset of C. Alocalization +C[s−1:s∈S] ofCatSis aC∞-ringD=C[s−1:s∈S] and a morphism +π:C→Dsuch thatπ(s) is invertible in Dfor alls∈S, with the universal +property that if Eis aC∞-ring andφ:C→Ea morphism with φ(s) invertible +inEfor alls∈S, then there is a unique morphism ψ:D→Ewithφ=ψ◦π. +A localization C[s−1:s∈S] always exists — it can be constructed by +adjoining an extra generator s−1and an extra relation s·s−1−1 = 0 for each +s∈S— and is unique up to unique isomorphism. When S={c}we have +an exact sequence 0 →I→C⊗∞C∞(R)π−→C[c−1]→0, where C⊗∞C∞(R) +is the coproduct of C,C∞(R) as in§2.1, with pushout morphisms ι1:C→ +C⊗∞C∞(R),ι2:C∞(R)→C⊗∞C∞(R), andIis the ideal in C⊗∞C∞(R) +generated by ι1(c)·ι2(x)−1, wherexis the generator of C∞(R). +AnR-pointxof aC∞-ringCis aC∞-ring morphism x:C→R, where +Ris regarded as a C∞-ring as in Example 2.3. By [54, Prop. I.3.6], a map +x:C→Ris a morphism of C∞-rings if and only if it is a morphism of the +underlying R-algebras, as in Definition 2.6. Define Cxto be the localization +Cx=C[s−1:s∈C,x(s)/\e}atio\slash= 0], with projection πx:C→Cx. ThenCxis a +localC∞-ring by [53, Lem. 1.1]. The R-points ofC∞(Rn) are just evaluation +at pointsx∈Rn. This also holds for C∞(X) for any manifold X. +In a new result, we can describe these local C∞-ringsCxexplicitly. Note +that the surjectivity of πx:C→Cxin the next proposition is surprising. It +doesnot hold forgenerallocalizationsof C∞-rings— forinstance, π:C∞(R)→ +C∞(R)[x−1] is injective but not surjective, as x−1/∈Imπ— or for localizations +πx:A→Axof rings or K-algebras in conventional algebraic geometry. +Proposition 2.14. LetCbe aC∞-ring,x:C→RanR-point of C,andCx +the localization, with projection πx:C→Cx. Thenπxis surjective with kernel +an idealI⊂C,so thatCx∼=C/I,where +I=/braceleftbig +c∈C:there exists d∈Cwithx(d)/\e}atio\slash= 0inRandc·d= 0inC/bracerightbig +.(2.2) +Proof.ClearlyIin (2.2) is closed under multiplication by elements of C. Let +c1,c2∈I, so there exist d1,d2∈Cwithx(d1)/\e}atio\slash= 0/\e}atio\slash=x(d2) andc1d1= 0 =c2d2. +Thend1d2∈Cwithx(d1d2) =x(d1)x(d2)/\e}atio\slash= 0, and(c1+c2)(d1·d2) =d2(c1d1)+ +d1(c2d2) = 0, soc1+c2∈I. HenceIis an ideal, and C/IaC∞-ring. +Supposec∈I, so there exists d∈Cwithx(d)/\e}atio\slash= 0 andcd= 0. Then πx(d) +is invertible in Cxby definition. Thus +πx(c) =πx(c)πx(d)πx(d)−1=πx(cd)πx(d)−1=πx(0)πx(d)−1= 0. +ThereforeI⊆Kerπx. Soπx:C→Cxfactorizes uniquely as πx=ı◦π, where +π:C→C/Iis the projection and ı:C/I→Cxis aC∞-ring morphism. +10Supposec∈Cwithx(c)/\e}atio\slash= 0, and write ǫ=1 +2|x(c)|. Choose smooth +functionsη:R→R\{0}, so thatη−1:R→R\{0}is also smooth, such that +η(t) =tfor allt∈(x(c)−ǫ,x(c)+ǫ), andζ:R→Rsuch thatζ(t) = 0 for all +t∈R\(x(c)−ǫ,x(c)+ǫ), so that (η−idR)·ζ= 0, andζ(x(c)) = 1. +Setc1= Φη(c),c2= Φη−1(c) andd= Φζ(c) inC, using the C∞-ring +operations from η,η−1,ζ. Thenc1c2= 1 inC, asη·η−1= 1, andx(d) = +x(Φζ(c)) =ζ(x(c)) = 1, asx:C→Ris aC∞-ring morphism. Also +(c1−c)·d=/parenleftbig +Φη(c)−ΦidR(c)/parenrightbig +Φζ(c) = Φ(η−idR)ζ(c) = Φ0(c) = 0. +Hencec1−c∈Iasx(d)/\e}atio\slash= 0, soc+I=c1+I. But then ( c+I)(c2+I) = +(c1+I)(c2+I) =c1c2+I= 1+IinC/I, soπ(c) =c+Iis invertible in C/I. +As this holds for all c∈Cwithx(c)/\e}atio\slash= 0, by the universal property of Cx +there exists a unique C∞-ring morphism :Cx→C/Iwithπ=◦πx. Since +πx,πare surjective, πx=ı◦πandπ=◦πximply that ı:C/I→Cxand +:Cx→C/Iare inverse, so both are isomorphisms. +Example 2.15. Forn/greaterorequalslant0 andp∈Rn, defineC∞ +p(Rn) to be the set of germs +of smooth functions c:Rn→Ratp∈Rn, made into a C∞-ring in the obvious +way. Then C∞ +p(Rn) is a localC∞-ring in the sense of Definition 2.10. Here are +three different ways to define C∞ +p(Rn), which yield isomorphic C∞-rings: +(a) Defining C∞ +p(Rn) asthe germsoffunctions ofsmoothfunctionsat pmeans +that points of C∞ +p(Rn) are∼-equivalence classes [( U,c)] of pairs ( U,c), +whereU⊆Rnis open with p∈Uandc:U→Ris smooth, and +(U,c)∼(U′,c′) if there exists p∈V⊆U∩U′open withc|V≡c′|V. +(b) As the localization ( C∞(Rn))p=C∞(Rn)[g∈C∞(Rn) :g(p)/\e}atio\slash= 0]. Then +points of (C∞(Rn))pare equivalence classes [ f/g] of fractions f/gfor +f,g∈C∞(Rn) withg(p)/\e}atio\slash= 0, and fractions f/g,f′/g′are equivalent if +there exists h∈C∞(Rn) withh(p)/\e}atio\slash= 0 andh(fg′−f′g)≡0. +(c) As the quotient C∞(Rn)/I, whereIis the ideal of f∈C∞(Rn) with +f≡0 nearp∈Rn. +One can show (a)–(c) are isomorphic using the fact that if Uis any open neigh- +bourhood of pinRnthen there exists smooth η:Rn→[0,1] such that η≡0 on +an open neighbourhood of Rn\UinRnandη≡1 on an open neighbourhood +ofpinU. By Moerdijk and Reyes [54, Prop. I.3.9], any finitely generated local +C∞-ring is a quotient of some C∞ +p(Rn). +2.4 Fair C∞-rings +We now discuss an important class of C∞-rings, which we call fairC∞-rings, +for brevity. Although our term ‘fair’ is new, we stress that the idea is already +well-known, being originally introduced by Dubuc [22], [23, Def. 11], who first +recognized their significance, under the name ‘ C∞-rings of finite type presented +by an ideal of local character’, and in more recent works would be re ferred to +as ‘finitely generated and germ-determined C∞-rings’. +11Definition 2.16. An idealIinC∞(Rn) is called fairif for eachf∈C∞(Rn), +flies inIif and only if πp(f) lies inπp(I)⊆C∞ +p(Rn) for allp∈Rn, where +C∞ +p(Rn) is as in Example 2.15 and πp:C∞(Rn)→C∞ +p(Rn) is the natural +projectionπp:c/ma√sto→[(Rn,c)]. AC∞-ringCis called fairif it is isomorphic +toC∞(Rn)/I, whereIis a fair ideal. Equivalently, Cis fair if it is finitely +generated and whenever c∈Cwithπp(c) = 0 in Cpfor allR-pointsp:C→R +thenc= 0, using the notation of Definition 2.13. +Dubuc [22], [23, Def. 11] calls fair ideals ideals of local character , and Mo- +erdijk and Reyes [54, I.4] call them germ determined , which has now become the +accepted term. Fair C∞-rings are also sometimes called germ determined C∞- +rings, a more descriptive term than ‘fair’, but the definition of germ deter mined +C∞-ringsCin [54, Def. I.4.1] does not require Cfinitely generated, so does not +equate exactly to our fair C∞-rings. By Dubuc [22, Prop. 1.8], [23, Prop. 12] +any finitely generated ideal Iis fair, so Cfinitely presented implies Cfair. We +writeC∞Ringsfafor the full subcategory of fair C∞-rings in C∞Rings. +Proposition 2.17. SupposeI⊂C∞(Rm)andJ⊂C∞(Rn)are ideals with +C∞(Rm)/I∼=C∞(Rn)/JasC∞-rings. Then Iis finitely generated, or fair, if +and only if Jis finitely generated, or fair, respectively. +Proof.Writeφ:C∞(Rm)/I→C∞(Rn)/Jfor the isomorphism, and x1,...,xm +for the generators of C∞(Rm), andy1,...,ynfor the generators of C∞(Rn). +Sinceφis an isomorphism we can choose f1,...,fm∈C∞(Rn) withφ(xi+I) = +fi+Jfori= 1,...,mandg1,...,gn∈C∞(Rm) withφ(gi+I) =yi+Jfor +i= 1,...,n. It is now easy to show that +I=/parenleftbig +xi−fi/parenleftbig +g1(x1,...,xm),...,gn(x1,...,xm)/parenrightbig +, i= 1,...,m, +andh/parenleftbig +g1(x1,...,xm),...,gn(x1,...,xm)/parenrightbig +, h∈J/parenrightbig +. +Hence, ifJisgeneratedby h1,...,hkthenIisgeneratedby xi−fi(g1,...,gn) +fori= 1,...,mandhj(g1,...,gn) forj= 1,...,k, soJfinitely generated +impliesIfinitelygenerated. Applyingthesameargumentto φ−1:C∞(Rn)/J→ +C∞(Rm)/I, we see that Iis finitely generated if and only if Jis. +SupposeIis fair, and let f∈C∞(Rn) withπq(f)∈πq(J)⊆C∞ +q(Rn) for +allq∈Rn. We will show that f∈J, so thatJis fair. Consider the function +f′=f(g1,...,gn)∈C∞(Rm). Ifp= (p1,...,pm) inRmandq= (q1,...,qn) =/parenleftbig +g1(p1,...,pm),...,gn(p1,...,pm)/parenrightbig +thenφ:C∞(Rm)/I→C∞(Rn)/Jlocalizes +to an isomorphism φp:C∞ +p(Rm)/πp(I)→C∞ +q(Rn)/πq(J) which maps φp: +πp(f′)+πp(I)/ma√sto→πq(f)+πq(J). Sinceπq(f)∈πq(J), this gives πp(f′)∈πp(I) +for allp∈Rm, sof′∈IasIis fair. But φ(f′+I) =f+J, sof′∈Iimplies +f∈J. Therefore Jis fair. Conversely, Jis fair implies Iis fair. +Example 2.18. The localC∞-ringC∞ +p(Rn) of Example 2.15 is the quotient of +C∞(Rn) by the ideal Iof functions fwithf≡0 nearp∈Rn. Forn>0 thisI +is fair, but not finitely generated. So C∞ +p(Rn) is fair, but not finitely presented, +by Proposition 2.17. +12The following example taken from Dubuc [24, Ex. 7.2] shows that localiz a- +tions of fair C∞-rings need not be fair: +Example 2.19. LetCbe the local C∞-ringC∞ +0(R), as in Example 2.15. Then +C∼=C∞(R)/I, whereIis the ideal of all f∈C∞(R) withf≡0 near 0 in R. +ThisIis fair, so Cis fair. Let c= [(x,R)]∈C. Then the localization C[c−1] +is theC∞-ring of germs at 0 in Rof smooth functions R\{0}→R. Taking +y=x−1as a generator of C[c−1], we see that C[c−1]∼=C∞(R)/J, whereJis +the ideal of compactly supported functions in C∞(R). ThisJis not fair, so by +Proposition 2.17, C[c−1] is not fair. +Recall from category theory that if Cis a subcategory of a category D, a +reflectionR:D→Cisaleft adjointtothe inclusion C֒→D. Thatis,R:D→C +is a functor with natural isomorphisms Hom C(R(D),C)∼=HomD(D,C) for all +C∈CandD∈D. We will define a reflection for C∞Ringsfa⊂C∞Ringsfg, +following Moerdijk and Reyes [54, p. 48–49] (see also Dubuc [23, Th. 13]). +Definition 2.20. LetCbe a finitely generated C∞-ring. LetICbe the ideal +of allc∈Csuch thatπp(c) = 0 in Cpfor allR-pointsp:C→R. ThenC/IC +is a finitely generated C∞-ring, with projection π:C→C/IC. It has the same +R-points as C, that is, morphisms p:C/IC→Rare in 1-1 correspondence +with morphisms p′:C→Rbyp′=p◦π, and the local rings ( C/IC)pandCp′ +are naturally isomorphic. It follows that C/ICis fair. Define a functor Rfa +fg: +C∞Ringsfg→C∞RingsfabyRfa +fg(C) =C/ICon objects, and if φ:C→D +is a morphism then φ(IC)⊆ID, soφinduces a morphism φ∗:C/IC→D/ID, +and we set Rfa +fg(φ) =φ∗. It is easy to see Rfa +fgis a reflection. +IfIis an ideal in C∞(Rn), write¯Ifor the set of f∈C∞(Rn) withπp(f)∈ +πp(I) for allp∈Rn. Then¯Iis the smallest fair ideal in C∞(Rn) containing I, +thegerm-determined closure ofI, andRfa +fg/parenleftbig +C∞(Rn)/I/parenrightbig∼=C∞(Rn)/¯I. +Example 2.21. Letη:R→[0,∞) be smooth with η(x)>0 forx∈(0,1) and +η(x) = 0 forx /∈(0,1). DefineI⊆C∞(R) by +I=/braceleftbig/summationtext +a∈Aga(x)η(x−a) :A⊂Zis finite,ga∈C∞(R),a∈A/bracerightbig +. +ThenIis an ideal in C∞(R), soC=C∞(R)/Iis aC∞-ring. The set of +f∈C∞(R) such that πp(f) lies inπp(I)⊆C∞ +p(R) for allp∈Ris +¯I=/braceleftbig/summationtext +a∈Zga(x)η(x−a) :ga∈C∞(R), a∈Z/bracerightbig +, +where the sum/summationtext +a∈Zga(x)η(x−a) makes sense as at most one term is nonzero +at any point x∈R. Since¯I/\e}atio\slash=I, we see that Iisnot fair, soC=C∞(R)/Iis +not a fairC∞-ring. In fact ¯Iis the smallest fair ideal containing I. We have +IC∞(R)/I=¯I/I, andRfa +fg/parenleftbig +C∞(R)/I) =C∞(R)/¯I. +Proposition 2.22. LetCbe aC∞-ring, andGa finite group acting on Cby +automorphisms. Then the fixed subset CGofGinChas the structure of a C∞- +ring in a unique way, such that the inclusion CG֒→Cis aC∞-ring morphism. +IfCis fair, or finitely presented, then CGis also fair, or finitely presented. +13Proof.For the first part, let f:Rn→Rbe smooth, and c1,...,cn∈CG. Then +γ·Φf(c1,...,cn) = Φf(γ·c1,...,γ·cn) = Φf(c1,...,cn) for eachγ∈G, so +Φf(c1,...,cn)∈CG. Define ΦG +f: (CG)n→CGby ΦG +f= Φf|(CG)n. It is now +trivial to check that the operations ΦG +ffor smooth f:Rn→RmakeCGinto a +C∞-ring, uniquely such that CG֒→Cis aC∞-ring morphism. +Suppose now that Cis finitely generated. Choose a finite set of generators +forC, and by adding the images of these generators under G, extend to a set +of (not necessarily distinct) generators x1,...,xnforC, on whichGacts freely +by permutation. This gives an exact sequence 0 ֒→I→C∞(Rn)→C→0, +whereC∞(Rn) is freely generated by x1,...,xn. HereRnis a direct sum of +copies of the regular representation of G, andC∞(Rn)→CisG-equivariant. +HenceIis aG-invariant ideal in C∞(Rn), which is fair, or finitely generated, +respectively. Taking G-invariant parts gives an exact sequence 0 ֒→IG→ +C∞(Rn)Gπ−→CG→0, whereC∞(Rn)G,CGare clearlyC∞-rings. +AsGacts linearly on Rnit acts by automorphisms on the polynomial ring +R[x1,...,xn]. By a classical theorem of Hilbert [70, p. 274], R[x1,...,xn]G +is a finitely presented R-algebra, so we can choose generators p1,...,plfor +R[x1,...,xn]G, which induce a surjective R-algebra morphism R[p1,...,pl]→ +R[x1,...,xn]Gwith kernel generated by q1,...,qm∈R[p1,...,pl]. +By results of Bierstone [6] for Ga finite group and Schwarz [63] for Ga +compact Lie group, any G-invariant smooth function on Rnmay be written +as a smooth function of the generators p1,...,plofR[x1,...,xn]G, giving a +surjective morphism C∞(Rl)→C∞(Rn)G, whose kernel is the ideal in C∞(Rl) +generated by q1,...,qm. ThusC∞(Rn)Gis finitely presented. +AlsoCGis generated by π(p1),...,π(pl), soCGis finitely generated, and we +have an exact sequence 0 ֒→J→C∞(Rl)π−→CG→0, whereJis the ideal in +C∞(Rl) generated by q1,...,qmand the lifts to C∞(Rl) of a generating set for +the idealIGinC∞(Rn)G∼=C∞(Rl)/(q1,...,qm). +Suppose now that Iis fair. Then for f∈C∞(Rn)G,flies inIGif and only +ifπp(f)∈πp(I)⊆C∞ +p(Rn) for allp∈Rn. IfHis the subgroup of Gfixing +pthenHacts onC∞ +p(Rn), andπp(f) isH-invariant as fisG-invariant, and +πp(I)H=πp(IG). Thus we may rewrite the condition as flies inIGif and only +ifπp(f)∈πp(IG)⊆C∞ +p(Rn) for allp∈Rn. Projecting from RntoRn/G, this +says thatflies inIGif and only if πp(f) lies inπp(IG)⊆/parenleftbig +C∞(Rn)G/parenrightbig +pfor all +p∈Rn/G. SinceC∞(Rn)Gis finitely presented, it follows as in [54, Cor. I.4.9] +thatJis fair, so CGis fair. +SupposeIis finitely generated in C∞(Rn), with generators f1,...,fk. As +Rnis a sum of copies of the regular representation of G, so that every irre- +ducible representation of Goccurs as a summand of Rn, one can show that IG +is generated as an ideal in C∞(Rn/G) by then(k+1) elements fG +iand (fixj)G +fori= 1,...,kandj= 1,...,n, wherefG=1 +|G|/summationtext +γ∈Gf◦γis theG-invariant +part off∈C∞(Rn). Therefore Jis finitely generated by q1,...,qmand lifts of +fG +i,(fixj)G. Hence if Cis finitely presented then CGis finitely presented. +142.5 Pushouts of C∞-rings +Proposition 2.5 shows that pushouts of C∞-rings exist. For finitely generated +C∞-rings, we can describe these pushouts explicitly. +Example 2.23. Suppose the following is a pushout diagram of C∞-rings: +Cβ/d47/d47 +α/d15/d15E +δ/d15/d15 +Dγ/d47/d47F, +so thatF=D∐CE, withC,D,Efinitely generated. Then we have exact +sequences +0→I ֒→C∞(Rl)φ−→C→0,0→J ֒→C∞(Rm)ψ−→D→0, +and 0→K ֒→C∞(Rn)χ−→E→0,(2.3) +whereφ,ψ,χare morphisms of C∞-rings, and I,J,Kare ideals in C∞(Rl), +C∞(Rm),C∞(Rn). Writex1,...,xlandy1,...,ymandz1,...,znfor the gen- +erators ofC∞(Rl),C∞(Rm),C∞(Rn) respectively. Then φ(x1),...,φ(xl) gen- +erateC, andα◦φ(x1),...,α◦φ(xl) lie inD, so we may write α◦φ(xi) =ψ(fi) +fori= 1,...,lasψis surjective, where fi:Rm→Ris smooth. Similarly +β◦φ(x1),...,β◦φ(xl) lie inE, so we may write β◦φ(xi) =χ(gi) fori= 1,...,l, +wheregi:Rn→Ris smooth. +Then from the explicit construction of pushouts of C∞-rings we obtain an +exact sequence with ξa morphism of C∞-rings +0 /d47/d47L /d47/d47C∞(Rm+n)ξ/d47/d47F /d47/d470, (2.4) +where we write the generators of C∞(Rm+n) asy1,...,ym,z1,...,zn, and then +Lis the ideal in C∞(Rm+n) generated by the elements d(y1,...,ym) ford∈ +J⊆C∞(Rm), ande(z1,...,zn) fore∈K⊆C∞(Rn), andfi(y1,...,ym)− +gi(z1,...,zn) fori= 1,...,l. +For the case of coproducts D⊗∞E, withC=R,l= 0 andI={0}, we have +/parenleftbig +C∞(Rm)/J/parenrightbig +⊗∞/parenleftbig +C∞(Rn)/K/parenrightbig∼=C∞(Rm+n)/(J,K). +Proposition 2.24. The subcategories C∞RingsfgandC∞Ringsfpare closed +under pushouts and all finite colimits in C∞Rings. +Proof.Firstweshow C∞Ringsfg,C∞Ringsfpareclosedunderpushouts. Sup- +poseC,D,Eare finitely generated, and use the notation of Example 2.23. Then +Fis finitely generated with generators y1,...,ym,z1,...,zn, soC∞Ringsfg +is closed under pushouts. If C,D,Eare finitely presented then we can take +J= (d1,...,dj) andK= (e1,...,ek), and then Example 2.23 gives +L=/parenleftbig +dp(y1,...,ym), p= 1,...,j, ep(z1,...,zn), p= 1,...,k, +fp(y1,...,ym)−gp(z1,...,zn), p= 1,...,l/parenrightbig +.(2.5) +15SoLis finitely generated, and F∼=C∞(Rm+n)/Lis finitely presented. Thus +C∞Ringsfpis closed under pushouts. +NowRis an initial object in C∞Ringsfg,C∞Ringsfp⊂C∞Rings, and +all finite colimits may be constructed by repeated pushouts involving the initial +object. Hence C∞Ringsfg,C∞Ringsfpare closed under finite colimits. +Here is an example from Dubuc [24, Ex. 7.1], Moerdijk and Reyes [54, p . 49]. +Example 2.25. Consider the coproduct C∞(R)⊗∞C∞ +0(R), whereC∞ +0(R) is +theC∞-ring of germs of smooth functions at 0 in Ras in Example 2.15. Then +C∞(R),C∞ +0(R) are fairC∞-rings, but C∞ +0(R) is not finitely presented. By +Example 2.23, C∞(R)⊗∞C∞ +0(R) =C∞(R)∐RC∞ +0(R)∼=C∞(R2)/L, whereL +is the ideal in C∞(R2) generated by functions f(x,y) =g(y) forg∈C∞(R) +withg≡0 near 0∈R. This ideal Lis not fair, since for example one can +findf∈C∞(R2) withf(x,y) = 0 if and only if |xy|/lessorequalslant1, and then f /∈Lbut +πp(f)∈πp(L)⊆C∞ +p(R2) for allp∈R2. HenceC∞(R)⊗∞C∞ +0(R) is not a fair +C∞-ring, by Proposition 2.17, and pushouts of fair C∞-rings need not be fair. +Our next result is referred to in the last part of Dubuc [23, Th. 13]. +Proposition 2.26. C∞Ringsfais not closed under pushouts in C∞Rings. +Nonetheless, pushouts and all finite colimits exist in C∞Ringsfa,although they +may not coincide with pushouts and finite colimits in C∞Rings. +Proof.Example 2.25 shows that C∞Ringsfais not closed under pushouts in +C∞Rings. To construct finite colimits in C∞Ringsfa, we first take the colimit +inC∞Ringsfg, which exists by Propositions 2.5 and 2.24, and then apply the +reflection functor Rfa +fg. By the universal properties of colimits and reflection +functors, the result is a colimit in C∞Ringsfa. +2.6 Flat ideals +The following class of ideals in C∞(Rn) is defined by Moerdijk and Reyes [54, +p. 47, p. 49] (see also Dubuc [22, §1.7(a)]), who call them flat ideals : +Definition 2.27. LetXbe a closed subset of Rn. Define m∞ +Xto be the ideal +of all functions g∈C∞(Rn) such that ∂kg|X≡0 for allk/greaterorequalslant0, that is,gand +all its derivatives vanish at each x∈X. If the interior X◦ofXinRnis dense +inX, that is (X◦) =X, then∂kg|X≡0 for allk/greaterorequalslant0 if and only if g|X≡0. In +this caseC∞(Rn)/m∞ +X∼=C∞(X) :=/braceleftbig +f|X:f∈C∞(Rn)/bracerightbig +. +Flat ideals are always fair. Here is an example from [54, Th. I.1.3]. +Example 2.28. TakeXtobethepoint{0}. Iff,f′∈C∞(Rn)thenf−f′liesin +m∞ +{0}if and only if f,f′have the same Taylor series at 0. Thus C∞(Rn)/m∞ +{0}is +theC∞-ring of Taylor series at 0 of f∈C∞(Rn). Since any formal power series +inx1,...,xnis the Taylorseries of some f∈C∞(Rn), we haveC∞(Rn)/m∞ +{0}∼= +R[[x1,...,xn]]. Thus the R-algebra of formal power series R[[x1,...,xn]] can +be made into a C∞-ring. +16The following nontrivial result is proved by Reyes and van Quˆ e [60, Th . 1], +generalizing an unpublished result of A.P. Calder´ on in the case X=Y={0}. +It can also be found in Moerdijk and Reyes [54, Cor. I.4.12]. +Proposition 2.29. LetX⊆RmandY⊆Rnbe closed. Then as ideals in +C∞(Rm+n)we have (m∞ +X,m∞ +Y) =m∞ +X×Y. +Moerdijk and Reyes [54, Cor. I.4.19] prove: +Proposition 2.30. LetX⊆Rnbe closed with X/\e}atio\slash=∅,Rn. Then the ideal m∞ +X +inC∞(Rn)is not countably generated. +We can use these to study C∞-rings of manifolds with corners. +Example 2.31. Let 00, we can +embedXasaclosedsubsetinan n-manifoldX′withoutboundary,suchthatthe +inclusionX ֒→X′is locally modelled on the inclusion of Rn +k= [0,∞)k×Rn−kin +(−ǫ,∞)k×Rn−kfork/lessorequalslantn. Choose a closed embedding i:X′֒→RNforN≫0 +as above, giving 0 →I′→C∞(RN)i∗ +−→C∞(X′)→0 withI′generated by +f1,...,fk∈C∞(RN). Leti(X′)⊂T⊂RNbe an open tubular neighbourhood +ofi(X′) inRN, with projection π:T→i(X′). SetU=π−1(i(X◦))⊂T⊂RN, +whereX◦is the interior of X. ThenUis open in RNwithi(X◦) =U∩i(X′), +and the closure ¯UofUinRNhasi(X) =¯U∩i(X′). +LetIbe the ideal ( f1,...,fk,m∞¯U) inC∞(RN). ThenIis fair, as (f1,...,fk) +andm∞¯Uare fair. Since Uis open in RNand dense in ¯U, as in Definition +2.27 we have g∈m∞ +¯Uif and only if g|¯U≡0. Therefore the isomorphism +(i∗)∗:C∞(RN)/I′→C∞(X′) identifies the ideal I/I′inC∞(X′) with the +ideal off∈C∞(X′) such that f|X≡0, sinceX=i−1(¯U). Hence +C∞(RN)/I∼=C∞(X′)//braceleftbig +f∈C∞(X′) :f|X≡0/bracerightbig∼=/braceleftbig +f|X:f∈C∞(X′)/bracerightbig∼=C∞(X). +AsIis a fair ideal, this implies that C∞(X) is a fairC∞-ring. If∂X/\e}atio\slash=∅then +using Proposition 2.30 we can show Iis not countably generated, so C∞(X) is +not finitely presented by Proposition 2.17. +Next we consider the transformation X/ma√sto→C∞(X) as a functor. +Definition 3.2. WriteC∞Ringsop, (C∞Ringsfp)op, (C∞Ringsfa)opfor the +opposite categories of C∞Rings,C∞Ringsfp,C∞Ringsfa(i.e. directions of +morphisms are reversed). Define functors +FC∞Rings +Man:Man−→(C∞Ringsfp)op⊂C∞Ringsop, +FC∞Rings +Manb:Manb−→(C∞Ringsfa)op⊂C∞Ringsop, +FC∞Rings +Manc:Manc−→(C∞Ringsfa)op⊂C∞Ringsop +asfollows. On objectsthe functors FC∞Rings +Man∗mapX/ma√sto→C∞(X), whereC∞(X) +is aC∞-ring as in Example 2.2. On morphisms, if f:X→Yis a smooth map +of manifolds then f∗:C∞(Y)→C∞(X) mapping c/ma√sto→c◦fis a morphism +18ofC∞-rings, so that f∗:C∞(Y)→C∞(X) is a morphism in C∞Rings, +andf∗:C∞(X)→C∞(Y) a morphism in C∞Ringsop, andFC∞Rings +Man∗maps +f/ma√sto→f∗. ClearlyFC∞Rings +Man,FC∞Rings +Manb,FC∞Rings +Mancare functors. +Iff:X→Yisonlyweakly smooth thenf∗:C∞(Y)→C∞(X)inDefinition +3.2 is still a morphism of C∞-rings. From [54, Prop. I.1.5] we deduce: +Proposition 3.3. LetX,Ybe manifolds with corners. Then the map f/ma√sto→f∗ +from weakly smooth maps f:X→Yto morphisms of C∞-ringsφ:C∞(Y)→ +C∞(X)is a1-1correspondence. +In the category of manifolds Man, the morphisms are weakly smooth maps. +SoFC∞Rings +Man is both injective on morphisms (faithful), and surjective on mor- +phisms (full), as in Moerdijk and Reyes [54, Th. I.2.8]. But in Manb,Manc +the morphisms are smooth maps, a proper subset of weakly smooth maps, so +the functors are injective but not surjective on morphisms. That is: +Corollary 3.4. The functor FC∞Rings +Man:Man→(C∞Ringsfp)opis full and +faithful. However, the functors FC∞Rings +Manb:Manb→(C∞Ringsfa)opand +FC∞Rings +Manc:Manc→(C∞Ringsfa)opare faithful, but not full. +Of course, if we defined Manb,Mancto have morphisms weakly smooth +maps, then FC∞Rings +Manb,FC∞Rings +Mancwould be full and faithful. +LetX,Y,Zbe manifolds and f:X→Z,g:Y→Zbe smooth maps. If +X,Y,Zare without boundary then f,gare called transverse if whenever x∈X +andy∈Ywithf(x) =g(y) =z∈Zwe haveTzZ= df(TxX)+dg(TyY). If +f,gare transverse then a fibre product X×ZYexists in Man. +For manifolds with boundary, or with corners, the situation is more c ompli- +cated, as explained in [35, §6], [40,§4.3]. In the definition of smoothf:X→Y +we impose extra conditions over ∂jX,∂kY, and in the definition of transverse +f,gwe impose extra conditions over ∂jX,∂kY,∂lZ. With these more restrictive +definitions of smooth and transverse maps, transverse fibre pro ducts exist in +Mancby [35, Th. 6.3] (see also [40, Th. 4.27]). The na¨ ıve definition of tran sver- +sality is not a sufficient condition for fibre products to exist. Note to o that a +fibre product of manifolds with boundary may be a manifold with corne rs, so +fibre products work best in ManorMancrather than Manb. +Our next theorem is given in [23, Th. 16] and [54, Prop. I.2.6] for manif olds +without boundary, and the special case of products Man×Manb→Manb +follows from Reyes [59, Th. 2.5], see also Kock [44, §III.9]. It can be proved +by combining the usual proof in the without boundary case, the pro of of [35, +Th. 6.3], and Proposition 2.29. +Theorem 3.5. The functors FC∞Rings +Man,FC∞Rings +Mancpreserve transverse fibre +products in Man,Manc,in the sense of [35,§6]. That is, if the following is a +Cartesian square of manifolds with g,htransverse +Wf/d47/d47 +e/d15/d15Y +h/d15/d15 +Xg/d47/d47Z,(3.1) +19so thatW=X×g,Z,hY,then we have a pushout square of C∞-rings +C∞(Z) +h∗/d47/d47 +g∗/d15/d15C∞(Y) +f∗/d15/d15 +C∞(X)e∗/d47/d47C∞(W),(3.2) +so thatC∞(W) =C∞(X)∐g∗,C∞(Z),h∗C∞(Y). +4C∞-ringed spaces and C∞-schemes +Inalgebraicgeometry,if Aisanaffineschemeand Rtheringofregularfunctions +onA, then we can recover Aas the spectrum of the ring R,A∼=SpecR. One of +the ideas of synthetic differential geometry, as in [54, §I], is to regard a manifold +Xas the ‘spectrum’ of the C∞-ringC∞(X) in Example 2.2. So we can try to +develop analogues of the tools of scheme theory for smooth manifo lds, replacing +rings byC∞-rings throughout. This was done by Dubuc [22,23]. The analogues +of the algebraic geometry notions [31, §II.2] of ringed spaces, locally ringed +spaces, and schemes, are called C∞-ringed spaces, local C∞-ringed spaces and +C∞-schemes. The material of §4.6–§4.9 is new. +4.1 Some basic topology +Later we will use several properties of topological spaces, e.g. se cond countable, +metrizable, Lindel¨ of, ..., so we now recall their definitions and som e relation- +ships between them. Let Xbe a topological space, with topology T. Then: +•AbasisforTis a familyB⊆Tsuch that every open set in Xis a union +of sets inB. We callXsecond countable ifThas a countable basis. +•An open cover{Ui:i∈I}ofXislocally finite if everyx∈Xhas an +open neighbourhood WwithW∩Ui/\e}atio\slash=∅for only finitely many i∈I. +An open cover{Vj:j∈J}ofXis arefinement of another open cover +{Ui:i∈I}if for allj∈Jthere exists i∈IwithVj⊆Ui⊆X. +We callXparacompact if every open cover {Ui:i∈I}ofXadmits a +locally finite refinement {Vj:j∈J}. +•We callXHausdorff if for allx,y∈Xwithx/\e}atio\slash=ythere exist open +U,V⊆Xwithx∈U,y∈VandU∩V=∅. +•We callXmetrizable if there exists a metric on Xinducing topology T. +•We callXregularif for every closed subset C⊆Xand eachx∈X\C +there exist disjoint open sets U,V⊆XwithC⊆Uandx∈V. +•We callXcompletely regular if for every closed C⊆Xandx∈X\C +there exists a continuous f:X→[0,1] withf|C= 0 andf(x) = 1. +•We callXseparable if it has a countable dense subset S⊆X. +20•We callXlocally compact if for allx∈Xthere exist x∈U⊆C⊆X +withUopen andCcompact. +•We callXLindel¨ of if every open cover of Xhas a countable subcover. +By well known results in topology, including Urysohn’s metrization the orem, +the following are equivalent: +(i)Xis Hausdorff, second countable and regular. +(ii)Xis second countable and metrizable. +(iii)Xis separable and metrizable. +Here are some useful implications: +•XHausdorff and locally compact imply Xis regular. +•Xmetrizable implies Xis Hausdorff, paracompact, and regular. +•Xsecond countable implies Xis Lindel¨ of. +•XLindel¨ of and regular imply Xis paracompact. +4.2 Sheaves on topological spaces +Sheaves are a fundamental concept in algebraic geometry. They a re necessary +even to define schemes, since a scheme is a topological space Xequipped with +a sheaf of ringsOX. In this book, sheaves of C∞-rings, and sheaves of modules +over a sheaf of C∞-rings, play a fundamental rˆ ole. +We now summarize some basics of sheaf theory, following Hartshorn e [31, +§II.1]. A more detailed reference is Godement [28]. We concentrate on sheaves +of abelian groups; to define sheaves of C∞-rings, etc., one replaces abelian +groups with C∞-rings, etc., throughout. This is justified since limits in all these +categories (including abelian groups) are computed at the level of u nderlying +sets, because they are all algebras for algebraic theories. +Definition 4.1. LetXbe a topological space. A presheaf of abelian groups E +onXconsists of the data of an abelian group E(U) for every open set U⊆X, +and a morphism of abelian groups ρUV:E(U)→E(V) called the restriction +mapfor every inclusion V⊆U⊆Xof open sets, satisfying the conditions that +(i)E(∅) = 0; +(ii)ρUU= idE(U):E(U)→E(U) for all open U⊆X; and +(iii)ρUW=ρVW◦ρUV:E(U)→E(W) for all open W⊆V⊆U⊆X. +That is, a presheaf is a functor E:Open(X)op→AbGp, whereOpen(X) is +the category of open subsets of Xwith morphisms inclusions, and AbGpis the +category of abelian groups. +A presheaf of abelian groups EonXis called a sheafif it also satisfies +(iv) IfU⊆Xis open,{Vi:i∈I}is an open cover of U, ands∈E(U) has +ρUVi(s) = 0 inE(Vi) for alli∈I, thens= 0 inE(U); and +21(v) IfU⊆Xis open,{Vi:i∈I}is an open cover of U, and we are given +elementssi∈E(Vi) for alli∈Isuch thatρVi(Vi∩Vj)(si) =ρVj(Vi∩Vj)(sj) +inE(Vi∩Vj) for alli,j∈I, then there exists s∈E(U) withρUVi(s) =si +for alli∈I. Thissis unique by (iv). +SupposeE,Fare presheavesor sheavesof abelian groups on X. Amorphism +φ:E→Fconsists of a morphism of abelian groups φ(U) :E(U)→F(U) for all +openU⊆X, suchthatthefollowingdiagramcommutesforallopen V⊆U⊆X +E(U) +φ(U)/d47/d47 +ρUV/d15/d15F(U) +ρ′ +UV/d15/d15 +E(V)φ(V)/d47/d47F(V), +whereρUVis the restriction map for E, andρ′ +UVthe restriction map for F. +Definition 4.2. LetEbe a presheaf of abelian groups on X. For eachx∈X, +thestalkExis the direct limit of the groups E(U) for allx∈U⊆X, via the +restriction maps ρUV. It is an abelian group. A morphism φ:E→Finduces +morphisms φx:Ex→Fxfor allx∈X. IfE,Fare sheaves then φis an +isomorphism if and only if φxis an isomorphism for all x∈X. +Sheaves of abelian groups on Xform an abelian category Sh(X). Thus we +have (category-theoretic) notions of when a morphism φ:E→Fin Sh(X) is +injective orsurjective (epimorphic ), and when a sequence E→F→G in Sh(X) +isexact. It turns out that φ:E→Fis injective if and only if φ(U) :E(U)→ +F(U) is injective for all open U⊆X. Howeverφ:E→Fsurjective does not +imply that φ(U) :E(U)→F(U) is surjective for all open U⊆X. Instead,φis +surjective if and only if φx:Ex→Fxis surjective for all x∈X. +Definition 4.3. LetEbe a presheaf of abelian groups on X. Asheafification +ofEis a sheaf of abelian groups ˆEonXand a morphism π:E→ˆE, such that +wheneverFis a sheaf of abelian groups on Xandφ:E→Fis a morphism, +there is a unique morphism ˆφ:ˆE→Fwithφ=ˆφ◦π. As in [31, Prop. II.1.2], +a sheafification always exists, and is unique up to canonical isomorph ism; one +can be constructed explicitly using the stalks ExofE. +Next we discuss pushforwards andpullbacks of sheaves by continuous maps. +Definition 4.4. Letf:X→Ybe a continuous map of topological spaces, and +Ea sheaf of abelian groups on X. Define the pushforward (direct image ) sheaf +f∗(E) onYby/parenleftbig +f∗(E)/parenrightbig +(U) =E/parenleftbig +f−1(U)/parenrightbig +for all open U⊆V, with restriction +mapsρ′ +UV=ρf−1(U)f−1(V):/parenleftbig +f∗(E)/parenrightbig +(U)→/parenleftbig +f∗(E)/parenrightbig +(V) for all open V⊆U⊆Y. +Thenf∗(E) is a sheaf of abelian groups on Y. +Ifφ:E→Fis a morphism in Sh( X) we define f∗(φ) :f∗(E)→f∗(F) by/parenleftbig +f∗(φ)/parenrightbig +(u) =φ/parenleftbig +f−1(U)/parenrightbig +for all open U⊆Y. Thenf∗(φ) is a morphism in +Sh(Y), andf∗is a functor Sh( X)→Sh(Y). It is a left exact functor between +abelian categories, but in general is not exact. For continuous map sf:X→Y, +g:Y→Zwe have (g◦f)∗=g∗◦f∗. +22Definition 4.5. Letf:X→Ybe a continuous map of topological spaces, +andEa sheaf of abelian groups on Y. Define a presheaf Pf−1(E) onXby/parenleftbig +Pf−1(E)/parenrightbig +(U) = limA⊇f(U)E(A) for open A⊆X, where the direct limit is +taken over all open A⊆Ycontaining f(U), using the restriction maps ρAB +inE. For open V⊆U⊆X, defineρ′ +UV:/parenleftbig +Pf−1(E)/parenrightbig +(U)→/parenleftbig +Pf−1(E)/parenrightbig +(V) as +the direct limit of the morphisms ρABinEforB⊆A⊆Ywithf(U)⊆A +andf(V)⊆B. Then we define the pullback (inverse image )f−1(E) to be the +sheafification of the presheaf Pf−1(E). +Pullbacksf−1(E) are only unique up to canonical isomorphism, rather than +unique. By convention we choose once and for all a pullback f−1(E) for all +X,Y,f,E, using the Axiom of Choice if necessary. If φ:E→Fis a morphism +in Sh(Y), one can define a pullback morphism f−1(φ) :f−1(E)→f−1(F). +Thenf−1: Sh(Y)→Sh(X) is an exact functor between abelian categories. +We compare pushforwards and pullbacks: +Remark 4.6. (a) There are two kinds of pullback, with slightly different no- +tation. The first kind, written f−1(E) as in Definition 4.5, is used for sheaves +of abelian groups or C∞-rings. The second kind, written f∗(E) orf∗(E) and +discussed in§5.3 and§8.3, is used for sheaves of OY-modulesE. +(b)The definition of pushforward sheaves f∗(E) is wholly elementary. In con- +trast, the definition of pullbacks f−1(E) is complex, involving a direct limit +followed by a sheafification, and includes arbitrary choices. +Pushforwards f∗are strictly functorial in the continuous map f:X→Y, +thatis, forcontinuous f:X→Y,g:Y→Zwehave(g◦f)∗=g∗◦f∗: Sh(X)→ +Sh(Z). However, pullbacks f−1are only weakly functorial in f: ifE∈Sh(Z) +then we need not have ( g◦f)−1(E) =f−1(g−1(E)). This is because pullbacks +are only natural up to canonical isomorphism, and we make an arbitr ary choice +for each pullback. So although f−1(g−1(E)) is a possible pullback for Ebyg◦f, +it may not be the one we chose. +Thus, thereisacanonicalisomorphism( g◦f)−1(E)∼=f−1(g−1(E)), whichwe +will write as If,g(E) : (g◦f)−1(E)→f−1(g−1(E)). TheIf,g(E) for allE∈Sh(Z) +comprise a natural isomorphism of functors If,g: (g◦f)−1⇒f−1◦g−1. Sim- +ilarly, forE ∈Sh(X) we may not have id−1 +X(E) =E, but instead there are +canonical isomorphisms δX(E) : id−1 +X(E)→E, which make up a natural iso- +morphismδX: id−1 +X⇒idSh(X). Many authors ignore the natural isomorphisms +If,g,δXentirely. +(c)Letf:X→Ybe a continuous map of topological spaces. Then we have +functorsf∗: Sh(X)→Sh(Y), andf−1: Sh(Y)→Sh(X). Asin[31,Ex.II.1.18], +f∗is right adjoint to f−1. That is, there is a natural bijection +HomX/parenleftbig +f−1(E),F/parenrightbig∼=HomY/parenleftbig +E,f∗(F)/parenrightbig +(4.1) +for allE∈Sh(Y) andF∈Sh(X), with functorial properties. +We define finesheaves, as in Godement [28, §II.3.7] or Voisin [69, Def. 4.35]. +They will be important in §4.7 and§5.3. +23Definition 4.7. LetXbe a topological space (usually paracompact), and Ea +sheaf of abelian groups on X, or more generally a sheaf of rings, or C∞-rings, +orOX-modules, or any other objects which are also abelian groups. We ca llE +fineif for any open cover {Ui:i∈I}ofX, a subordinate locally finite partition +of unity{ζi:i∈I}exists in the sheaf Hom(E,E). +Hereζi:E→Eis a morphism of sheaves of abelian groups (or rings, C∞- +rings, ...) for each i∈I. For{ζi:i∈I}to besubordinate to{Ui:i∈I} +means that ζiis supported in Uifor eachi∈I, that is, there exists open Vi⊆X +withζi|Vi= 0 andUi∪Vi=X. For{ζi:i∈I}to belocally finite means that +eachx∈Xhas an open neighbourhood Wwithζi|W/\e}atio\slash= 0 for only finitely many +i∈I. For{ζi:i∈I}to be apartition of unity means that/summationtext +i∈Iζi= idE, +where the sum makes sense as {ζi:i∈I}is locally finite. +IfE=OXis a sheaf of commutative rings or C∞-rings, then writing ηi= +ζi(1) inOX(X), we see that ζi=ηi·is multiplication by ηi. So we can regard +the partition of unity as living in OX(X) rather thanHom(OX,OX). +4.3C∞-ringed spaces and local C∞-ringed spaces +Definition 4.8. AC∞-ringed space X= (X,OX) is a topological space X +with a sheafOXofC∞-rings onX. That is, for each open set U⊆Xwe are +given aC∞ringOX(U), and for each inclusion of open sets V⊆U⊆Xwe are +given a morphism of C∞-ringsρUV:OX(U)→OX(V), called the restriction +maps, and all this data satisfies the sheaf axioms in Definition 4.1. +Equivalently,OXis a presheaf of C∞-rings onX, that is, a functor +OX:Open(X)op−→C∞Rings, +whose underlying presheaf of abelian groups, or of sets, is a sheaf . The sheaf +axioms Definition 4.1(iv),(v) do not use the C∞-ring structure. +Amorphismf= (f,f♯) : (X,OX)→(Y,OY) ofC∞ringed spaces is a +continuous map f:X→Yand a morphism f♯:f−1(OY)→OXof sheaves of +C∞-rings onX, forf−1(OY) as in Definition 4.5. Since f∗is right adjoint to +f−1, as in (4.1) there is a natural bijection +HomX/parenleftbig +f−1(OY),OX/parenrightbig∼=HomY/parenleftbig +OY,f∗(OX)/parenrightbig +. (4.2) +Writef♯:OY→f∗(OX) for the morphism of sheaves of C∞-rings onYcorre- +sponding to f♯under (4.2), so that +f♯:f−1(OY)−→OX/squiggleleftrightf♯:OY−→f∗(OX). (4.3) +Iff:X→Yandg:Y→ZareC∞-scheme morphisms, the composition is +g◦f=/parenleftbig +g◦f,(g◦f)♯/parenrightbig +=/parenleftbig +g◦f,f♯◦f−1(g♯)◦If,g(OZ)/parenrightbig +, +whereIf,g(OZ) : (g◦f)−1(OZ)→f−1(g−1(OZ)) is the canonical isomorphism +from Remark 4.6(b). In terms of f♯:OY→f∗(OX), composition is +(g◦f)♯=g∗(f♯)◦g♯:OZ−→(g◦f)∗(OX) =g∗◦f∗(OX). +24AlocalC∞-ringed space X= (X,OX) is aC∞-ringed space for which the +stalksOX,xofOXatxare localC∞-rings for all x∈X. As in Definition +2.10, since morphisms of local C∞-rings are automatically local morphisms, +morphisms of local C∞-ringed spaces ( X,OX),(Y,OY) are just morphisms of +C∞-ringedspaces,without anyadditionallocalitycondition. Moerdijk, vanQuˆ e +and Reyes [52,§3] call our local C∞-ringed spaces Archimedean C∞-spaces. +WriteC∞RSfor the category of C∞-ringed spaces, and LC∞RSfor the +full subcategory of local C∞-ringed spaces. +For brevity, we will use the notation that underlined upper case lett ers +X,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) and +x∈X. When we write ‘ Uis open in X’ we mean that U= (U,OU) and +X= (X,OX) withU⊆Xan open set andOU=OX|U. +Remark 4.9. As above, there are two equivalent ways to write morphisms +ofC∞-ringed spaces ( X,OX)→(Y,OY), either using pullbacks as ( f,f♯) for +f♯:f−1(OY)→OX, or using pushforwards as ( f,f♯) forf♯:OY→f∗(OX). +Each definition has advantages and disadvantages. We choose to r egardf♯: +f−1(OY)→OXas the primary object, and so define morphisms of C∞-ringed +spaces as (f,f♯) rather than ( f,f♯), although we will use f♯in a few places. We +can always switch between the two points of view using (4.3). +Example 4.10. LetXbe a manifold, which may have boundary or corners. +Define aC∞-ringed space X= (X,OX) to have topological space Xand +OX(U) =C∞(U) for each open subset U⊆X, whereC∞(U) is theC∞- +ring of smooth maps c:U→R, and ifV⊆U⊆Xare open we define +ρUV:C∞(U)→C∞(V) byρUV:c/ma√sto→c|V. +It is easyto verify that OXis a sheaf of C∞-ringsonX(not just a presheaf), +soX= (X,OX) is aC∞-ringed space. For each x∈X, the stalkOX,xis the +localC∞-ring of germs [( c,U)] of smooth functions c:X→Ratx∈X, as in +Example 2.15, with unique maximal ideal mX,x=/braceleftbig +[(c,U)]∈OX,x:c(x) = 0/bracerightbig +andOX,x/mX,x∼=R. HenceXis a localC∞-ringed space. +LetX,Ybe manifolds and f:X→Ya weakly smooth map. Define +(X,OX),(Y,OY) as above. For all open U⊆Ydefinef♯(U) :OY(U) = +C∞(U)→OX(f−1(U)) =C∞(f−1(U)) byf♯(U) :c/ma√sto→c◦ffor allc∈C∞(U). +Thenf♯(U) is a morphism of C∞-rings, and f♯:OY→f∗(OX) is a morphism +of sheaves of C∞-rings onY. Letf♯:f−1(OY)→OXcorrespond to f♯un- +der (4.3). Then f= (f,f♯) : (X,OX)→(Y,OY) is a morphism of (local) +C∞-ringed spaces. +As the category Topof topological spaces has all finite limits, and the con- +structionof C∞RSinvolvesTopinacovariantwayandthecategory C∞Rings +in a contravariant way, using Proposition 2.5 one may prove: +Proposition 4.11. All finite limits exist in the category C∞RS. +Dubuc [23, Prop. 7] proves: +25Proposition 4.12. The full subcategory LC∞RSof localC∞-ringed spaces in +C∞RSis closed under finite limits in C∞RS. +4.4 The spectrum functor +We now define a spectrum functor Spec :C∞Ringsop→LC∞RS. It is +equivalent to those constructed by Dubuc [22,23] and Moerdijk, v an Quˆ e and +Reyes [52,§3], but our presentation is closer to that of Hartshorne [31, p. 70]. +Definition 4.13. LetCbe aC∞-ring, and use the notation of Definition 2.13. +WriteXCfor the set of all R-pointsxofC. LetTCbe the topology on XC +generated by the basis of open sets Uc=/braceleftbig +x∈XC:x(c)/\e}atio\slash= 0/bracerightbig +for allc∈C. +For eachc∈Cdefinec∗:XC→Rto mapc∗:x/ma√sto→x(c). +Example 4.14. Suppose Cis a finitely generated C∞-ring, with exact sequence +0→I ֒→C∞(Rn)φ−→C→0. Define a map φ∗:XC→Rnbyφ∗:x/ma√sto→/parenleftbig +x◦φ(x1),...,x◦φ(xn)/parenrightbig +, wherex1,...,xnare the generators of C∞(Rn). Then +φ∗gives a homeomorphism +φ∗:XC∼=−→Xφ +C=/braceleftbig +(x1,...,xn)∈Rn:f(x1,...,xn) = 0 for all f∈I/bracerightbig +,(4.4) +where the right hand side is a closed subset of Rn. So the topological spaces +(XC,TC) for finitely generated Care homeomorphic to closed subsets of Rn. +Recall that a topological space Xisregularif whenever S⊆Xis closed and +x∈X\Sthen there exist open U,V⊆Xwithx∈U,S⊆VandU∩V=∅. +Lemma 4.15. In Definition 4.13,the topologyTCis also generated by the basis +of open sets c−1 +∗(V)for allc∈Cand openV⊆R. That is,TCis the weakest +topology on XCsuch thatc∗:XC→Ris continuous for all c∈C. Also +(XC,TC)is a Hausdorff, regular topological space. +Proof.Supposec∈CandV⊆Ris open. Then there exists smooth f:R→R +withV={x∈R:f(x)/\e}atio\slash= 0}. Setc′= Φf(c), using the C∞-ring operation +Φf:C→C. Thenc′ +∗=f◦c∗asc:C→Ris aC∞-ring morphism, so +Uc′= (c′ +∗)−1(R\{0}) = (f◦c∗)−1(R\{0}) =c−1 +∗[f−1(0)] =c−1 +∗(V). +Soc−1 +∗(V) is of the form Uc′. Conversely Uc=c−1 +∗(V) forV=R\{0}⊆R. So +the two given bases for TCare the same, proving the first part. +Letx,ybe distinct points of XC. Then there exists c∈Cwithx(c)/\e}atio\slash=y(c), +asx/\e}atio\slash=y. Setǫ=1 +2|x(c)−y(c)|>0 andU=c−1 +∗/parenleftbig +(x(c)−ǫ,x(c) +ǫ)/parenrightbig +, +V=c−1 +∗/parenleftbig +(y(c)−ǫ,y(c) +ǫ)/parenrightbig +. ThenU,V⊆XCare disjoint open sets with +x∈U,y∈V, so (XC,TC) is Hausdorff. +SupposeS⊆XCis closed, and x∈X\S. Then there exists c∈Cwithx∈ +Uc⊆XC\S, sinceXC\Sis open inXCand theUcare a basis forTC. Therefore +c∗(x)/\e}atio\slash= 0 andc∗|S= 0. Setǫ=1 +2|c∗(x)|>0,U=c−1 +∗/parenleftbig +(c∗(x)−ǫ,c∗(x) +ǫ)/parenrightbig +andV=c−1 +∗/parenleftbig +(−ǫ,ǫ)/parenrightbig +. ThenU,V⊆XCare disjoint open sets with x∈U, +S⊆V, so (XC,TC) is regular. +26Definition 4.16. LetCbe aC∞-ring, and XCthe topological space from +Definition4.13. Foreachopen U⊆XC, defineOXC(U) tobe thesetoffunctions +s:U→/coproducttext +x∈UCxwiths(x)∈Cxfor allx∈U, and such that Umay be covered +by open sets W⊆U⊆XCfor which there exist c∈Cwiths(x) =πx(c)∈Cx +for allx∈W. Define operations Φ fonOXC(U) pointwise in x∈Uusing the +operations Φ fonCx. This makesOXC(U) into aC∞-ring. IfV⊆U⊆XCare +open, the restriction map ρUV:OXC(U)→OXC(V) mappingρUV:s/ma√sto→s|Vis +a morphism of C∞-rings. +ClearlyOXCis a sheaf of C∞-rings onXC. Lemma 4.18 shows that the stalk +OXC,xatx∈XCisCx, which is a local C∞-ring. Hence ( XC,OXC) is a local +C∞-ringed space, which we call the spectrum ofC, and write as Spec C. +Now letφ:C→Dbe a morphism of C∞-rings. Define fφ:XD→ +XCbyfφ(x) =x◦φ. Thenfφis continuous. For U⊆XCopen define +(fφ)♯(U) :OXC(U)→OXD(f−1 +φ(U)) by (fφ)♯(U)s:x/ma√sto→φx(s(fφ(x))), where +φx:Cfφ(x)→Dxis the induced morphism of local C∞-rings. Then ( fφ)♯: +OXC→(fφ)∗(OXD) is a morphism of sheaves of C∞-rings onXC. Letf♯ +φ: +f−1 +φ(OXC)→OXDbe the corresponding morphism of sheaves of C∞-rings on +XDunder (4.3). Then fφ= (fφ,f♯ +φ) : (XD,OXD)→(XC,OXC) is a morphism +of localC∞-ringed spaces. Define Spec φ: SpecD→SpecCby Specφ=fφ. +Then Spec is a functor C∞Ringsop→LC∞RS, thespectrum functor . +Example 4.17. LetXbe a manifold. Then it followsfrom Theorem 4.41below +that the local C∞-ringed space Xconstructed in Example 4.10 is naturally +isomorphic to Spec C∞(X). +Lemma 4.18. In Definition 4.16,the stalkOXC,xofOXCatx∈XCis nat- +urally isomorphic to Cx. +Proof.Elements ofOXC,xare∼-equivalence classes [ U,s] of pairs (U,s), where +Uis an open neighbourhood of xinXCands∈OXC(U), and (U,s)∼(U′,s′) if +there exists open x∈V⊆U∩U′withs|V=s′|V. Define aC∞-ring morphism +Π :OXC,x→Cxby Π : [U,s]/ma√sto→s(x). +Supposecx∈Cx. Thencx=πx(c) for some c∈Cby Proposition 2.14. +The maps:XC→/coproducttext +x′∈XCCx′mappings:x′/ma√sto→πx′(c) lies inOXC(XC), and +Π : [XC,s]/ma√sto→πx(c) =cx. Hence Π :OXC,x→Cxis surjective. +Suppose [U,s]∈OXC,xwith Π([U,s]) = 0∈Cx. Ass∈OXC(U), there exist +openx∈V⊆Uandc∈Cwiths(x′) =πx′(c)∈Cx′for allx′∈V. Then +πx(c) =s(x) = Π([U,s]) = 0, soclies in the ideal Iin (2.2) by Proposition +2.14. Thus there exists d∈Cwithx(d)/\e}atio\slash= 0 inRandcd= 0 inC. Set +W={x′∈V:x′(d)/\e}atio\slash= 0}, so thatWis an open neighbourhood of xinU. If +x′∈Wthenx′(d)/\e}atio\slash= 0, soπx′(d) is invertible in Cx′. Thus +s(x′) =πx′(c) =πx′(c)πx′(d)πx′(d)−1=πx′(cd)πx′(d)−1=πx′(0)πx′(d)−1= 0. +Hence [U,s] = [W,s|W] = [W,0] = 0 inOXC,x, so Π :OXC,x→Cxis injective. +Thus Π :OXC,x→Cxis an isomorphism. +27Definition 4.19. Theglobal sections functor Γ :LC∞RS→C∞Ringsop +acts on objects ( X,OX) by Γ : (X,OX)/ma√sto→OX(X) and on morphisms ( f,f♯) : +(X,OX)→(Y,OY) by Γ : (f,f♯)/ma√sto→f♯(Y), forf♯:OY→f∗(OX) as in (4.3). +Then Γ◦Spec is a functor C∞Ringsop→C∞Ringsop, or equivalently a +functorC∞Rings→C∞Rings. For eachC∞-ringCandc∈C, define Ψ C(c) : +XC→/coproducttext +x∈XCCxby ΨC(c) :x/ma√sto→πx(c)∈Cx. Then Ψ C(c)∈OXC(XC) = +Γ◦SpecCby Definition 4.16, soΨ C:C→Γ◦SpecCis amap. Since πx:C→Cx +is aC∞-ring morphism and the C∞-ring operations on OXC(XC) are defined +pointwise in the Cx, this Ψ Cis aC∞-ring morphism. It is functorial in C, so +that the Ψ Cfor allCdefine a natural transformation Ψ : id C∞Rings⇒Γ◦Spec +of functors id C∞Rings,Γ◦Spec :C∞Rings→C∞Rings. +Theorem 4.20. The functor Spec :C∞Ringsop→LC∞RSisright adjoint +toΓ :LC∞RS→C∞Ringsop. That is, for all C∈C∞RingsandX∈ +LC∞RSthere are inverse bijections +HomC∞Rings(C,Γ(X))LC,X/d47/d47HomLC∞RS(X,SpecC), +RC,X/d111/d111 (4.5) +which are functorial in the sense that if λ:C→Dis a morphism in C∞Rings +ande:X→Ya morphism in LC∞RSthen the following commutes: +HomC∞Rings(D,Γ(Y))LD,Y/d47/d47 +φ/mapsto→Γ(e)◦φ◦λ/d15/d15HomLC∞RS(Y,SpecD) +RD,Y/d111/d111 +f/mapsto→Specλ◦f◦e/d15/d15 +HomC∞Rings(C,Γ(X))LC,X/d47/d47HomLC∞RS(X,SpecC). +RC,X/d111/d111(4.6) +WhenX= SpecCwe have ΨC=RC,X(idX),so thatΨCis the unit of the +adjunction between ΓandSpec. +Proof.LetC∈C∞RingsandX∈LC∞RS. WriteY= (Y,OY) = Spec C. +DefineRC,Xin (4.5) by, for each morphism f:X→YinLC∞RS, taking +RC,X(f) :C→Γ(X) to be the composition +CΨC/d47/d47Γ◦SpecC= Γ(Y)Γ(f)/d47/d47Γ(X). (4.7) +For the last part, if X= SpecCthen Ψ C=RC,X(idX) as Γ(idX) = idΓ(X). +Letφ:C→Γ(X) be a morphism in C∞Rings. We will construct a +morphismg= (g,g♯) :X→YinLC∞RS, and setLC,X(φ) =g. For any +x∈Xwe have an R-algebra morphism x∗: Γ(X)→Rby composing the stalk +mapσx: Γ(X)→OX,xwith the unique morphism π:OX,x→R, asOX,xis a +localC∞-ring. Then x∗◦φ:C→RisanR-algebramorphism, andhenceapoint +ofY. Defineg:X→Ybyg(x) =x∗◦φ. Ifc∈CthenUc={y∈Y:y(c)/\e}atio\slash= 0} +is open inY, andg−1(Uc) ={x∈X:x∗(φ(c))/\e}atio\slash= 0}is open inX, asx/ma√sto→x∗(d) +is a continuous map X→Rfor anyd∈Γ(X). Since the Ucforc∈Care a +basis for the topology of Yby Definition 4.13, g:X→Yis continuous. +28Letx∈Xwithg(x) =y∈Y. Consider the diagram of C∞-rings +C +πy/d15/d15φ/d47/d47Γ(X) +σx/d15/d15 +Cy∼=OY,yφx/d47/d47OX,x.(4.8) +HereCy∼=OY,yby Lemma 4.18. If c∈Cwithy(c)/\e}atio\slash= 0 thenσx◦φ(c)∈OX,x +withπ[σx◦φ(c)]/\e}atio\slash= 0, soσx◦φ(c) is invertible inOX,xasOX,xis a localC∞- +ring. Thus by the universal property of πy:C→Cythere is a unique morphism +φx:OY,y→OX,xmaking (4.8) commute. +For each open V⊆YwithU=g−1(V)⊆X, defineg♯(V) :OY(V)→ +g∗(OX)(V) =OX(U) byg♯(V)s:x/ma√sto→φx(s(g(x))) fors∈OY(V) andx∈U⊆ +X, so thatg(x)∈V, ands(g(x))∈OY,g(x), andφx(s(g(x)))∈OX,x. Here as +OXisasheafwemayidentify elementsof OX(U)withmaps t:U→/coproducttext +x∈UOX,x +witht(x)∈OX,xforx∈U, such that tsatisfies certain local conditions in U. +Ifs∈OY(V) andx∈U⊆Xwithg(x) =y∈V⊆Y, then by Definition 4.16 +there is an open neighbourhood WyofyinVandc∈Cwiths(y′) =πy′(c)∈ +Cy′∼=OY,y′for ally′∈Wy. Therefore g♯(V)smapsx′/ma√sto→σx′(φ(c)) for allx′in +the open neighbourhood g−1(Wy) ofxinU, by (4.8). Since the open subsets +g−1(Wy) coverU,g♯(V)sis a section ofOX|U, andg♯(V) is well defined. +As theφxareC∞-ring morphisms, this defines a morphism g♯:OY→ +g∗(OX) of sheaves of C∞-rings onY. Letg♯:g−1(OY)→OXbe the corre- +sponding morphism of sheaves on Xunder (4.3). The stalk g♯ +x:OY,y→OX,x +ofg♯atx∈Xwithg(x) =y∈Yisg♯ +x=φx. Theng= (g,g♯) is a morphism +inLC∞RS. SetLC,X(φ) =g. This defines LC,Xin (4.5). +Forφ,gas above,c∈C, andx∈Xwithg(x) =y=x∗◦φ∈Y, we have +σx/bracketleftbig/parenleftbig +RC,X◦LC,X(φ)/parenrightbig +(c)/bracketrightbig +=σx/bracketleftbig +Γ(g)◦ΨC(c)/bracketrightbig +=g♯ +x◦σy[ΨC(c)] +=φx◦σy[ΨC(c)] =φx◦πy(c) =σx◦φ(c), +usingLC,X(φ) =gand the definition (4.7) of RC,X(g) in the first step, σx◦ +Γ(g) =g♯ +x◦σy: Γ(Y)→OX,xin the second, g♯ +x=φxin the third, σy◦ΨC=πy +as maps C→OY,y∼=Cyin the fourth, and (4.8) in the fifth. As/producttext +x∈Xσx: +Γ(X)→/producttext +x∈XOX,xis injective, this implies that/parenleftbig +RC,X◦LC,X(φ)/parenrightbig +(c) =φ(c) +for allc∈C, soRC,X◦LC,X(φ) =φ, andRC,X◦LC,X= id. +Supposef:X→Yis a morphism in LC∞RS, and setφ=RC,X(f) and +g=LC,X(φ). Letx∈Xwithf(x) =y∈Y. Then we have a commutative +diagram in C∞Rings +C +y +/d38/d38◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆φ +/d46/d46 +πy/d15/d15ΨC/d47/d47Γ◦SpecC= Γ(Y) +σy/d15/d15Γ(f)/d47/d47Γ(X) +σx/d15/d15x∗ +/d119/d119♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ +Cy +π/d43/d43❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲∼=/d47/d47OY,y +π +/d15/d15f♯ +x/d47/d47OX,x +π +/d115/d115❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ +R,(4.9) +29where the isomorphism Cy∼=OY,ycomes from Lemma 4.18. Since g(x) = +x∗◦φ:C→R, this proves that g(x) =y=f(x), sof=g. Also by definition +the stalkg♯ +x:OY,y→OX,xisφxin (4.8), so comparing (4.8) and (4.9) and +usingπy:C→Cysurjective by Proposition 2.14 shows that f♯ +x=g♯ +x. As +this holds for all x∈Xwe havef♯=g♯, sof= (f,f♯) = (g,g♯) =g. Thus +LC,X◦RC,X(f) =ffor allf:X→Y, soLC,X◦RC,X= id. Therefore +LC,X,RC,Xin (4.5) are inverse bijections. +It is easy to see that the rectangle in (4.6) involving RD,Y,RC,Xcommutes +using (4.7) and functoriality of the Ψ Cand Γ. Then the rectangle involving +LD,Y,LC,Xcommutes as LD,Y=R−1 +D,YandLC,X=R−1 +C,X. So (4.6) commutes. +This completes the proof. +Remark 4.21. (a) The fact in Theorem 4.20 that Spec : C∞Ringsop→ +LC∞RSis right adjoint to Γ : LC∞RS→C∞Ringsopdetermines Spec +uniquely up to natural isomorphism, by properties of adjoint funct ors. +Dubuc [23] and Moerdijk, van Quˆ e and Reyes [52, §3] both prove the ex- +istence of a right adjoint to Γ : LC∞RS→C∞Ringsop, which is therefore +naturally isomorphic to our functor Spec in Definition 4.16. But they s how Spec +exists by category theory, without constructing it explicitly as we d o. +Moerdijk et al. [52, §3] call our functor Spec the Archimedean spectrum . +They also give a nonequivalent definition [52, §1] of the spectrum Spec C, in +which the points are not R-points, but ‘ C∞-radical prime ideals’. +(b)Since Spec is a right adjoint functor, it preserves limits, as in [23, p. 687]. +Equivalently, Spec takes colimits in C∞Ringsto limits in LC∞RS. So, for +example, a pushout C=D∐FEof morphisms φ:F→D,ψ:F→Ein +C∞Ringsis mapped to a fibre product Spec C∼=SpecD×SpecFSpecEof +morphisms Spec φ: SpecD→SpecF, Specψ: SpecE→SpecFinLC∞RS. +Here are some properties of finitely generated and fair C∞-rings, due to +Dubuc [23, Th. 13]. The reflection functor Rfa +fgis as in Definition 2.20. +Theorem 4.22. (a) IfCis a finitely generated C∞-ring, there is a natural +isomorphism Γ◦SpecC∼=Rfa +fg(C),which identifies ΨC:C→Γ(SpecC)with the +natural surjective projection C→Rfa +fg(C). +These isomorphisms for all Cform a natural isomorphism Rfa +fg∼=Γ◦Spec +of functors Rfa +fg,Γ◦Spec :C∞Ringsfg→C∞Ringsfa. +Hence, if Cis fair then ΨC:C→Γ(SpecC)∼=Rfa +fg(C)is an isomorphism. +(b)IfCis finitely generated then SpecΨ C: SpecC→SpecΓ(Spec C)∼= +SpecRfa +fg(C)is an isomorphism in LC∞RS. +(c)The functor Spec|···: (C∞Ringsfa)op→LC∞RSis full and faithful, and +takes finite limits in (C∞Ringsfa)opto finite limits in LC∞RS. +To see that Spec is full and faithful on ( C∞Ringsfa)opin (c), let C,Dbe +fairC∞-rings. Then putting X= SpecDin (4.5) and using D∼=Γ◦SpecDby +(a) shows that the following is a bijection. +Spec : Hom C∞Rings(C,D)−→HomLC∞RS(SpecD,SpecC). +30Note that Spec is neither full nor faithful on ( C∞Ringsfg)oporC∞Ringsop. +This is a contrast to conventional algebraic geometry, where Γ(Sp ecR)∼=Rfor +arbitrary rings R, as in [31, Prop. II.2.2], so that Spec is full and faithful. In +§4.6 we will generalize Theorem 4.22 to non-finitely-generated C∞-rings. +4.5 Affine C∞-schemes and C∞-schemes +As for the usual definitions of affine schemes and schemes, we defin e: +Definition 4.23. A localC∞-ringed space Xis called an affineC∞-scheme +if it is isomorphic in LC∞RSto SpecCfor someC∞-ringC. We callXa +finitely presented , orfair, affineC∞-scheme ifX∼=SpecCforCthat kind of +C∞-ring. Write AC∞Sch,AC∞Schfp,AC∞Schfafor the full subcategories +of affineC∞-schemes and of finitely presented, and fair, affine C∞-schemes in +LC∞RSrespectively. +We do not define finitely generated affineC∞-schemes, because Theorem +4.22(b) implies that they coincide with fair affine C∞-schemes. +LetX= (X,OX) be a local C∞-ringed space. We call XaC∞-schemeif +Xcan be covered by open sets U⊆Xsuch that ( U,OX|U) is an affine C∞- +scheme. We call a C∞-schemeXlocally fair , orlocally finitely presented , ifX +can be covered by open U⊆Xwith (U,OX|U) a fair, or finitely presented, +affineC∞-scheme, respectively. +We call aC∞-schemeXHausdorff ,second countable ,Lindel¨ of,compact, +locally compact ,paracompact ,metrizable ,regular, orseparable , if the topological +spaceXis. AffineC∞-schemes are Hausdorff and regular by Lemma 4.15. +WriteC∞Schlf,C∞Schlfp,C∞Schfor the full subcategories in LC∞RS +of locally fair C∞-schemes, locally finitely presented C∞-schemes, and all C∞- +schemes, respectively. +Remark 4.24. Ordinary schemes are a much larger class than ordinary affine +schemes, and central examples such as CPnare not affine schemes. However, +affineC∞-schemes are already general enough for many purposes. For ex ample, +all second countable, metrizable C∞-schemes are affine, as in §4.8, including +manifolds and manifolds with corners. Affine C∞-schemes are Hausdorff and +regular, so any non-Hausdorff or non-regular C∞-scheme is not affine. +For the next theorem, part (a) follows from Propositions 2.5, 2.24 a nd +2.26, Remark 4.21(b), and Theorem 4.22(c). Part (b) holds as finite lim- +its inC∞Schlfp,C∞Schlf,C∞Schare locally modelled on finite limits in +AC∞Schfp,AC∞SchfaandAC∞Sch. +Theorem 4.25. (a) The full subcategories AC∞Schfp,AC∞Schfa,AC∞Sch +are closed under all finite limits in LC∞RS. Hence, fibre products and all finite +limits exist in each of these subcategories. +(b)The full subcategories C∞Schlfp,C∞SchlfandC∞Schare closed under +all finite limits in LC∞RS. Hence, fibre products and all finite limits exist in +each of these subcategories. +31Definition 4.26. Define functors +FC∞Sch +Man:Man−→AC∞Schfp⊂AC∞Sch, +FC∞Sch +Manb:Manb−→AC∞Schfa⊂AC∞Sch, +FC∞Sch +Manc:Manc−→AC∞Schfa⊂AC∞Sch, +byFC∞Sch +Man∗= Spec◦FC∞Rings +Man∗, in the notation of Definitions 3.2 and 4.16. +By Example 4.17, if Xis a manifold with corners then FC∞Sch +Manc(X) is nat- +urally isomorphic to the local C∞-ringed space Xin Example 4.10. +IfX,Y,... are manifolds, or f,g,...are (weakly) smooth maps, we may use +X,Y,...,f,g,...to denote the images of X,Y,...,f,g,... underFC∞Sch +Manc. So +for instance we will write Rnand [0,∞)forFC∞Sch +Man(Rn) andFC∞Sch +Manb/parenleftbig +[0,∞)/parenrightbig +. +Our categories of spaces so far are related as follows: +Man +FC∞Sch +Man/d15/d15⊂/d47/d47Manb +FC∞Sch +Manb/d15/d15⊂/d47/d47Manc +FC∞Sch +Manc/d118/d118♥♥♥♥♥♥♥♥♥♥ +AC∞Schfp +⊂/d47/d47 +⊂/d15/d15AC∞Schfa +⊂/d47/d47 +⊂/d15/d15AC∞Sch +⊂/d15/d15⊂ +/d39/d39❖❖❖❖❖❖❖❖❖❖ +C∞Schlfp⊂/d47/d47C∞Schlf⊂/d47/d47C∞Sch⊂/d47/d47LC∞RS⊂/d47/d47C∞RS. +By Corollary 3.4 and Theorems 3.5 and 4.22(c), we find as in [23, Th. 16]: +Corollary 4.27. FC∞Sch +Man:Man֒→AC∞Schfp⊂AC∞Schis a full and +faithful functor, and FC∞Sch +Manb:Manb→AC∞Schfa⊂AC∞Sch, FC∞Sch +Manc: +Manc→AC∞Schfa⊂AC∞Schare both faithful functors, but are not full. +Also these functors take transverse fibre products in Man,Mancto fibre prod- +ucts inAC∞Schfp,AC∞Schfa. +We study open subspaces of C∞-schemes. The definition of Spec Cimplies: +Lemma 4.28. LetCbe aC∞-ring, andc∈C. WriteSpecC= (X,OX)and +Uc={x∈X:x(c)/\e}atio\slash= 0}. ThenUc⊆Xis open with (Uc,OX|Uc)∼=SpecC[c−1]. +Corollary 4.29. LetX= (X,OX)be aC∞-scheme and V⊆Xbe open. +ThenV= (V,OX|V)is also aC∞-scheme. +Proof.Letx∈V. Then there exists an open x∈Y⊆XwithY∼=SpecCfor +someC∞-ringC, asXas aC∞-scheme. Identify Ywith Spec C. AsV∩Yis +open inY=XC, and the topology on XCis generated by subsets Uc={˜x∈ +XC: ˜x(c)/\e}atio\slash= 0}forc∈C, there exists c∈Csuch thatx∈Uc⊆V∩Y. Then +(Uc,OX|Uc)∼=SpecC[c−1] by Lemma 4.28. So every x∈Vhas an affine open +neighbourhood, and Vis aC∞-scheme. +Lemma 4.30. LetCbe a finitely generated C∞-ring and (X,OX) = Spec C. +SupposeV⊆Xis open. Then there exists c∈CwithV={x∈X:x(c)/\e}atio\slash= 0}. +We callcacharacteristic function forV. +32Proof.AsCis a finitely generated C∞-ring it fits into an exact sequence 0 → +I ֒→C∞(Rn)φ−→C→0. Example 4.14 gives a homeomorphism φ∗:X→Xφ +C +with a closed subset Xφ +CinRngiven in (4.4). Then φ∗(V) is open in Xφ +C, so +there exists an open U⊆RnwithU∩Xφ +C=φ∗(V). By [54, Lem. I.1.4] there +existsf∈C∞(Rn) withU=/braceleftbig +x∈Rn:f(x)/\e}atio\slash= 0/bracerightbig +. Thenc=φ(f)∈Cis a +characteristic function for V. +Example 4.31. LetIbe an infinite set, and write C∞(RI) for the free C∞- +ring with generators xifori∈I. ThenX= SpecC∞(RI) has topological space +X=RIwith points ( xi)i∈Iforxi∈R. Elements of C∞(RI) are functions +c:RI→Rdepending only on xjforjin afinitesubsetJ⊆I, and which are +smooth functions of these xj,j∈J. +LetV=RI\{0}. ThenVis open inX. But no characteristic function c +exists forVinC∞(RI), sincecwould depend only on xjforjin a finite subset +J⊆I, butVdepends on xifor alli∈I. Thus, infinitely generated C∞-rings +need not admit characteristic functions, in contrast to Lemma 4.30 . +IfCis a finitely generated (or finitely presented) C∞-ring andc∈Cthen +C[c−1] is also finitely generated (or finitely presented), since C[c−1]∼=C[x]/(c· +x−1) is the result of adding one extra generator and one extra relatio n toC. +Thus from Lemmas 4.28 and 4.30 we deduce: +Corollary 4.32. (a) Let(X,OX)be a fair (or finitely presented) affine C∞- +scheme, and U⊆Xbe an open subset. Then (U,OX|U)is also a fair (or +finitely presented) affine C∞-scheme. +(b)Let(X,OX)be a locally fair (or locally finitely presented) C∞-scheme, and +U⊆Xbe an open subset. Then (U,OX|U)is also a locally fair (or locally +finitely presented) C∞-scheme. +Our next result describes the sheaf of C∞-ringsOXin SpecCforCa finitely +generatedC∞-ring. It is a version of [31, Prop. I.2.2(b)] in algebraic geometry, +and reduces to Moerdijk and Reyes [54, Prop. I.1.6] when C=C∞(Rn). +Proposition 4.33. LetCbe a finitely generated C∞-ring, write (X,OX) = +SpecC,and letU⊆Xbe open. By Lemma 4.30we may choose a character- +istic function c∈CforU. Then there is a canonical isomorphism OX(U)∼= +Rfa +fg(C[c−1]),in the notation of Definitions 2.13and2.20. IfCis finitely pre- +sented thenOX(U)∼=C[c−1]. +Proof.We have morphisms of C∞-ringsc∗:C∞(R)→Candi∗:C∞(R)→ +C∞(R\{0}),andC∞(R),C∞(R\{0})arefinitelypresented C∞-ringsbyPropo- +sition 3.1(a). So as Spec preserves limits in ( C∞Ringsfg)opwe have +Spec/parenleftbig +C∐c∗,C∞(R),i∗C∞(R\{0})/parenrightbig∼=SpecC×f,R,iR\{0}∼=(U,OX|U). +ButC∐C∞(R)C∞(R\{0})∼=C[c−1] for formal reasons. Thus Theorem 4.22(a) +givesOX(U)∼=Γ/parenleftbig +(U,OX|U)/parenrightbig∼=Rfa +fg(C[c−1]). IfCis finitely presented then +C[c−1] is too, as in Corollary 4.32, so C[c−1] is fair and Rfa +fg/parenleftbig +C[c−1]/parenrightbig +=C[c−1], +and thereforeOX(U)∼=C[c−1]. +334.6 Complete C∞-rings +The material of this section appears to be new. +Proposition 4.34. LetCbe aC∞-ring, and ΨCbe as in Definition 4.19. Then +SpecΨ C: Spec◦Γ◦SpecC→SpecCis an isomorphism in LC∞RS. +Proof.WriteD= Γ◦SpecC,X= SpecC,Y= SpecD, andf= SpecΨ C:Y→ +X. Letx∈X, and define y=π◦Πx:D→Rto be the composition of the +projection Π x:D→Cx, noting that D⊆/producttext +˜x∈XC˜xby Definition 4.19, and the +unique morphism π:Cx→R, asCxis a localC∞-ring. Then f(y) =π◦πx= +x:C→Rforπx:C→Cx, sof:Y→Xis surjective. +Suppose now that y∈Ywithf(y) =x, so thaty:D→Ris anR-algebra +morphism. We will prove that y=π◦Πxas above. Let d∈D. By definition of +D=OXC(XC) there exist an open neighbourhood WofxinXandc1∈Csuch +thatd(˜x) =π˜x(c1) inC˜xfor all ˜x∈W. By definition of the topology TC, there +existsc2∈Csuch thatUc2={˜x∈X: ˜x(c2)/\e}atio\slash= 0}is an open neighbourhood of +xinW⊆X. Hencex(c2)/\e}atio\slash= 0 and ˜x(c2) = 0 for all ˜ x∈X\W. +Choose smooth functions g,h:R→Rwithg(x(c2)) = 1 and g= 0 in an +open neighbourhood ( −ǫ,ǫ) of 0 in R, andh(0)/\e}atio\slash= 0 andh= 0 outside (−ǫ,ǫ), +so thatg·h= 0. Setc3= Φg(c2) andc4= Φh(c2), with Φ g,Φh:C→Cthe +C∞-ring operations. Then x(c3) = 1, andπ˜x(c3) = 0 inC˜xfor all ˜x∈X\W, as +π˜x(c3)·π˜x(c4) =π˜x/parenleftbig +Φg(c2)·Φh(c2)/parenrightbig +=π˜x◦Φgh(c2) =π˜x◦Φ0(c2) = 0, +butπ˜x(c4) is invertible in C˜xas ˜x(c4) =h(˜x(c2)) =h(0)/\e}atio\slash= 0. Thus we have +d·ΨC(c3) = ΨC(c1)·ΨC(c3) = ΨC(c1·c3) inD, asd(˜x) = ΨC(c1)˜xfor all ˜x∈W, +and Ψ C(c3)˜x= 0 for all ˜x∈X\W. Therefore +y(d) =y(d)·1 =y(d)·x(c3) =y(d)·y(ΨC(c3)) =y/parenleftbig +d·ΨC(c3)/parenrightbig +=y/parenleftbig +ΨC(c1·c3)/parenrightbig +=x(c1·c3)=x(c1)·x(c3)=/parenleftbig +π◦Πx(d)/parenrightbig +·1=π◦Πx(d). +As this holds for all d∈D, we see that y∈Ywithf(y) =ximplies that +y=π◦Πx. Hencef:Y→Xis injective, and so bijective. +From above f:Y→Xis continuous. To show f−1:X→Yis continuous, +note that the topology on Yis generated by the basis of open sets Vd={y∈ +Y:y(d)/\e}atio\slash= 0}for alld∈D. So it is enough to show that f(Vd) ={x∈X: +π◦Πx(d) = 0}is open inXfor alld. For fixed d, by definition we may cover +Xby openW⊆Xfor which there exist c∈Cwithd(x) =πx(c)∈Cxfor all +x∈W. But then W∩f(Vd) =W∩Uc, whereUc={x∈X:x(c)/\e}atio\slash= 0}is open +inX. So we can cover Xby openW⊆XwithW∩f(Vd) open, and f(Vd) is +open. Therefore f−1is continuous, and f:Y→Xis a homeomorphism. +Lety∈Ywithf(y) =x. Taking stalks of f♯:f−1(OX)→OYatygives a +morphismf♯ +y:OX,x→OY,y, whereOX,x∼=CxandOY,y∼=Dyby Lemma 4.18, +and we have a commutative diagram +C +πx/d15/d15ΨC/d47/d47D +πy/d15/d15Πx/d114/d114❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ +Cx∼=OX,xΨC,x∼=f♯ +y/d47/d47OY,y∼=Dy.(4.10) +34Here the outer rectangle and top left triangle obviously commute. T o see that +the bottom right triangle commutes, we use that any d∈D=OXC(XC) has +d(˜x) = Ψ C(c)˜xfor somec∈Cand all ˜xin an open neighbourhood Wofxin +X. As in the first part of the proof, we can find c3∈Cwithx(c3) = 1 and +π˜x(c3) = 0 inC˜xfor all ˜x∈X\W. Then evaluating at ˜ x∈Wand ˜x∈X\W +we see that Ψ C(c)·ΨC(c3) =d·ΨC(c3), which forces πy(d) =πy(ΨC(c)), since +πy◦ΨC(c3) is invertible in Dyasπ◦πy◦ΨC(c3) =x(c3) = 1>0. Thus +πy(d) =πy◦ΨC(c) =f♯ +y◦πx(c) =f♯ +y◦Πx◦ΨC(c) =f♯ +y◦Πx(d). +Sinceπy:D→Dyis surjective by Proposition 2.14, the bottom right +trianglein(4.10)impliesthat f♯ +y:OX,x→OY,yissurjective. Suppose cx∈OX,x +withf♯ +y(cx) = 0 inOY,y. Asπxis surjective by Proposition 2.14 we may +writecx=πx(c) forc∈C. Thenπy◦ΨC(c) =f♯ +y◦πx(c) =f♯ +y(cx) = 0, so +ΨC(c)∈Kerπy. WriteI⊂CandJ⊂Dfor the ideals in (2.2) for x,y. Then +J= Kerπy, so ΨC(c)∈J, and thus there exists d∈Dwithy(d) =π◦Πx(d)/\e}atio\slash= 0 +inRand Ψ C(c)·d= 0 inD. Applying Π xgives +cx·Πx(d) =πx(c)·Πx(d) = Πx(ΨC(c))·Πx(d) = Πx(ΨC(c)·d) = Πx(0) = 0. +But Πx(d) is invertible in Cxasπ◦Πx(d)/\e}atio\slash= 0 inR, socx= 0. Thusf♯ +y:OX,x→ +OY,yis injective, and so an isomorphism. +We have shown that f:Y→Xis a homeomorphism, and f♯ +y:OX,f(y)→ +OY,yis an isomorphism on stalks at all y∈Y. Hence SpecΨ C= (f,f♯) is an +isomorphism in LC∞RS, as we have to prove. +Definition 4.35. We call aC∞-ringCcomplete if the morphism Ψ C:C→ +Γ◦SpecCin Definition 4.19 is an isomorphism. Write C∞Ringscofor the full +subcategory of complete C∞-ringsCinC∞Rings. +IfCis anyC∞-ring, applying Γ to SpecΨ Cin Proposition 4.34 shows that +Γ◦SpecΨ C= ΨΓ◦SpecC: Γ◦SpecC−→Γ◦Spec(Γ◦SpecC) +is an isomorphism in C∞Rings, where we check that Γ ◦SpecΨ C= ΨΓ◦SpecC +from Definitions 4.16 and 4.19. Hence Γ ◦SpecCis a complete C∞-ring. Define +a functorRco +all:C∞Rings→C∞RingscobyRco +all= Γ◦Spec. +The next result extends Definition 2.20 and Theorem 4.22 from C∞Ringsfa +⊂C∞RingsfgtoC∞Ringsco⊂C∞Rings. +Theorem 4.36. (a) LetXbe an affine C∞-scheme. Then X∼=SpecOX(X), +whereOX(X)is a complete C∞-ring. +(b)Spec|(C∞Ringsco)op: (C∞Ringsco)op→LC∞RSis full and faithful, and +an equivalence of categories Spec|···: (C∞Ringsco)op→AC∞Sch. +(c)Rco +all:C∞Rings→C∞Ringscois left adjoint to the inclusion functor +inc :C∞Ringsco֒→C∞Rings. That is,Rco +allis areflection functor . +(d)All small colimits exist in C∞Ringsco,although they may not coincide with +the corresponding small colimits in C∞Rings. +35(e)Spec|(C∞Ringsco)op= Spec◦inc : (C∞Ringsco)op→LC∞RSis right +adjoint toRco +all◦Γ :LC∞RS→(C∞Ringsco)op. ThusSpec|···takes limits in +(C∞Ringsco)op(equivalently, colimits in C∞Ringsco) to limits in LC∞RS. +Proof.For (a), ifXis an affine C∞-scheme then X∼=SpecCfor someC∞-ring +C, soOX(X)∼=Γ◦SpecC, and thus X∼=SpecOX(X) by Proposition 4.34. +Also, applying Γ to SpecΨ Cin Proposition 4.34 shows that +Γ◦SpecΨ C= ΨΓ◦SpecC: Γ◦SpecC−→Γ◦Spec(Γ◦SpecC) +is an isomorphism in C∞Rings, where Γ◦SpecΨ C= ΨΓ◦SpecCfollows from +the definitions. Hence Γ ◦SpecC∼=OX(X) is complete, proving (a). +For (b), if C,Dare complete C∞-ringsthen putting X= SpecDin Theorem +4.20 and using Γ ◦SpecD∼=D, equation (4.5) shows that +Spec =LC,X: Hom C∞Rings(C,D)−→HomLC∞RS(SpecD,SpecC) +is a bijection, where the definition of LC,Xagrees with the definition of Spec on +morphisms in this case. Thus Spec is full and faithful on complete C∞-rings. +Therefore Spec|···: (C∞Ringsco)op→LC∞RSis an equivalence of categories +from (C∞Ringsco)opto its essential image in LC∞RS, which is AC∞Sch. +For (c), let C,DbeC∞-rings with Dcomplete. Then we have bijections +HomC∞Ringsco/parenleftbig +Rco +all(C),D/parenrightbig∼=HomC∞Rings/parenleftbig +Γ◦SpecC,Γ◦SpecD/parenrightbig +∼=HomLC∞RS/parenleftbig +SpecD,Spec◦Γ◦SpecC/parenrightbig∼=HomLC∞RS/parenleftbig +SpecD,SpecC/parenrightbig +∼=HomC∞Rings/parenleftbig +C,Γ◦SpecD/parenrightbig∼=HomC∞Rings/parenleftbig +C,D/parenrightbig += Hom C∞Rings/parenleftbig +C,inc(D)/parenrightbig +, (4.11) +usingD∼=Γ◦SpecDasDis complete in the first and fifth steps, Theorem +4.20 in the second and fourth, and Proposition 4.34 in the third. The b ijections +(4.11) are functorial in C,Das each step is. Hence Rco +allis left adjoint to inc. +For (d), note that Rco +all:C∞Rings→C∞Ringscotakes colimits to colim- +its, asit isaleft adjointfunctor by(a). Sogivenafunctor F:J→C∞Ringsco +forJa small category, we may take the colimit C= colim JFinC∞Rings, +which exists by Proposition 2.5, and then D=Rco +all(C) is the colimit of Rco +all◦F +inC∞Ringsco. ButRco +all◦F∼=FasRco +all|C∞Ringsco∼=id. Hence D= colim JF +inC∞Ringsco, and all small colimits exist in C∞Ringsco. In Example 2.25, +the colimits in C∞RingscoandC∞Ringsare different. +The first part of (e) holds by composing (c) and Theorem 4.20, and t he +second part follows as right adjoint functors preserve limits. This c ompletes the +proof of Theorem 4.36. +Remark 4.37. LetCbe aC∞-ring, so that Ψ C:C→Rco +all(C) is a morphism of +C∞-rings. If Cis finitely generated then Theorem 4.22(a) gives an isomorphism +Rco +all(C)∼=Rfa +fg(C) identifying Ψ Cwith the surjective projection π:C→Rfa +fg(C), +forRfa +fgas in Definition 2.20. Thus Ψ C:C→Rco +all(C) is surjective in this case, +andRco +all,Rfa +fgagree on finitely generated C∞-rings up to natural isomorphism. +36ForCinfinitely generated, Ψ C:C→Rco +all(C) need not be surjective, and +Rco +all(C) can be much larger than C. For example, if Iis an infinite set and +C=C∞(RI) is as in Example 4.31, then elements of Care functions c:RI→R +which depend smoothly only on xjforjin a finite subset J⊆I, but elements +ofRco +all(C) are functions c:RI→Rwhichlocally in RIdepend smoothly only +onxjforjin a finite subset J⊆I, but globally may depend on xifor infinitely +manyi∈I. So Ψ C:C→Rco +all(C) is injective but not surjective. +4.7 Partitions of unity +We now study the existence of smooth partitions on unity on C∞-schemes and +localC∞-ringed spaces. We will need the next definition. +Definition 4.38. LetX= (X,OX) be a local C∞-ringed space. Then each +c∈OX(X) defines a continuous map c∗:X→Rmappingx/ma√sto→π◦πx(c), for +πx:OX(X)→OX,xandπ:OX,x→Rthe natural C∞-ring morphisms. Thus +Uc={x∈X:c∗(x)/\e}atio\slash= 0}is open inX. We say that the topology on Xis +smoothly generated if{Uc:c∈OX(X)}is a basis for the topology on X. +This implies Xis a regular (and completely regular) topological space. +Example 4.39. (a) LetXbeacompletelyregulartopologicalspace, anddefine +a sheaf ofC∞-ringsOXonXby takingOX(U) =C0(U) to be the C∞-ring of +continuous functions c:U→Rfor all open U⊆X. ThenX= (X,OX) is a +localC∞-ringed space, and the topology on Xis smoothly generated. +(b)LetXbe an affine C∞-scheme. Then X∼=SpecOX(X) by Theorem +4.36(a). So the definition of the topology on Xin Definition 4.13 implies that +the topology on Xis smoothly generated. +(c)SupposeXis a regular C∞-scheme, and let T⊆Xbe open and x∈T. +Thenxhas an affine open neighbourhood YinX. SinceXis regular, there +exist disjoint open neighbourhoods VofxandWofX\YinX. +Thenx∈T∩V⊆Y, and the topology on Yis smoothly generated by (b), +so there exists a∈OY(Y) withx∈UY +a⊆T∩V. Nowa∗(x)/\e}atio\slash= 0 anda∗(y) = 0 +for ally∈Y\UY +a, but this does not imply that ais supported in UY +a, as we +could have πy(a)/\e}atio\slash= 0 inOY,yeven though π◦πy(a) = 0 in R. Choose smooth +f:R→Rwithf(a∗(x))/\e}atio\slash= 0 andf(t) = 0 fortin an open neighbourhood of 0 +inR. Setb= Φf(a), for Φf:OY(Y)→OY(Y) theC∞-ring operation. +Thenb∗(x)/\e}atio\slash= 0, andUY +b⊆UY +a⊆T, andbis supported in UY +a⊆V⊆Y. +SinceWis open inXwithX\Y⊆W⊆Y\V, there exists a unique c∈OX(X) +withc|Y=bandc|W= 0. We have x∈UX +c=UY +b⊆T. Thus, for each open +T⊆Xandx∈Twe can find c∈OX(X) withx∈UX +c⊆T. So the topology +onXis smoothly generated. +(d)LetXbe an infinite-dimensional Banach space or Banach manifold, and +makeXinto a local C∞-ringed space X= (X,OX) as in Example 4.10. The +question of when the topology of Xis smoothly generated (framed in terms +of the existence of ‘smooth bump functions’ on X) is very well understood, as +in Bonic and Frampton [10] and Deville, Godefroy and Zizler [18, §V]. For +37example, if Xis a Hilbert manifold, or modelled on Lq(Y) orℓqfor evenq/greaterorequalslant2, +then the topology on Xis smoothly generated, but if Xis modelled on Lq(Y) +orℓqforq∈[1,∞] not even, the topology on Xis not smoothly generated. +For the next theorem, §4.1 defined Lindel¨ of spaces, and explained their rela- +tion to other topological assumptions. Second countable implies Lind el¨ of, and +Lindel¨ of and regular imply paracompact (note that Xis regular as its topology +is smoothly generated). It is easy to see that OXfine implies that the topology +onXis smoothly generated. +The proof of Theorem 4.40 is based on the proof of the existence of smooth +partitions on unity on suitable separableBanachmanifolds in Bonic and Framp- +ton [10, Th. 1] (see also Lang [45, §II.3] and Deville et al. [18, §VIII.3]). +Theorem 4.40 applies to a very large class of C∞-schemes, showing that +partitions of unity exist on most interesting examples of C∞-schemes. +Theorem 4.40. LetX= (X,OX)be a Lindel¨ of local C∞-ringed space, and +suppose the topology on Xis smoothly generated. Then OXisfine, as in +Definition 4.7. That is, for every open cover {Vi:i∈I}ofXthere exists a +subordinate locally finite partition of unity {ηi:i∈I}inOX(X). +Proof.Forc∈OX(X) andx∈Xwe haveπx(c)∈OX,xandc∗(x) =π◦ +πx(c)∈R, whereπx:OX(X)→OX,xandπ:OX,x→Rare the natural C∞- +morphisms. Then c∗:X→Ris continuous. Write Uc={x∈X:c∗(x)/\e}atio\slash= 0}, +so thatUcis open inX. Thesupportofcis suppc={x∈X:πx(c)/\e}atio\slash= 0}. +Then suppcis closed in XwithUc⊆suppc, but suppcmay be larger than +the closure of Uc. Note that an infinite sum/summationtext +j∈JcjinOX(X) is defined, as +a section of the sheaf OX, if{suppcj:j∈J}is locally finite (that is, each +x∈Xhas an open neighbourhood Wxintersecting supp cjfor only finitely +manyj∈J), but may not make sense if only {Ucj:j∈J}is locally finite. +Because of this, we are careful to keep track of both Ucjand suppcjin the +following proof. +Let{Vi:i∈I}be an open cover of X. Supposei∈Iandx∈Vi. As the +topology on Xis smoothly generated there exists c∈OX(X) withx∈Uc⊆Vi. +Soc∗(x)/\e}atio\slash= 0 andc∗|X\Vi= 0. We do not know that supp c⊆Vi, but we can +correct this as follows. Choose smooth f:R→Rsuch thatf(c∗(x))/\e}atio\slash= 0 and +f= 0 in a neighbourhood of 0 in R. Setc′= Φf(c), where Φ f:OX(X)→ +OX(X) is theC∞-ring operation. Then x∈Uc′⊆suppc′⊆Uc⊆Vi⊆X. +Thus, we can choose a family {cj:j∈J}such thatcj∈OX(X), and +Ucj⊆suppcj⊆Vij⊆Xfor eachj∈Jand someij∈I, and{Ucj:j∈J} +is an open cover of X. SinceXis Lindel¨ of we can take Jto be countable, and +chooseJ=N. +Replacingcjbyc2 +jwe have (cj)∗/greaterorequalslant0 onX. For eachj∈N, choose smooth +fj:Rj+1→Rsuch thatfj(t0,t1,...,tj)>0 ifti<1/jfori= 0,1,...,j−1 +andtj>0, andfj(t0,t1,...,tj) = 0 otherwise. Define dj= Φfj(c0,c1,...,cj), +38with Φfj:OX(X)j+1→OX(X) theC∞-ring operation. Then +Udj=/braceleftbig +x∈X: (dj)∗(x)/\e}atio\slash= 0/bracerightbig +=/braceleftbig +x∈X: (ci)∗(x)<1/j, i= 0,...,j−1,(cj)∗(x)/\e}atio\slash= 0/bracerightbig +⊆Vij, +suppdj⊆/braceleftbig +x∈X: (ci)∗(x)/lessorequalslant1/j, i= 1,...,j−1/bracerightbig +∩suppcj⊆Vij.(4.12) +Fixx∈X. Thenx∈Ucjfor somej∈Nas{Ucj:j∈J}coversX. +Letj∈Nbe least with x∈Ucj. Then (cj)∗(x)>0 and (ci)∗(x) = 0 for +i= 0,1,...,j−1. Thusx∈Udj, so{Udj:j∈N}is an open cover of X. Define +Tx={y∈X: (cj)∗(y)>1 +2(cj)∗(x)}. ThenTxis an open neighbourhood of x +inX, andTx∩Udk=∅=Tx∩suppdkprovidedk >max/parenleftbig +j,2(cj)∗(x)−1/parenrightbig +by +(4.12). Thus, both {Udj:j∈N}and{suppdj:j∈N}are locally finite. +For eachi∈I, defineei=/summationtext +j∈N:ij=idjinOX(X). This is well defined +as{suppdj:j∈N}is locally finite. We have Uei⊆suppei⊆Vi, since +Udj⊆suppdj⊆Vifor eachj∈Nwithij=i. Both{Uei:i∈I}and +{suppei:i∈I}are locally finite, as {Udj:j∈N}and{suppdj:j∈N}are. +Thuse=/summationtext +i∈Ieiis well defined in OX(X). Ifx∈Xthen +e∗(x) =/summationtext +i∈I(ei)∗(x) =/summationtext +i∈I/summationtext +j∈N:ij=i(dj)∗(x) =/summationtext +j∈N(dj)∗(x)>0, +where each sum has only finitely many nonzero terms, and/summationtext +j∈N(dj)∗(x)>0 as +{Udj:j∈N}coversXwith (dj)∗>0 onUdjand (dj)∗= 0 onX\Udj. Since +e∗is positive on X,eis invertible inOX(X). Setηi=e−1·eifori∈I. Then +suppηi⊆Vi, assuppei⊆Vi, and{ηi:i∈I}islocallyfinite, as {suppei:i∈I} +is, and/summationtext +i∈Iηi=/summationtext +i∈Ie−1·ei=e−1·e= 1. Hence{ηi:i∈I}is a locally +finite partition of unity subordinate to {Vi:i∈I}, soOXis fine. +4.8 A criterion for affine C∞-schemes +Here are sufficient conditions for a local C∞-ringed space Xto be an affine C∞- +scheme. Note that affine C∞-schemes are Hausdorff with smoothly generated +topology by Lemma 4.15 and Example 4.39(b), so Lindel¨ of is the only co ndition +in the theorem which is not also necessary. +Theorem 4.41. LetX= (X,OX)be a Hausdorff, Lindel¨ of, local C∞-ringed +space, with smoothly generated topology. Then Xis an affine C∞-scheme. +Proof.LetXbe as in the theorem. Note that Theorem 4.40 shows that OX +is fine. Write C=OX(X) = Γ(X), andY= SpecC. Define a morphism +f:X→Ybyf=LC,X(idC), using the notation of Theorem 4.20. We will +showfis an isomorphism, so that X∼=SpecCis an affine C∞-scheme. +Pointsx∈XinduceC∞-ring morphisms π◦πx:C=OX(X)→R, where +πx:OX(X)→OX,xandπ:OX,x→Rare the natural projections. Points +y∈YareC∞-ring morphisms y:C→R, andf:X→Yisf(x) =π◦πx. +Supposex,x′∈Xwithx/\e}atio\slash=x′, and setf(x) =yandf(x′) =y′. SinceXis +Hausdorff there exists open U⊆Xwithx∈Uandx′/∈U. As the topology on +Xissmoothlygeneratedthereexists c∈OX(X)withc∗(x)/\e}atio\slash= 0andc∗|X\U= 0, +39so thatc∗(x′) = 0. Then y(c) =c∗(x)/\e}atio\slash= 0 andy′(c) =c∗(x′) = 0, soy/\e}atio\slash=y′. +Hencef:X→Yis injective. +Suppose for a contradiction that y∈Y, butf(x)/\e}atio\slash=yfor allx∈X. Then +for eachx∈X, there exists a∈Cwithy(a)/\e}atio\slash=π◦πx(a). Choose smooth +g:R→Rwithg(y(a)) = 0 andg= 1 in an open neighbourhood of π◦πx(a) in +R. Setb= Φg(a), where Φ g:C→Cis theC∞-ring operation. Then y(b) = 0 +andπ◦π˜x(b) = 1 for ˜xin an open neighbourhood VofxinX. +Thus we may choose a family of pairs {(Vj,bj) :j∈J}such that for each +j∈Jwe haveVj⊆Xopen andbj∈Cwithy(bj) = 0 andπ◦πx(bj) = 1 for +x∈Vj, and{Vj:j∈J}is an open cover of X. AsXis Lindel¨ of we can suppose +Jis countable, and so take J=N. By Theorem 4.40 there exists a locally finite +partition of unity {ηj:j∈N}inCsubordinate to{Vj:j∈N}. +Setc=/summationtext +j∈Nj·ηj·bjinC=OX(X), which makes sense in global sections +ofOXas{ηj:j∈N}is locally finite. Choose n∈Nwithn>y(c), and define +d=c−y(c)·1X+/summationtextn−1 +j=0(n−j)·ηj·bjinC, where 1X∈Cis the identity. Then +y(d) =y(c)−y(c)·y(1X)+/summationtextn−1 +j=0(n−j)·y(ηj)·y(bj) = 0, +asy(1X) = 1 andy(bj) = 0. And if x∈Xthen +π◦πx(d) =π◦πx/bracketleftbig/summationtext +j∈Nj·ηj·bj−y(c)·/summationtext +j∈Nηj+/summationtextn−1 +j=0(n−j)·ηj·bj/bracketrightbig +=/summationtext +j∈N/parenleftbig +max(j,n)−y(c)/parenrightbig +π◦πx(ηj)>0, +whereeachsumhasonlyfinitelymanynonzeroterms,andweuse/summationtext +j∈Nηj= 1X, +π◦πx(bj) = 1, and max( j,n)−y(c)>0,π◦πx(ηj)/greaterorequalslant0 forj∈N. +Sinceπ◦πx(d)>0 for allx∈X, we see that dis invertible in C=OX(X), +but this contradicts y(d) = 0. Hence each y∈Yhasy=f(x) for somex∈X, +andfis surjective, so f:X→Yis a bijection. By definition of Y= SpecC, +the topology on Yis generated by the open sets Uc={y∈Y:y(c)/\e}atio\slash= 0}for +allc∈C. As the topology on Xis smoothly generated, it is generated by the +open setsf−1(Uc) ={x∈X:c∗(x)/\e}atio\slash= 0}forc∈C. Therefore f:X→Yis a +bijection identifying bases for the topologies of X,Y, sofis a homeomorphism. +Letx∈Xwithf(x) =y∈Y. Taking stalks of f♯:f−1(OY)→OXatx +gives a morphism f♯ +x:OY,y→OX,x. By the definition of f=LC,X(idC) in the +proof of Theorem 4.20, f♯ +xagrees with φxin (4.8), and is the unique morphism +making the following commute, where Cy∼=OY,yby Lemma 4.18: +C +πy/d15/d15y +/d34/d34OX(X) +πx/d15/d15 +Cy∼=OY,y +π/d42/d42❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱f♯ +x/d47/d47OX,x +π/d116/d116✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ +R.(4.13) +Supposeay∈OY,ywithf♯ +x(ay) = 0. Then ay=πy(a) for some a∈C= +OX(X), asπyis surjective by Proposition 2.14, and then πx(a) = 0 inOX,x, as +40(4.13) commutes. Hence there exists an open neighbourhood UofxinXwith +a|U= 0 inOX(U). As the topology on Xis smoothly generated, there exists +b∈OX(X) withb∗(x)/\e}atio\slash= 0 andb∗|X\U= 0. Choose smooth g:R→Rwith +g(b∗(x))/\e}atio\slash= 0 andg= 0 near 0 in R, and setc= Φg(b), where Φ g:OX(X)→ +OX(X) is theC∞-ring operation. Then y(c) =c∗(x)/\e}atio\slash= 0, andcis supported in +U. Asa|U= 0 we see that a·c= 0 inOX(X). Thusalies in the ideal Iin (2.2) +which is the kernel of πy:C→Cy, by Proposition 2.14, and so ay=πy(a) = 0. +Thereforef♯ +x:OY,y→OX,xis injective. +Supposeax∈OX,x. Thenby definition of OX,xthereexists open x∈U⊆X +anda∈OX(U) withπx(a) =ax. As the topology on Xis smoothly generated +there exists b∈ OX(X) withb∗(x)/\e}atio\slash= 0 andb∗|X\U= 0. Choose smooth +g:R→Rwithg= 1 nearb∗(x) inRandg= 0 near 0 in R. Setc= Φg(b), +where Φg:OX(X)→OX(X) is theC∞-ring operation. Then cis supported +inU, and there exists an open neighbourhood VofxinUwithc|V= 1. Since +cis supported in U, the section c|U·a∈OX(U) can be extended by zero over +X\Uto give a unique d∈OX(X) supported in Uwithd|U=c|U·a. +Thend|V=c|V·a|V= 1·a|V=a|V. Hencef♯ +x◦πy(d) =πx(d) =ax, +sof♯ +x:OY,y→ OX,xis surjective, and an isomorphism. This proves that +f♯:f−1(OY)→OXis an isomorphism on stalks at every x∈X, sof♯is +an isomorphism. As fis a homeomorphism, f= (f,f♯) :X→SpecCis an +isomorphism. This completes the proof of Theorem 4.41. +Corollary 4.42. LetX= (X,OX)be a localC∞-ringed space. Then the +following are equivalent: +(i)Xis Hausdorff and second countable, with smoothly generated t opology. +(ii)Xis separable and metrizable, with smoothly generated topol ogy. +(iii)Xis a Hausdorff, second countable, regular C∞-scheme. +(iv)Xis a separable, metrizable C∞-scheme. +(v)Xis a second countable, affine C∞-scheme. +When these hold, Xis regular, normal, and paracompact, and OXis fine. +Proof.Section 4.1 implies that (i),(ii) are equivalent (as Xsmoothly generated +topology implies Xregular), and (iii),(iv) are equivalent. Also (v) implies (iii) +by Lemma 4.15, and (iii) implies (i) by Example 4.39(b), and (i) implies (v) +by Theorem 4.41 (as second countable implies Lindel¨ of). Hence (i)–( v) are +equivalent. The last part follows from §4.1 and Theorem 4.40. +In comparison to Theorem 4.41, we have strengthened the Lindel¨ o f assump- +tion to second countable. The category of C∞-schemes in Corollary 4.42 is very +large, and convenient to work in. They are closed under products, fibre prod- +ucts, and arbitrary subspaces (Lindel¨ of spaces are none of the se). They have +partitions of unity, and as they are affine we can argue globally using C∞-rings. +41Example 4.43. LetX= (X,OX) be a second countable, affine C∞-scheme, +and letY⊆Xbeanysubset, not necessarily open or closed. Then Y= +(Y,OX|Y) is also a second countable, affine C∞-scheme by Corollary 4.42, as +being Hausdorff, second countable, and of smoothly generated to pology, are all +preserved under passing to subspaces, so Ysatisfies Corollary4.42(i) as Xdoes. +Example 4.44. LetXbe a separable Banach manifold modelled locally on +separableBanach spaces Bwhich admit ‘smooth bump functions’ (that is, there +exists a nonzero smooth function f:B→Rwith bounded support in B). See +Deville et al. [18, §V] for results on when a Banach space Bhas a smooth bump +function, for example, every Hilbert space does. +MakeXinto a local C∞-ringed space X= (X,OX) as in Example 4.10. +Then the topology on Xis smoothly generated as in Example 4.39(d), so Xis +an affineC∞-scheme by Corollary 4.42(ii),(v). +4.9 Quotients of C∞-schemes by finite groups +Finally we discuss quotients of C∞-schemes by finite groups. +Definition 4.45. LetX= (X,OX) be a local C∞-ringed space, Ga finite +group, and r:G→Aut(X) an action of GonX. We will define a local +C∞-ringed space Y=X/G. +SetY=X/r(G) to be the quotient topological space. Open sets V⊆Yare +of the form U/GforU⊆Xopen andG-invariant. Then γ/ma√sto→r♯(γ)(U) gives an +action ofGon theC∞-ringOX(U), so as in Proposition2.22we havea C∞-ring +OX(U)G, theG-invariant subspace in OX(U). DefineOY(V) =OX(U)G. +IfV2⊆V1⊆Yare open then V1=U1/G,V2=U2/GforU2⊆U1⊆X +open andG-invariant. The restriction morphism ρU1U2:OX(U1)→OX(U2) in +OXisG-equivariant, and so restricts to ρU1U2|OX(U1)G:OX(U1)G→OX(U2)G. +SetρV1V2=ρU1U2|OX(U1)G:OY(V1)→OY(V2). It is now easy to check that +OYis a sheaf of C∞-rings onY, soY= (Y,OY) is aC∞-ringed space. +Ifx∈Xandy=xG∈Y, the stalkOY,yofOYatyis (OX,x)H, where +OX,xis a localC∞-ring, andH=/braceleftbig +γ∈G:γ(x) =x/bracerightbig +is the stabilizer group +ofxinG, which acts onOX,xin the obvious way. As OX,xis local there is +anR-algebra morphism π:OX,x→R, such that c∈OX,xis invertible if and +only ifπ(c)/\e}atio\slash= 0. Thusπ|(OX,x)H: (OX,x)H→Ris anR-algebra morphism, and +c∈(OX,x)His invertible inOX,xif and only if π(c)/\e}atio\slash= 0. But if c∈(OX,x)H +is invertible inOX,xthenc−1isH-invariant, so cis invertible in (OX,x)H. +ThereforeOY,y∼=(OX,x)His a localC∞-ring, and Yis a localC∞-ringed +space. Write X/G=Y. +Defineπ:X→X/Gto be the natural projection. Define a morphism +π♯:OY→π∗(OX) of sheaves of C∞-rings onY=X/Gby +π♯(V) = inc :OY(V) =OX(U)G−→OX(U) =π∗(OX)(V) +for all open V=U/G⊆Y=X/G, where inc :OX(U)G֒→OX(U) is the +inclusion. Let π♯:π−1(OY)→OXbe the morphism of sheaves of C∞-rings on +42Xcorrespondingto π♯under (4.3). Then π= (π,π♯) :X→X/Gis a morphism +of localC∞-ringed spaces. +It is easy to see that X/G,πhave the universal property that if f:X→Z +is a morphism in LC∞RSwithf◦r(γ) =ffor allγ∈Gthenf=g◦πfor a +unique morphism g:X/G→ZinLC∞RS. +Proposition 4.46. LetX= (X,OX)be an affine C∞-scheme,Ga finite +group, and r:G→Aut(X)an action of GonX. SupposeXis Lindel¨ of. +ThenX= SpecCforC=OX(X)a complete C∞-ring, andr= Specsfor +s:G→Aut(C)a unique action of GonC. Form the G-invariantC∞-ring +CG⊆Cas in Proposition 2.22. ThenCGis complete, and there is a canonical +isomorphism X/G∼=SpecCGinLC∞RS. +Proof.Theorem 4.36(a) shows that X∼=SpecC, whereC=OX(X) is a com- +pleteC∞-ring. As Spec is full and faithful on complete C∞-rings by Theorem +4.36(b), Spec : Aut( C)→Aut(X) is an isomorphism, so there is a unique action +s:G→Aut(C) withr= Specs. +LetY=X/Gbe as in Definition 4.45. Then Y=X/Gis Hausdorff, as X +is Hausdorff and Gis finite. Suppose {Vi:i∈I}is an open cover of Y. Then +Vi=Ui/Gfor{Ui:i∈I}an open cover of X. AsXis Lindel¨ of there exists a +subcover{Ui:i∈S}for countable S⊆I, and then{Vi:i∈S}is a countable +subcover of{Vi:i∈I}. HenceYis Lindel¨ of. +SupposeV⊆Yis open and y∈V. ThenV=U/Gandy=xGforG- +invariantopen U⊆Xwithx∈U. As the topologyon Xis smoothly generated, +there exists c∈Cwithc∗(x)/\e}atio\slash= 0 andc∗(x′) = 0 for all x′∈X\U. Define +d=/summationtext +γ∈Gγ∗(c2) inC. ThendisG-invariant with d∗(x)>0 andd∗(x′) = 0 +for allx′∈X\U. Henced∈OY(Y) =OX(X)G=CG, withd∗(y)>0 and +d∗(y′) = 0 for all y′∈Y\V. Thus the topology of Yis smoothly generated. +Theorem 4.41 now implies that Y=X/Gis an affine C∞-scheme, and +Theorem4.36(a)givesacanonicalisomorphism X/G∼=SpecOY(Y) = Spec CG, +whereCGis complete. +Proposition 4.47. SupposeXis a Hausdorff, second countable C∞-scheme, +Ga finite group, and r:G→Aut(X)an action of GonX. Then the quotient +X/Gis also a Hausdorff, second countable C∞-scheme. If Xis locally fair, or +locally finitely presented, then so is X/G. +Proof.Letx∈X, and write H=/braceleftbig +γ∈G:γ(x) =x/bracerightbig +. Then the G-orbitxG +is|G|/|H|points. Since Xis Hausdorff and Gis finite, we can find an open +neighbourhood RofxinXsuch thatRisH-invariant and R∩γ·R=∅for all +γ∈G\H. AsXis aC∞-scheme, there is an open neighbourhood SofxinR +with (S,OX|S) an affineC∞-scheme. Then T=/intersectiontext +γ∈Hγ·Sis anH-invariant +open neighbourhood of xinS. Choose an open neighbourhood UofxinTwith +(U,OX|U) an affineC∞-scheme. +DefineV=/intersectiontext +γ∈Hγ·U. ThenVis anH-invariant open neighbourhood +ofxinU⊆T⊆S⊆R⊆X. It is the intersection of the |H|affineC∞- +subschemes ( γ·U,OX|γ·U) forγ∈Hinside the affine C���-scheme (S,OX|S). +43Finite intersections of affine C∞-subschemes in an affine C∞-scheme are affine, +as such intersections are fibre products and Spec : C∞Ringsop→LC∞RS +preserves limits by Remark 4.21(b). Thus ( V,OX|V) is an affine C∞-scheme. +SetW=/uniontext +γH∈G/Hγ·V. ThenWis aG-invariant open neighbourhood +ofxinX, and (W,OX|W) is the disjoint union of |G|/|H|affineC∞-schemes +isomorphic to ( V,OX|V), so it is affine. We have shown that every x∈Xhas +aG-invariant open neighbourhood W⊆XwithW= (W,OX|W) affine. Then +W/Gis an open neighbourhood of xGinX/G. AsXis second countable, W +is second countable and so Lindel¨ of. Thus W/Gis an affine C∞-scheme by +Proposition 4.46. As we can cover X/Gby such open W/G, it is aC∞-scheme. +IfXis locally fair, or locally finitely presented, we can do the argument +above with S,U,V,W,W/Gfair, or finitely presented, using Proposition 2.22 +forW/G, soX/Gis also locally fair, or locally finitely presented. +5 Modules over C∞-rings and C∞-schemes +Nextwediscussmodulesover C∞-rings,andsheavesofmoduleson C∞-schemes. +The author knows of no previous work on these, so all this section m ay be new, +although much of it is a straightforward generalization of well known facts. +5.1 Modules over C∞-rings +Definition 5.1. LetCbe aC∞-ring. A moduleMoverC, orC-module, is a +module over Cregarded as a commutative R-algebra as in Definition 2.6, and +morphisms of C-modules are morphisms of R-algebra modules. We will write +µM:C×M→Mfor the multiplication map, and also write µM(c,m) =c·m +forc∈Candm∈M. ThenC-modules form an abelian category, which we +write as C-mod. +The action of Con itself by multiplication makes Cinto aC-module, and +moregenerally C⊗RVisaC-moduleforany R-vectorspace V. AC-moduleMis +finitely generated ifitfitsintoanexactsequence C⊗Rn→M→0inC-mod, and +finitely presented if it fits into an exact sequence C⊗Rm→C⊗Rn→M→0. +BecauseC∞-rings such as C∞(Rn) are not noetherian, finitely generated +C-modules generally need not be finitely presented. +Now letφ:C→Dbe a morphism of C∞-rings. IfMis aC-module then +φ∗(M) =M⊗CDis aD-module, and this induces a functor φ∗:C-mod→ +D-mod. Also,any D-moduleNmayberegardedasa C-moduleφ∗(N) =Nwith +C-actionµφ∗(N)(c,n) =µN(φ(c),n), and this defines a functor φ∗:D-mod→ +C-mod. Note that φ∗:C-mod→D-mod takes finitely generated (or finitely +presented) C-modules to finitely generated (or finitely presented) D-modules, +butφ∗:D-mod→C-mod generally does not. +Vector bundles Eover manifolds Xgive examples of modules over C∞(X). +Example 5.2. LetXbe a manifold and E→Xbe a vector bundle, and write +Γ∞(E) for the vector space of smooth sections eofE. This is a module over +44theC∞-ringC∞(X), multiplying functions on Xby sections of E. +LetE,F→Xbe vector bundles over Xandλ:E→Fa morphism of +vector bundles. Then λ∗: Γ∞(E)→Γ∞(F) defined by λ∗:e/ma√sto→λ◦eis a +morphism of C∞(X)-modules. +Now letX,Ybe manifolds and f:X→Ya (weakly) smooth map. Then +f∗:C∞(Y)→C∞(X) is a morphism of C∞-rings. IfE→Yis a vector +bundle over Y, thenf∗(E) is a vector bundle over X. Under the functor ( f∗)∗: +C∞(Y)-mod→C∞(X)-mod of Definition 5.1, we see that ( f∗)∗/parenleftbig +Γ∞(E)/parenrightbig += +Γ∞(E)⊗C∞(Y)C∞(X) is isomorphic as a C∞(X)-module to Γ∞/parenleftbig +f∗(E)/parenrightbig +. +IfE→Xis any vector bundle over a manifold Xthen by choosing sections +e1,...,en∈Γ∞(E) forn≫0 such that e1|x,...,en|xspanE|xfor allx∈X +we obtain a surjective morphism of vector bundles ψ:X×Rn→E, whose +kernel is another vector bundle F. By choosing another surjective morphism +φ:X×Rm→Fwe obtain an exact sequence of vector bundles +X×Rmφ/d47/d47X×Rnψ/d47/d47E /d47/d470, +which induces an exact sequence of C∞(X)-modules +C∞(X)⊗RRmφ∗/d47/d47C∞(X)⊗RRnψ∗/d47/d47Γ∞(E) /d47/d470. +Thus Γ∞(E) is a finitely presented C∞(X)-module. +5.2 Cotangent modules of C∞-rings +Given aC∞-ringC, we will define the cotangent module ΩCofC. Although +our definition of C-module only used the commutative R-algebra underlying the +C∞-ringC, our definition of the particular C-module Ω Cdoes use the C∞-ring +structure in a nontrivial way. It is a C∞-ring version of the module of relative +differential forms orK¨ ahler differentials in Hartshorne [31, p. 172], and is an +example of a construction for Fermat theories by Dubuc and Kock [ 25]. +Definition 5.3. Suppose Cis aC∞-ring, andMaC-module. A C∞-derivation +is anR-linear map d : C→Msuch that whenever f:Rn→Ris a smooth map +andc1,...,cn∈C, we have +dΦf(c1,...,cn) =n/summationtext +i=1Φ∂f +∂xi(c1,...,cn)·dci. (5.1) +Note that d is nota morphism of C-modules. We call such a pair M,d acotan- +gent module forCif it has the universal property that for any C∞-derivation +d′:C→M′, there exists a unique morphism of C-modulesλ:M→M′ +with d′=λ◦d. +There is a natural construction for a cotangent module: we take Mto +be the quotient of the free C-module with basis of symbols d cforc∈C +by theC-submodule spanned by all expressions of the form dΦ f(c1,...,cn)−/summationtextn +i=1Φ∂f +∂xi(c1,...,cn)·dciforf:Rn→Rsmooth and c1,...,cn∈C. Thus +45cotangent modules exist, and are unique up to unique isomorphism. W hen we +speak of ‘the’ cotangent module, we mean that constructed abov e. We write +dC:C→ΩCfor the cotangent module of C. +LetC,DbeC∞-rings with cotangent modules Ω C,dC, ΩD,dD, andφ:C→ +Dbe a morphism of C∞-rings. Then we may regard Ω D=φ∗(ΩD) as aC- +module, and d D◦φ:C→ΩDas aC∞-derivation. Thus by the universal +property of Ω C, there exists a unique morphism of C-modules Ω φ: ΩC→ΩD +with d D◦φ= Ωφ◦dC. This then induces a morphism of D-modules (Ω φ)∗: +ΩC⊗CD→ΩD. Ifφ:C→D,ψ:D→Eare morphisms of C∞-rings +then Ωψ◦φ= Ωψ◦Ωφ: ΩC→ΩE. +Example 5.4. LetXbeamanifold. Thenthecotangentbundle T∗Xisavector +bundle over X, so as in Example 5.2 it yields a C∞(X)-module Γ∞(T∗X). The +exterior derivative d : C∞(X)→Γ∞(T∗X), d :c/ma√sto→dcis then aC∞-derivation, +since equation (5.1) follows from +d/parenleftbig +f(c1,...,cn)/parenrightbig +=/summationtextn +i=1∂f +∂xi(c1,...,cn)dcn +forf:Rn→Rsmooth and c1,...,cn∈C∞(X), which holds by the chain rule. +It is easy to show that Γ∞(T∗X),d have the universal property in Definition +5.3, and so form a cotangent module for C∞(X). +Now letX,Ybe manifolds, and f:X→Ya smooth map. Then f∗(T∗Y), +T∗Xare vector bundles over X, and the derivative of fgives a vector bundle +morphism d f:f∗(T∗Y)→T∗X. This induces a morphism of C∞(X)-modules +(df)∗: Γ∞(f∗(T∗Y))→Γ∞(T∗X). This (df)∗is identified with (Ω f∗)∗under +the natural isomorphism Γ∞(f∗(T∗Y))∼=Γ∞(T∗Y)⊗C∞(Y)C∞(X), where we +identifyC∞(Y),C∞(X),f∗withC,D,φin Definition 5.3. +The importance of Definition 5.3 is that it abstracts the notion of cot angent +bundle of a manifold in a way that makes sense for any C∞-ring. +Remark 5.5. There is a second way to define a cotangent-type module for a +C∞-ringC, namely the module Kd CofK¨ ahler differentials of the underlying +R-algebra of C. This is defined as for Ω C, but requiring (5.1) to hold only when +f:Rn→Ris a polynomial. Since we impose many fewer relations, Kd Cis +generally much larger than Ω C, so that Kd C∞(Rn)is not a finitely generated +C∞(Rn)-module for n>0, for instance. +Proposition 5.6. IfCis a finitely generated C∞-ring then ΩCis a finitely +generated C-module. If Cis finitely presented, then ΩCis finitely presented. +Proof.IfCis finitely generated we have an exact sequence +0 /d47/d47I /d47/d47C∞(Rn)φ/d47/d47C /d47/d470. (5.2) +Writex1,...,xnfor the generators of C∞(Rn). Then any c∈Cmay be written +asφ(f) for somef∈C∞(Rn), and (5.1) implies that +dc= dΦf/parenleftbig +φ(x1),...,φ(xn)/parenrightbig +=/summationtextn +i=1Φ∂f +∂xi(φ(x1),...,φ(xn))·d◦φ(xi). +46Hence the generators d cof ΩCforc∈CareC-linear combinations of d ◦φ(xi), +i= 1,...,n, so ΩCis spanned by the d ◦φ(xi), and is finitely generated. +Suppose Cis finitely presented. Then we have an exact sequence (5.2) with +idealI= (f1,...,fm). We will define an exact sequence of C-modules +C⊗RRmα/d47/d47C⊗RRnβ/d47/d47ΩC/d47/d470. (5.3) +Write (a1,...,am), (b1,...,bn) for bases of Rm,Rn. AsC⊗RRm,C⊗RRnare +freeC-modules, the C-module morphisms α,βare specified uniquely by giving +α(ai) fori= 1,...,mandβ(bj) forj= 1,...,n, which we define to be +α:ai/ma√sto−→/summationtextn +j=1Φ∂fi +∂xj/parenleftbig +φ(x1),...,φ(xn)/parenrightbig +·bjandβ:bj/ma√sto−→dC/parenleftbig +φ(xj)/parenrightbig +. +Then fori= 1,...,mwe have +β◦α(ai) =/summationtextn +j=1Φ∂fi +∂xj/parenleftbig +φ(x1),...,φ(xn)/parenrightbig +·dC/parenleftbig +φ(xj)/parenrightbig += dC/parenleftbig +Φfi/parenleftbig +φ(x1),...,φ(xn)/parenrightbig/parenrightbig += dC◦φ/parenleftbig +Φfi(x1,...,xn)/parenrightbig += dC◦φ/parenleftbig +fi(x1,...,xn)) = d C(0) = 0, +using (5.1) in the second step, φa morphism of C∞-rings in the third, the +definition of C∞(Rn) asaC∞-ringin the fourth, and fi(x1,...,xn)∈I= Kerφ +in the fifth. Hence β◦α= 0, and (5.3) is a complex. +Thusβinducesβ∗: (C⊗RRn)/α(C⊗RRm)→ΩC. We will show β∗is an +isomorphism, so that (5.3) is exact. Define d : C→(C⊗RRn)/α(C⊗RRm) by +d/parenleftbig +φ(h)/parenrightbig +=/summationtextn +j=1Φ∂h +∂xj/parenleftbig +φ(x1),...,φ(xn)/parenrightbig +·bj+α(C⊗RRm).(5.4) +Hereeveryc∈Cmaybe written as φ(h) forsomeh∈C∞(Rn) asφis surjective. +To show (5.4) is well-defined we must show the right hand side is indepen dent +of the choice of hwithφ(h) =c, that is, we must show that the right hand side +is zero ifh∈I. It is enough to check this when his a generator f1,...,fmof +I, and this holds by definition of α. Hence d in (5.4) is well-defined. +It is easy to see that d is a C∞-derivation, and that β∗◦d = d C. So by +the universal property of Ω C, there is a unique C-module morphism ψ: ΩC→ +(C⊗RRn)/α(C⊗RRm)withd =ψ◦dC. Thusβ∗◦ψ◦dC=β∗◦d = dC= idΩC◦dC, +so as Imd Cgenerates Ω Cas anC-module we see that β∗◦ψ= idΩC. Similarly +ψ◦β∗is the identity, so ψ,β∗are inverse, and β∗is an isomorphism. Therefore +(5.3) is exact, and Ω Cis finitely presented. +Cotangent modules behave well under localization. +Proposition 5.7. LetCbe aC∞-ring,S⊆C,andD=C[s−1:s∈S]be the +localization of CatSwith projection π:C→D,as in Definition 2.13. Then +(Ωπ)∗: ΩC⊗CD→ΩDis an isomorphism of D-modules. +47Proof.Let ΩC,ΩDbe constructed as in Definition 5.3. As D=C[s−1:s∈S] +isCtogether with an extra generator s−1and an extra relation s·s−1= 1 for +eachs∈S, we see that the D-module Ω Dmay be constructed from Ω C⊗CD +by adding an extra generator d( s−1) and an extra relation d( s·s−1−1) = 0 for +eachs∈S. But using (5.1) and s·s−1= 1 inD, we see that this extra relation +is equivalent to d( s−1) =−(s−1)2ds. Thus the extra relations exactly cancel +the effect of adding the extra generators, so (Ω π)∗is an isomorphism. +Here is a useful exactness property of cotangent modules. +Theorem 5.8. Suppose we are given a pushout diagram of C∞-rings: +Cβ/d47/d47 +α/d15/d15E +δ/d15/d15 +Dγ/d47/d47F,(5.5) +so thatF=D∐CE. Then the following sequence of F-modules is exact: +ΩC⊗C,γ◦αF(Ωα)∗⊕−(Ωβ)∗/d47/d47ΩD⊗D,γF⊕ +ΩE⊗E,δF(Ωγ)∗⊕(Ωδ)∗/d47/d47ΩF/d47/d470.(5.6) +Here(Ωα)∗: ΩC⊗C,γ◦αF→ΩD⊗D,γFis induced by Ωα: ΩC→ΩD,and so +on. Note the sign of −(Ωβ)∗in(5.6). +Proof.By Ωψ◦φ= Ωψ◦Ωφin Definition 5.3 and commutativity of (5.5) we have +Ωγ◦Ωα= Ωγ◦α= Ωδ◦β= Ωδ◦Ωβ: ΩC→ΩF. Tensoring with Fthen gives +(Ωγ)∗◦(Ωα)∗= (Ωδ)∗◦(Ωβ)∗: ΩC⊗CF→ΩF. Asthe compositionofmorphisms +in (5.6) is (Ω γ)∗◦(Ωα)∗−(Ωδ)∗◦(Ωβ)∗, this implies (5.6) is a complex. +For simplicity, first suppose C,D,E,Fare finitely presented. Use the nota- +tion of Example 2.23 and the proof of Proposition 2.24, with exact seq uences +(2.3) and (2.4), where I= (h1,...,hi)⊂C∞(Rl),J= (d1,...,dj)⊂C∞(Rm) +andK= (e1,...,ek)⊂C∞(Rn). ThenLis given by (2.5). Applying the proof +of Proposition 5.6 to (2.3)–(2.4) yields exact sequences of F-modules +F⊗RRiǫ1/d47/d47F⊗RRlζ1/d47/d47ΩC⊗CF /d47/d470,(5.7) +F⊗RRjǫ2/d47/d47F⊗RRmζ2/d47/d47ΩD⊗DF /d47/d470,(5.8) +F⊗RRkǫ3/d47/d47F⊗RRnζ3/d47/d47ΩE⊗EF /d47/d470,(5.9) +F⊗RRj+k+lǫ4/d47/d47F⊗RRm+n=F⊗RRm⊕F⊗RRnζ4/d47/d47ΩF/d47/d470,(5.10) +where for (5.7)–(5.9) we have tensored (5.3) for C,D,EwithF. +DefineF-module morphisms θ1:F⊗RRl→F⊗RRm,θ2:F⊗RRl→F⊗RRn +byθ1(a1,...,al) = (b1,...,bm),θ2(a1,...,al) = (c1,...,cn) with +bq=l/summationdisplay +p=1Φ∂fp +∂yq(ξ(y1),...,ξ(ym))·ap, cr=l/summationdisplay +p=1Φ∂gp +∂yr(ξ(z1),...,ξ(zn))·ap, +48forap,bq,cr∈F. Now consider the diagram +F⊗RRj⊕ +F⊗RRk⊕ +F⊗RRl ǫ 4=/parenleftBigǫ20θ1 +0ǫ3−θ2/parenrightBig/d47/d47 +(0 0ζ1) +/d15/d15F⊗RRm⊕ +F⊗RRnζ4/d47/d47 +/parenleftBigζ20 +0ζ3/parenrightBig +/d15/d15ΩF/d47/d47 +idΩF0 +ΩC⊗CF/parenleftbigg +(Ωα)∗ +−(Ωβ)∗/parenrightbigg +/d47/d47ΩD⊗DF⊕ +ΩE⊗EF((Ωγ)∗(Ωδ)∗)/d47/d47ΩF/d47/d470,(5.11) +using matrix notation. The top line is the exact sequence (5.10), whe re the sign +in−θ2comes fromthe sign of gpin the generators fp(y1,...,ym)−gp(z1,...,zn) +ofLin (2.5). The bottom line is the complex (5.6). +The left hand square commutes as ζ2◦ǫ2=ζ3◦ǫ3= 0 by exactness of (5.8)– +(5.9)andζ2◦θ1= (Ωα)∗◦ζ1followsfrom α◦φ(xp) =ψ(fp), andζ3◦θ2= (Ωβ)∗◦ζ1 +follows from β◦φ(xp) =χ(gp). The right hand square commutes as ζ4and +(Ωγ)∗◦ζ2act onF⊗RRmby (a1,...,am)/ma√sto→/summationtextm +q=1aqdF◦ξ(yq), andζ4and +(Ωδ)∗◦ζ3act onF⊗RRnby (b1,...,bn)/ma√sto→/summationtextn +r=1brdF◦ξ(zr). Hence (5.11) +is commutative. The columns are surjective since ζ1,ζ2,ζ3are surjective as +(5.7)–(5.9) are exact and identities are surjective. +The bottom right morphism/parenleftbig +(Ωγ)∗(Ωδ)∗/parenrightbig +in (5.11) is surjective as ζ4is +and the right hand square commutes. Also surjectivity of the middle column +implies that it maps Ker ζ4surjectively onto Ker/parenleftbig +(Ωγ)∗(Ωδ)∗/parenrightbig +. But Kerζ4= +Imǫ4as the top row is exact, so as the left hand square commutes we see that/parenleftbig +(Ωα)∗−(Ωβ)∗/parenrightbigTsurjects onto Ker/parenleftbig +(Ωγ)∗(Ωδ)∗/parenrightbig +, and the bottom row of (5.11) +is exact. This proves the theorem for C,D,E,Ffinitely presented. For the +general case we can use the same proof, but allowing i,j,k,l,m,n infinite. +Here is an example of the situation of Theorem 5.8 for manifolds. +Example 5.9. LetW,X,Y,Z,e,f,g,h be as in Theorem 3.5, so that (3.1) is +a Cartesian square of manifolds and (3.2) a pushout square of C∞-rings. We +have the following sequence of morphisms of vector bundles on W: +0 /d47/d47(g◦e)∗(T∗Z)e∗(dg∗)⊕−f∗(dh∗)/d47/d47e∗(T∗X)⊕f∗(T∗Y)de∗⊕df∗ +/d47/d47T∗W /d47/d470.(5.12) +Here dg:TX→g∗(TZ) is a morphism of vector bundles over X, and dg∗: +g∗(T∗Z)→T∗Xis the dual morphism, and e∗(dg∗) : (g◦e)∗(T∗Z)→e∗(T∗X) +is the pullback of this dual morphism to W. +Sinceg◦e=h◦f, we have de∗◦e∗(dg∗) = df∗◦f∗(dh∗), and so (5.12) is a +complex. As g,haretransverseand(3.1)isCartesian,(5.12)isexact. Sopassing +to smooth sections in (5.12) we get an exact sequence of C∞(W)-modules: +0 /d47/d47Γ∞/parenleftbig +(g◦e)∗(T∗Z)/parenrightbig(e∗(dg∗)⊕ +−f∗(dh∗))∗/d47/d47Γ∞/parenleftbig +e∗(T∗X) +⊕f∗(T∗Y)/parenrightbig(de∗⊕ +df∗)∗/d47/d47Γ∞(T∗W) /d47/d470. +The final four terms are the exact sequence (5.6) for the pushou t diagram (3.2). +495.3 Sheaves of OX-modules on a C∞-ringed space (X,OX) +We define sheaves of OX-modules on a C∞-ringed space, following [31, §II.5]. +Definition 5.10. Let (X,OX) be aC∞-ringed space. A sheaf ofOX-modules , +or simply anOX-module,EonXassigns a module E(U) over the C∞-ring +OX(U) for each open set U⊆X, and a linear map EUV:E(U)→E(V) for +each inclusion of open sets V⊆U⊆X, such that the following commutes +OX(U)×E(U) +ρUV×EUV/d15/d15µE(U)/d47/d47E(U) +EUV/d15/d15 +OX(V)×E(V)µE(V)/d47/d47E(V),(5.13) +and all this data E(U),EUVsatisfies the sheaf axioms in Definition 4.1. +Amorphism of sheaves of OX-modulesφ:E→Fassigns a morphism of +OX(U)-modulesφ(U) :E(U)→F(U) for each open set U⊆X, such that +φ(V)◦EUV=FUV◦φ(U) for each inclusion of open sets V⊆U⊆X. Then +OX-modules form an abelian category, which we write as OX-mod. +AnOX-moduleEis called a vector bundle of rank nif we may cover Xby +openU⊆XwithE|U∼=OX|U⊗RRn. +In Definition 4.7 we defined finesheavesEon a topological space X. In§4.7 +we gave sufficient conditions for when a C∞-ringed space X= (X,OX) hasOX +fine, which hold if Xis an affine C∞-scheme with XLindel¨ of. Now if OXis +fine, then anyOX-moduleEis also fine, since partitions of unity in OXinduce +partitions of unity in Hom(E,E). +As in Voisin [69, Prop.4.36], a fundamental propertyof fine sheaves Eis that +their cohomology groups Hi(E) are zero for all i >0. This means that H0is +an exact functor on fine sheaves, rather than just left exact, s inceH1measures +the failure of H0to be right exact. If Xis second countable then ( U,OX|U) is +a Lindel¨ of affine C∞-scheme for all open U⊆X. Thus we deduce: +Proposition 5.11. Let(X,OX)be an affine C∞-scheme with XLindel¨ of, and +··· /d47/d47Eiφi/d47/d47Ei+1φi+1/d47/d47Ei+2 /d47/d47··· +be an exact sequence in OX-mod. Then +··· /d47/d47Ei(X)φi(X)/d47/d47Ei+1(X)φi+1(X)/d47/d47Ei+2(X) /d47/d47··· +is an exact sequence of OX(X)-modules. If Xis also second countable then the +following is an exact sequence of OX(U)-modules for all open U⊆X: +··· /d47/d47Ei(U)φi(U)/d47/d47Ei+1(U)φi+1(U)/d47/d47Ei+2(U) /d47/d47···. +Remark 5.12. Recall that a C∞-ringChas an underlying commutative R- +algebra, and a module over Cis a module over this R-algebra, by Definitions 2.6 +and 5.1. Thus, by truncating the C∞-ringsOX(U) to commutative R-algebras, +50regarded as rings, a C∞-ringed space ( X,OX) has an underlying ringed space +in the usual sense of algebraic geometry [31, p. 72], [30, §0.4]. Our definition +ofOX-modules are simply OX-modules on this underlying ringed space [31, +§II.5], [30,§0.4.1]. Thus we can apply results from algebraic geometry without +change, for instance that OX-mod is an abelian category, as in [31, p. 202]. +Definition 5.13. Letf= (f,f♯) : (X,OX)→(Y,OY) be a morphism of +C∞-ringed spaces, and Ebe anOY-module. Define the pullbackf∗(E) by +f∗(E) =f−1(E)⊗f−1(OY)OX, wheref−1(E) is as in Definition 4.5, a sheaf of +modules over the sheaf of C∞-ringsf−1(OY) onX, and the tensor product uses +the morphism f♯:f−1(OY)→OX. Ifφ:E→Fis a morphism of OY-modules +we have a morphism of OX-modulesf∗(φ) =f−1(φ)⊗idOX:f∗(E)→f∗(F). +Remark 5.14. Pullbacksf∗(E) are a kind of fibre product, and may be char- +acterized by a universal property in OX-mod. So they should be regarded as +beingunique up to canonical isomorphism , rather than unique. One can give +an explicit construction for pullbacks, or use the Axiom of Choice to c hoose +f∗(E) for allf,E, and so speak of ‘the’ pullback f∗(E). However, it may not be +possible to make these choices strictly functorial in f. +That is, if f:X→Y,g:Y→Zare morphisms and E∈OZ-mod then +(g◦f)∗(E),f∗(g∗(E)) are canonically isomorphic in OX-mod, but may not be +equal. We will write If,g(E) : (g◦f)∗(E)→f∗(g∗(E)) for these canonical +isomorphisms, as in Remark 4.6(b). Then If,g: (g◦f)∗⇒f∗◦g∗is a natural +isomorphism of functors. It is common to ignore this point and identif y (g◦f)∗ +withf∗◦g∗. Vistoli [68] makes careful use of natural isomorphisms ( g◦f)∗⇒ +f∗◦g∗in his treatment of descent theory. +Whenfis the identity id X:X→XandE∈OX-mod we do not require +id∗ +X(E) =E, but asEis a possible pullback for id∗ +X(E) there is a canonical +isomorphism δX(E) : id∗ +X(E)→E, and then δX: id∗ +X⇒idOX-modis a natural +isomorphism of functors. +By Grothendieck [30, §0.4.3.1] we have: +Proposition 5.15. LetX,YbeC∞-ringed spaces and f:X→Ya morphism. +Then pullback f∗:OY-mod→OX-modis aright exact functor between +abelian categories. That is, if Eφ−→Fψ−→G → 0is exact inOY-modthen +f∗(E)f∗(φ)−→f∗(F)f∗(ψ)−→f∗(G)→0is exact inOX-mod. +In generalf∗is not exact, or left exact, unless f:X→Yis flat. +5.4 Sheaves on affine C∞-schemes, MSpecandΓ +In§4.4 we defined Spec : C∞Ringsop→LC∞RS. In a similar way, if Cis a +C∞-ring and (X,OX) = Spec Cwe can define MSpec : C-mod→OX-mod, a +spectrum functor for modules. +Definition 5.16. Let (X,OX) = Spec Cfor someC∞-ringCandMbe aC- +module. We will define an OX-moduleE= MSpecM. For each open U⊆X, +51defineE(U) to be the R-vector space of functions e:U→/coproducttext +x∈U(M⊗CCx) with +e(x)∈M⊗CCxfor allx∈U, and such that Umay be covered by open sets +W⊆U⊆Xfor which there exist m∈Mwithe(x) =m⊗1∈M⊗CCxfor all +x∈W. Here the Cx-moduleM⊗CCxis defined using the C-module structure +onMand the projection πx:C→Cx. +Definition 4.16 defines OX(U) as a set of functions U→/coproducttext +x∈UCx. Define +anOX(U)-module structure µE(U):OX(U)×E(U)→E(U) onE(U) by +µE(U)(s,e) :x/ma√sto−→s(x)·e(x), +for alls∈ OX(U),e∈ E(U) andx∈U. For open V⊆U⊆X, define +EUV:E(U)→E(V) byEUV:e/ma√sto→e|V. It is now easy to check that Eis a sheaf +ofOX-modules on X. Define MSpec M=EinOX-mod. +An equivalent way to define MSpec Mis as the sheafification of the presheaf +U/ma√sto→M⊗COX(U). The definition above performs the sheafification explicitly. +Now letα:M→Nbe a morphism in C-mod, and setE= MSpecMand +F= MSpecN. For each open U⊆X, defineλ(U) :E(U)→F(U) by +λ(U)(e) :x/ma√sto→(α⊗id)(e(x)) forx∈U, +whereα⊗id mapsM⊗CCx→N⊗CCx. It is easy to check that λ(U) is an +OX(U)-module morphism and λ(V)◦EUV=FUV◦λ(U) :E(U)→F(V) for +all openV⊆U⊆X. Henceλ:E→Fis a morphism in OX-mod. Define +MSpecα=λ, so that MSpec α: MSpecM→MSpecN. This defines a functor +MSpec : C-mod→OX-mod. It is an exact functor of abelian categories, since +M/ma√sto→M⊗CCxis an exact functor C-mod→Cx-mod for each x∈X, as the +localization πx:C→Cxis a flat morphism of R-algebras. +Definition 5.17. LetCbe aC∞-ring, and ( X,OX) = Spec C. IfEis anOX- +module thenE(X) is a module over OX(X), so using Ψ C:C→Γ(SpecC) = +OX(X) we may regard E(X) as aC-module. Define Γ( E) to be the C-module +E(X). Ifα:E → F is a morphism of OX-modules then Γ( α) :=α(X) : +E(X)→F(X) is a morphism Γ( α) : Γ(E)→Γ(F) inC-mod. This defines the +global sections functor Γ :OX-mod→C-mod. +In general Γ is a left exact functor of abelian categories, but may n ot be +right exact. However, if Xis Lindel¨ of (for example, if Cis finitely or countably +generated) then Proposition 5.11 shows that Γ is an exact functor . +Now Γ◦MSpec is a functor C-mod→C-mod. For each C-moduleMand +m∈M, define Ψ M(m) :X→/coproducttext +x∈XM⊗CCxby ΨM(m) :x/ma√sto→m⊗1Cx∈ +M⊗CCx. Then Ψ M(m)∈MSpecM(X) = Γ◦MSpecMby Definition 5.16, so +ΨM:M→Γ◦MSpecMis a linear map, and in fact a C-module morphism. +Itisfunctorialin M,sothattheΨ MforallMdefineanaturaltransformation +Ψ : idC-mod⇒Γ◦MSpec of functors id C-mod,Γ◦MSpec : C-mod→C-mod. +Here are the analogues of Lemma 4.18 and Theorem 4.20: +Lemma 5.18. In Definition 5.16,the stalk (MSpecM)x=ExofMSpecMat +x∈Xis naturally isomorphic to M⊗CCx,as modules over Cx∼=OX,x. +52Proof.Elements ofExare∼-equivalence classes [ U,e] of pairs (U,e), whereU +is an open neighbourhood of xinXande∈E(U), and (U,e)∼(U′,e′) if there +exists open x∈V⊆U∩U′withe|V=e′|V. Define a Cx-module morphism +Π :Ex→M⊗CCxby Π : [U,e]/ma√sto→e(x). +Proposition 2.14 shows that Cx∼=C/IforIthe ideal in (2.2). Hence M⊗C +Cx∼=M/(I·M), and thus every element of M⊗CCxis of the form m⊗1Cx +for somem∈M. But ΨM(m)∈E(X), so that [ X,ΨM(m)]∈Ex, with Π : +[X,ΨM(m)]/ma√sto→m⊗1Cx. Hence Π :Ex→M⊗CCxis surjective. +Suppose [U,e]∈Exwith Π([U,e]) = 0∈M⊗CCx. Ase∈E(U), there exist +openx∈V⊆Uandm∈Mwithe(x′) =m⊗1Cx′∈M⊗CCx′for allx′∈V. +Thenm⊗1Cx=e(x) = Π([U,e]) = 0 inM⊗CCx, som∈I·M⊆M, and we +may writem=/summationtextk +a=1ia·maforia∈Iandma∈M. By (2.2) we may choose +d1,...,dk∈Cwithx(da)/\e}atio\slash= 0 andia·da= 0 inCfora= 1,...,k. +SetW={x′∈V:x′(da)/\e}atio\slash= 0, a= 1,...,k}, so thatWis an open +neighbourhood of xinU. Ifx′∈Wthenx′(da)/\e}atio\slash= 0, soπx′(da) is invertible in +Cx′. Butia·da= 0, soπx′(ia) = 0 inCx′fora= 1,...,k. Asm=/summationtextk +a=1ia·ma +it follows that e(x′) =m⊗1Cx′= 0 inM⊗CCx′for allx′∈W. Thuse|W= 0 in +E(W), so [U,e] = [W,e|W] = 0 inEx. Therefore Π :Ex→M⊗CCxis injective, +and so an isomorphism. +Theorem 5.19. LetCbe aC∞-ring, and (X,OX) = Spec C. Then Γ : +OX-mod→C-modisright adjoint toMSpec : C-mod→OX-mod. That +is, for allM∈C-modandE∈OX-modthere are inverse bijections +HomC-mod(M,Γ(E))LM,E/d47/d47HomOX-mod(MSpecM,E), +RM,E/d111/d111 (5.14) +which are functorial in M,E. WhenE= MSpecMwe have ΨM=RM,E(idE), +so thatΨMis the unit of the adjunction between ΓandMSpec. +Proof.LetM∈C-mod andE∈OX-mod, and setD= MSpecM. DefineRM,E +in (5.14) by, for each morphism α:D→EinOX-mod, taking RM,E(α) :M→ +Γ(E) to be the composition +MΨM/d47/d47Γ◦MSpecM= Γ(D)Γ(α)/d47/d47Γ(E). +For the last part, if E= MSpecMthen ΨM=RM,E(idE) as Γ(id E) = idΓ(E). +Letβ:M→Γ(E) be a morphism in C-mod. We will construct a morphism +λ:D→EinOX-mod, and set LM,E(β) =λ. Letx∈X. Consider the diagram +M⊗CC=M +id⊗πx/d15/d15β/d47/d47Γ(E) +σx/d15/d15 +M⊗CCx∼=Dxβx/d47/d47Ex(5.15) +inC-mod, where the isomorphism M⊗CCx∼=Dxcomes from Lemma 5.18. +HereExis the stalk ofEatx, andσx: Γ(E) =E(X)→Extakes stalks at +53x. TheC-action on Γ(E) factors via CΨC−→OX(X), and the C-action onEx +factors via CΨC−→OX(X)π−→OX,x, andβ,σxare both C-module morphisms. +ButOX,x∼=Cxby Lemma 4.18, so σx◦β:M→Exis aC-module morphism, +where the C-action onExfactors via Cπx−→Cx. Hence there is a unique OX,x- +module morphism βx:Dx→Exmaking (5.15) commute. +For each open U⊆X, defineλ(U) :D(U)→E(U) byλ(U)d:x/ma√sto→βx(d(x)) +ford∈D(U) andx∈U⊆X, andd(x)∈Dx, andβx(d(x))∈Ex. Here asE +is a sheaf we may identify elements of E(U) with maps e:U→/coproducttext +x∈UExwith +e(x)∈Exforx∈U, such that esatisfies certain local conditions in U. +Ifd∈D(U) = MSpec M(U) andx∈Uthen by Definition 5.16 we may +coverUby openW⊆Ufor which there exist m∈Mwithd(x) =m⊗1Cxin +M⊗CCxfor allx∈W. Therefore λ(U)dmapsx/ma√sto→σx(β(m)) for allx∈Wby +(5.15), soλ(U)dis a section β(m)|WofEonW. Henceλ(U)dis a section of +E|U, as suchWcoverU, andλ(U) :D(U)→E(U) is well defined. +Asβxis anOX,x-module morphism for all x∈U,λ(U) :D(U)→E(U) is +anOX(U)-module morphism. The definition of λ(U) is clearly compatible with +restriction to open V⊆U⊆X. Thus the λ(U) for all open U⊆Xdefine a +sheaf morphism λ:D→EinOX-mod. SetLM,E(β) =λ. This defines LM,Ein +(5.14). A very similar proof to that of Theorem 4.20 shows that LM,E,RM,Eare +inverse maps, so they are bijections, and that they are functoria l inM,E. +We show that Γ is a right inverse for MSpec: +Proposition 5.20. LetCbe aC∞-ring, and (X,OX) = Spec C,andEbe +anOX-module. Set M= Γ(E)inC-mod,and write ΨE=LM,E(idM). Then +ΨE: MSpec◦Γ(E)→Eis an isomorphism in OX-mod,for anyE. +These isomorphisms ΨEare functorial inE,and so define a natural isomor- +phismΨ : MSpec◦Γ⇒idOX-modof functorsOX-mod→OX-mod. +Proof.SetD= MSpecM= MSpec◦Γ(E), and letx∈X. Then by definition +of ΨE=LM,E(idM) :D→Ein the proof of Theorem 5.19, as in (5.15) the +stalk map Ψ E,x:Dx→Exis the unique morphism of modules over Cx∼=OX,x +making the following diagram of C-modules commute: +M⊗CC=M +id⊗πx/d15/d15idM/d47/d47M= Γ(E) +σx/d15/d15 +M⊗CCx∼=DxΨE,x/d47/d47Ex.(5.16) +Let [U,e]∈Ex, so thatx∈U⊆Xis open and e∈E(U). By Definition +4.13 there exists c∈Csuch thatx(c)/\e}atio\slash= 0 andy(c) = 0 for all y∈X\U. +Choose smooth f:R→Rsuch thatf= 0 near 0 in Randf= 1 nearx(c) +inR. Setc′= Φf(c), where Φ f:C→Cis theC∞-ring operation. Then +η= ΨC(c′)∈OX(X), and there exist open neighbourhoods VofX\UandW +ofxinXwithη|V= 0 andη|W= 1. Clearly V∩W=∅, sox∈W⊆U. We +haveη|U·e∈E(U), with (η|U·e)|U∩V= 0 and (η|U·e)|W=e|W. +54Since{U,V}is an open cover of Xand (η|U·e)|U∩V= 0 = 0|U∩V, by the +sheaf property of Ethere is a unique e′∈E(X) withe′|U=η|U·eande′|V= 0. +Thene′|W= (η|U·e)|W=e|W. Thus +σx(e′) = [X,e′] = [W,e′|W] = [W,e|W] = [U,e] +inEx. Henceσx: Γ(E)→Exis surjective, so Ψ E,x:Dx→Exis surjective by +(5.16), asπx:C→Cxis surjective by Proposition 2.14. +Supposed∈Dxwith Ψ E,x(d) = 0. We may write m⊗1Cx∼=dunder +the isomorphism M⊗CCx∼=Dxfor somem∈M, and then (5.16) gives +σx(m) = ΨE,x(d) = 0. Hence there exists open x∈U⊆Xwithm|U= 0. As +above we may construct η∈OX(X) and open V,W⊆XwithX\U⊆V, +x∈W⊆U,η|V= 0 andη|W= 1. Then η·m= 0 inMasm|U= 0,η|V= 0 +withU∪V=X, andπx(η) = 1CxinCxasη= 1 nearxinX. Hence +m⊗1Cx=1Cx·(m⊗1Cx)=πx(η)·(m⊗1Cx)=(η·m)⊗1Cx=0⊗1Cx=0 +inM⊗CCx. Therefore d= 0 inDx, and Ψ E,x:Dx→Exis injective, and so +an isomorphism. As this holds for all x∈X, ΨE:D→Eis an isomorphism, +proving the first part of the proposition. The second part follows f romLM,E +functorial in M,Ein Theorem 5.19. +As for quasicoherent sheaves in conventional algebraic geometry , we define: +Definition 5.21. LetX= (X,OX) be aC∞-scheme, andEbe anOX-module. +We callEquasicoherent if we may cover Xwith open U⊆Xsuch that +(U,OX|U)∼=SpecCandE|U∼=MSpecMfor someC∞-ringCandC-moduleM. +We write qcoh( X) for the category of quasicoherent sheaves on X. +If (X,OX) is aC∞-scheme andEanOX-module, we can cover Xby open +U⊆Xwith (U,OX|U)∼=SpecCaffine, and then Proposition 5.20 shows that +E|U∼=MSpecMforM=E(U). Thus we have: +Corollary 5.22. LetX= (X,OX)be aC∞-scheme. Then every OX-module +Eis quasicoherent, so that qcoh(X) =OX-mod. +Remark 5.23. (a) In conventional algebraic geometry, as in Hartshorne [31, +§II.5], ifRis a ring and ( X,OX) = SpecRthe corresponding affine scheme, we +also have functors MSpec : R-mod→OX-mod and Γ :OX-mod→R-mod. In +C∞-algebraic geometry, as in Proposition 5.20, Γ is a right inverse for MS pec, +but may not be a left inverse. But in algebraic geometry the opposite happens, +as Γ is a left inverse for MSpec [31, Cor. II.5.5], but may not be a right in verse. +The fact that Γ is a right inverseforMSpec in C∞-algebraicgeometrymeans +that allOX-modules on a C∞-scheme (X,OX) are quasicoherent, so quasico- +herence is not a very useful idea. But in algebraic geometry, as Γ is n ot a right +inverse for MSpec, this is false: there are many examples of schemes ( X,OX) +andOX-modulesEwhich are not quasicoherent. For instance, we may take +X=A1andE(U) = 0 if 0∈U,E(U) =OX(U) if 0/∈Ufor all open U⊆X. +55In§5.5 we will define a module Mover aC∞ringCto becomplete if +M∼=Γ◦MSpecM. Then Γ is a left inverse for MSpec on the subcategory +C-modco⊂C-mod of complete C-modules. In general C-modules need not be +complete. But in conventional algebraic geometry, as Γ is a left inver se for +MSpec allR-modules are complete, so completeness is not a useful idea. +(b)Inconventionalalgebraicgeometryonedefines coherent sheaves [31,§II.5]to +be quasicoherent sheaves Elocally modelled on MSpec MforMa finitely gener- +atedC-module. However, coherent sheaves are only well behaved on noetherian +schemes, and most interesting C∞-rings, such as C∞(Rn) forn >0, are not +noetherian R-algebras. Because of this, coherent sheaves do not seem to be a +useful idea in C∞-algebraic geometry (for instance, coh( X) is not closed under +kernelsin qcoh( X), and is not an abelian category), and we do not discussthem. +We can understand the pullback functor f∗in Definition 5.13 explicitly in +terms of modules over the corresponding C∞-rings: +Proposition 5.24. LetC,DbeC∞-rings,φ:D→Ca morphism, M,NbeD- +modules, and α:M→Na morphism of D-modules. Write X= SpecC, Y= +SpecD, f= Specφ:X→Y,andE= MSpecM,F= MSpecNinqcoh(Y). +Then there are natural isomorphisms f∗(E)∼=MSpec(M⊗DC)andf∗(F)∼= +MSpec(N⊗DC)inqcoh(X). These identify MSpec(α⊗idC) : MSpec(M⊗D +C)→MSpec(N⊗DC)withf∗(MSpecα) :f∗(E)→f∗(F). +Proof.WriteX= (X,OX),Y= (Y,OY) andf= (f,f♯). ThenEis the +sheafificationofthepresheaf V/ma√sto→M⊗DOY(V), andf−1(E) isthe sheafification +of the presheaf U/ma√sto→limV⊇f(U)E(V), andf−1(OY) is the sheafification of the +presheafU/ma√sto→limV⊇f(U)OY(V). Inf∗(E) =f−1(E)⊗f−1(OY)OX, these three +sheafifications combine into one, so f∗(E) is the sheafification of the presheaf +U/ma√sto→limV⊇f(U)(M⊗DOY(V))⊗OY(V)OX(U). But +(M⊗DOY(V))⊗OY(V)OX(U)∼=M⊗DOX(U)∼=(M⊗DC)⊗COX(U), +sothis iscanonicallyisomorphictothepresheaf U/ma√sto→(M⊗DC)⊗COX(U) whose +sheafification is MSpec( M⊗DC). This gives a natural isomorphism f∗(E)∼= +MSpec(M⊗DC). The same holds for N. The identification of MSpec( α⊗idC) +andf∗(MSpecα)followsbypassingfrommorphismsofpresheavestomorphisms +of the associated sheaves. +5.5 Complete modules over C∞-rings +Here are the module analogues of Definition 4.35 and Theorem 4.36(b) ,(c). +Definition 5.25. LetCbe aC∞-ring, andMaC-module. We call Mcomplete +if ΨM:M→Γ◦MSpecMin Definition 5.17 is an isomorphism. +WriteC-modcofor the full subcategory of complete C-modules in C-mod. +IfMis aC-module then applying Γ to Proposition 5.20 shows that +Γ(ΨMSpecM) : Γ◦MSpec(Γ◦MSpecM)−→Γ◦MSpecM +56is an isomorphism. From the definitions we can show that Ψ Γ◦MSpecM= +Γ(ΨMSpecM)−1. Thus Γ◦MSpecMis complete, for any C-moduleM. De- +fine a functor Rco +all= Γ◦MSpec : C-mod→C-modco. +Theorem 5.26. LetCbe aC∞-ring, andX= (X,OX) = Spec C. Then +(a)MSpec|C-modco:C-modco→qcoh(X)is an equivalence of categories. +(b)Rco +all:C-mod→C-modcois left adjoint to the inclusion functor inc : +C-modco֒→C-mod. That is,Rco +allis areflection functor . +Proof.For (a), ifM,Nare complete C-modules then putting E= MSpecNin +Theorem 5.19 and using Γ ◦MSpecN∼=N, equation (5.14) shows that +MSpec =LM,E: Hom C-modco(M,N)−→HomOX-mod(MSpecM,MSpecN) +isabijection, wherethedefinitionof LM,EagreeswiththedefinitionofMSpecon +morphisms in this case. Thus MSpec is full and faithful on complete C-modules. +IfE ∈OX-mod = qcoh( X) thenE∼=MSpec◦Γ(E) by Proposition 5.20. +Thus Γ(E)∼=Γ◦MSpec◦Γ(E), so Γ(E) is complete by Definition 5.25. Hence +E∼=MSpec|C-modco[Γ(E)], and the essential image of MSpec |C-modcois qcoh(X). +Therefore MSpec |C-modcois an equivalence of categories. +For (b), let M,NbeC-modules with Ncomplete. Then we have bijections +HomC-modco/parenleftbig +Rco +all(M),N/parenrightbig∼=HomC-mod/parenleftbig +Γ◦MSpecM,Γ◦MSpecN/parenrightbig +∼=HomOX-mod/parenleftbig +MSpec◦Γ◦MSpecM,MSpecN/parenrightbig +∼=HomOX-mod/parenleftbig +MSpecM,MSpecN/parenrightbig (5.17) +∼=HomC-mod/parenleftbig +M,Γ◦MSpecN/parenrightbig∼=HomC-mod/parenleftbig +M,N/parenrightbig +=Hom C-mod/parenleftbig +M,inc(N)/parenrightbig +, +usingN∼=Γ◦MSpecNasNis complete in the first and fifth steps, Theorem +5.19 in the second and fourth, and Proposition 5.20 in the third. The b ijections +(5.17) arefunctorial in M,Naseach step is. Hence Rco +allis left adjoint to inc. +Proposition 5.27. LetCbe aC∞-ring and (X,OX) = Spec C,and suppose +Xis Lindel¨ of. Then C-modcois closed under kernels, cokernels and extensions +inC-mod,that is,C-modcois an abelian subcategory of C-mod. +Proof.As in§5.4, MSpec : C-mod→OX-mod is an exact functor, and as Xis +Lindel¨ of Γ :OX-mod→C-mod is also exact by Proposition 5.11. Hence Rco +all= +Γ◦MSpec : C-mod→C-mod is an exact functor. Let 0 →M1→M2→M3be +exact in C-mod withM2,M3complete. Then we have a commutative diagram +0 /d47/d47M1 +ΨM1/d15/d15/d47/d47M2 +ΨM2∼=/d15/d15/d47/d47M3 +ΨM3∼=/d15/d15 +0 /d47/d47Rco +all(M1) /d47/d47Rco +all(M2) /d47/d47Rco +all(M3) +inC-mod, where both rows are exact as Rco +allis an exact functor, and the second +and third columns are isomorphisms. Hence the first column is also an is omor- +phism, and M1is complete, so C-modcois closed under kernels in C-mod. It is +closed under cokernels and extensions by very similar arguments. +57Example 5.28. LetCbe aC∞-ring with ( X,OX) = Spec C. Then: +(a)Considering Cas aC-module, we have Γ ◦MSpecC= Γ◦SpecC=OX(X), +and Ψ C:C→OX(X) in Definitions 4.19 and 5.17 coincide. Hence Cis +complete as a C-module if and only if it is complete as a C∞-ring, in the +sense of§4.6. So, if Cis a finitely generated but not fair C∞-ring, as in +Examples 2.19 and 2.21, then Cis a non-complete C-module. +(b)Suppose Cis complete and Xis Lindel¨ of. Let Mbe a finitely presented +C-module, so we have an exact sequence C⊗Rm→C⊗Rn→M→0 +inC-mod. Here C⊗Rm,C⊗Rnare complete as Cis by(a), soMis +complete by Proposition 5.27 as C-mod is closed under cokernels. +(c)Suppose Cis complete, Xis Lindel¨ of, and I⊆Cis a finitely generated +ideal. Choose generators i1,...,inforI. Then we have an exact sequence +C⊗Rn→C→C/I→0 inC-mod with C⊗Rn,Ccomplete, so C/Iis a +complete C-module by Proposition 5.27. Also we have an exact sequence +0→I→C→C/IwithC,C/Icomplete, so Iis a complete C-module. +(d)LetCbe complete and Vbe an infinite-dimensional R-vector space. One +can show that C⊗RVis a complete C-module if and only if Xis compact. +5.6 Cotangent sheaves of C∞-schemes +We nowdefine cotangent sheaves , the sheafversionofcotangentmodules in §5.2. +Definition 5.29. LetX= (X,OX) be aC∞-ringed space. Define PT∗Xto +associate to each open U⊆Xthe cotangent module Ω OX(U)of Definition 5.3, +regarded as a module over the C∞-ringOX(U), and to each inclusion of open +setsV⊆U⊆Xthe morphism of OX(U)-modules Ω ρUV: ΩOX(U)→ΩOX(V) +associated to the morphism of C∞-ringsρUV:OX(U)→OX(V). Then as we +want for (5.13) the following commutes: +OX(U)×ΩOX(U) +ρUV×ΩρUV/d15/d15µOX(U)/d47/d47ΩOX(U) +ΩρUV/d15/d15 +OX(V)×ΩOX(V)µOX(V)/d47/d47ΩOX(V). +Using this and functoriality of cotangent modules Ω ψ◦φ= Ωψ◦Ωφin Definition +5.3, we see thatPT∗Xis a presheaf ofOX-modules on X. Define the cotangent +sheafT∗XofXto be the sheaf of OX-modules associated to PT∗X. +IfU⊆Xis open then we have an equality of sheaves of OX|U-modules +T∗(U,OX|U) =T∗X|U. +As in Example 5.4, if f:X→Yis a smooth map of manifolds we have a +morphism d f:f∗(T∗Y)→T∗Xof vector bundles over X. Here is an analogue +forC∞-ringed spaces. Let f:X→Ybe a morphism of C∞-ringed spaces. +Then by Definition 5.13, f∗(T∗Y) =f−1(T∗Y)⊗f−1(OY)OX,whereT∗Yis the +58sheafification of the presheaf V/ma√sto→ΩOY(V), andf−1(T∗Y) the sheafification of +the presheaf U/ma√sto→limV⊇f(U)(T∗Y)(V), andf−1(OY) the sheafification of the +presheafU/ma√sto→limV⊇f(U)OY(V). These three sheafifications combine into one, +so thatf∗(T∗Y) is the sheafification of the presheaf P(f∗(T∗Y)) acting by +U/ma√sto−→P(f∗(T∗Y))(U) = limV⊇f(U)ΩOY(V)⊗OY(V)OX(U). +Define a morphism of presheaves PΩf:P(f∗(T∗Y))→PT∗XonXby +(PΩf)(U) = limV⊇f(U)(Ωρf−1(V)U◦f♯(V))∗, +where (Ω ρf−1(V)U◦f♯(V))∗: ΩOY(V)⊗OY(V)OX(U)→ΩOX(U)= (PT∗X)(U) is +constructed as in Definition 5.3 from the C∞-ring morphisms f♯(V) :OY(V)→ +OX(f−1(V)) fromf♯:OY→f∗(OX) corresponding to f♯infas in (4.3), and +ρf−1(V)U:OX(f−1(V))→OX(U) inOX. Define Ω f:f∗(T∗Y)→T∗Xto be +the induced morphism of the associated sheaves. +Remark 5.30. There is an alternative definition of the cotangent sheaf T∗X +following Hartshorne [31, p. 175]. We can form the product X×XinC∞RS, +and there is a natural diagonal morphism ∆ X:X→X×X. WriteIXfor +the sheaf of ideals in OX×Xvanishing on the closed C∞-ringed subspace ∆ X. +ThenT∗X∼=∆∗ +X(IX/I2 +X). This can be proved using the equivalence of two +definitions of cotangent module in [31, Prop. II.8.1A]. An affine version of this +also appears in Dubuc and Kock [25]. +Proposition 5.31. LetCbe aC∞-ring andX= SpecC. Then there is a +canonical isomorphism T∗X∼=MSpecΩ C. +Proof.By Definitions 5.16 and 5.29, MSpecΩ CandT∗Xare sheafifications of +presheavesPMSpecΩ C,PT∗X, where for open U⊆Xwe have +PMSpecΩ C(U) = ΩC⊗COX(U) andPT∗X(U) = ΩOX(U). +We haveC∞-ringmorphismsΨ C:C→OX(X) fromDefinition 4.19andrestric- +tionρXU:OX(X)→OX(U) fromOX, and so as in Definition 5.3 a morphism +ofOX(U)-modulesPρ(U) := (ρXU◦ΨC)∗: ΩC⊗COX(U)→ΩOX(U). This de- +fines a morphism of presheaves Pρ:PMSpecΩ C→PT∗X, and so sheafifying +induces a morphism ρ: MSpecΩ C→T∗X. +The induced morphism on stalks at x∈Xisρx= (πx)∗: ΩC⊗CCx→ΩCx, +whereπx:C→Cxisprojectiontothelocal C∞-ringCx, notingthatOX,x∼=Cx. +ButCxis the localization C[c−1:c∈C,c(x)/\e}atio\slash= 0], so Proposition 5.7 implies +that (πx)∗: ΩC⊗CCx→ΩCxis an isomorphism. Hence ρ: MSpecΩ C→T∗X +is a sheaf morphism which induces isomorphisms on stalks at all x∈X, soρis +an isomorphism. +Here are some properties of the morphisms Ω fin Definition 5.29. Equation +(5.20) is an analogue of (5.6) and (5.12). +59Theorem 5.32. (a) Letf:X→Yandg:Y→Zbe morphisms of C∞- +schemes. Then +Ωg◦f= Ωf◦f∗(Ωg)◦If,g(T∗Z) (5.18) +as morphisms (g◦f)∗(T∗Z)→T∗Xinqcoh(X). HereΩg:g∗(T∗Z)→T∗Yis a +morphism in qcoh(Y),so applying f∗givesf∗(Ωg) :f∗(g∗(T∗Z))→f∗(T∗Y)in +qcoh(X),andIf,g(T∗Z) : (g◦f)∗(T∗Z)→f∗(g∗(T∗Z))is as in Remark 5.14. +(b)Suppose we are given a Cartesian square in C∞Sch +Wf/d47/d47 +e/d15/d15Y +h/d15/d15 +Xg/d47/d47Z,(5.19) +so thatW=X×ZY. Then the following is exact in qcoh(W): +(g◦e)∗(T∗Z)e∗(Ωg)◦Ie,g(T∗Z)⊕ +−f∗(Ωh)◦If,h(T∗Z)/d47/d47e∗(T∗X) +⊕f∗(T∗Y)Ωe⊕Ωf/d47/d47T∗W /d47/d470.(5.20) +Proof.Combining several sheafifications into one as in the proof of Propos ition +5.24, we see that the sheaves T∗X,f∗(T∗Y),f∗(g∗(T∗Z)) and (g◦f)∗(T∗Z) on +Xare isomorphic to the sheafifications of the following presheaves: +T∗X/squigglerightU/ma√sto−→ΩOX(U), (5.21) +f∗(T∗Y)/squigglerightU/ma√sto−→lim +V⊇f(U)ΩOY(V)⊗OY(V)OX(U), (5.22) +f∗(g∗(T∗Z))/squigglerightU/ma√sto−→lim +V⊇f(U)lim +W⊇g(V)/parenleftbig +ΩOZ(W)⊗OZ(W)OY(V)/parenrightbig +⊗OY(V)OX(U),(5.23) +(g◦f)∗(T∗Z)/squigglerightU/ma√sto−→lim +W⊇g◦f(U)ΩOZ(W)⊗OZ(W)OX(U). (5.24) +Then Ωf,Ωg◦f,f∗(Ωg),If,g(T∗Z) are the morphisms of sheaves associated +to the following morphisms of the presheaves in (5.21)–(5.24): +Ωf/squigglerightU/ma√sto−→lim +V⊇f(U)(Ωρf−1(V)U◦f♯(V))∗, (5.25) +Ωg◦f/squigglerightU/ma√sto−→lim +W⊇g◦f(U)(Ωρ(g◦f)−1(W)U◦(g◦f)♯(W))∗,(5.26) +f∗(Ωg)/squigglerightU/ma√sto−→lim +V⊇f(U)lim +W⊇g(V)(Ωρg−1(W)V◦g♯(W))∗,(5.27) +If,g(T∗Z)/squigglerightU/ma√sto−→lim +V⊇f(U)lim +W⊇g(V)IUVW, (5.28) +by Definition 5.29, where IUVW: ΩOZ(W)⊗OZ(W)OX(U)→/parenleftbig +ΩOZ(W)⊗OZ(W) +OY(V)/parenrightbig +⊗OY(V)OX(U) is the natural isomorphism. +Now ifU⊆X,V⊆Y,W⊆Zare open with V⊇f(U),W⊇g(V) then +ρ(g◦f)−1(W)U◦(g◦f)♯(W) =/bracketleftbig +ρf−1(V)U◦f♯(V)/bracketrightbig +◦/bracketleftbig +ρg−1(W)V◦g♯(W)/bracketrightbig +60as morphismsOZ(W)→OX(U), so Ωφ◦ψ= Ωφ◦Ωψin Definition 5.3 implies +(Ωρ(g◦f)−1(W)U◦(g◦f)♯(W))∗= (Ωρf−1(V)U◦f♯(V))∗◦(Ωρg−1(W)V◦g♯(W))∗◦IUVW. +Taking limits lim V⊇f(U)limW⊇g(V)implies that the morphisms of presheaves in +(5.25)–(5.28) satisfy the analogue of (5.18), so passing to sheave s proves (a). +For (b), first observe that as (5.19) is commutative, by (a) we hav e +Ωe◦e∗(Ωg)◦Ie,g(T∗Z) = Ωg◦e= Ωh◦f= Ωf◦f∗(Ωh)◦If,h(T∗Z), +so Ωe◦/parenleftbig +e∗(Ωg)◦Ie,g(T∗Z)/parenrightbig +−Ωf◦/parenleftbig +f∗(Ωh)◦If,h(T∗Z)/parenrightbig += 0, +and (5.20) is a complex. To show it is exact, note that as in the first pa rt +of the proof, (5.20) is the sheafification of a complex of presheave s, and the +presheaves are defined as direct limits. Let S⊆Wbe open. Then the complex +ofpresheavescorrespondingto (5.20) evaluatedat S⊆Wis the directlimit over +all openT⊆X,U⊆Y,V⊆Zwithe(S)⊆T,f(S)⊆U,g(T)⊆V,h(U)⊆V +of equation (5.6) with OZ(V),OX(T),OY(U),OW(S) in place of C,D,E,F. +Since (5.6) is exact by Theorem 5.8 and direct limits are exact, the com plex +ofpresheaveswhose sheafificationis (5.20) is exact when evaluate d on each open +S⊆W, so it is exact. As sheafification is an exact functor, this implies that +equation (5.20) is exact. This completes the proof. +6C∞-stacks +We now discuss C∞-stacks, that is, geometric stacks over the site ( C∞Sch,J) +ofC∞-schemes with the open cover topology. The author knows of no pr evious +work on these. For the rest of the book, we will assume the reader has some +familiarity with stacks in algebraicgeometry. Appendix A summarizes t he main +definitions and results on stacks that we will use, but it is too brief to help +someone learn about stacks for the first time. Readers with little ex perience +of stacks are advised to first consult an introductory text such a s Vistoli [68], +Gomez [29], Laumon and Moret-Bailly [46], or the online ‘Stacks Project ’ [34]. +The author found Metzler [49] and Noohi [55] useful in writing this section. +6.1C∞-stacks +We use the material of §A.2–§A.5. +Definition 6.1. Define a Grothendieck pretopology PJon the category of +C∞-schemes C∞Schto have coverings {ia:Ua→U}a∈AwhereVa=ia(Ua) +is open inUwithia:Ua→(Va,OU|Va) and isomorphism for all a∈A, and +U=/uniontext +a∈AVa. Using Corollary 4.29 we see that up to isomorphisms of the +Ua, the coverings{ia:Ua→U}a∈AofUcorrespond exactly to open covers +{Va:a∈A}ofU. WriteJfor the associated Grothendieck topology. +It is a straightforward exercise in sheaf theory to prove: +61Proposition6.2. The site(C∞Sch,J)has descent for objects and morphisms, +in the sense of§A.3. Thus it is subcanonical. +The point here is that since coverings of UinJare just open covers of the +underlying topological space U, rather than something more complicated like +´ etale covers in algebraic geometry, proving descent is easy: for o bjects, we glue +the topological spaces XaofXatogether in the usual way to get a topological +spaceX, then we glue the OXatogether to get a presheaf of C∞-rings˜OXon +Xisomorphic toOXaonXa⊆Xfor alla∈A, and finally we sheafify ˜OXto a +sheaf ofC∞-ringsOXonX, which is still isomorphic to OXaonXa⊆X. +Definition 6.3. AC∞-stackXis a geometric stack on the site ( C∞Sch,J). +WriteC∞Stafor the 2-category of C∞-stacks,C∞Sta=GSta(C∞Sch,J). +As in Definition A.13, we will very often use the notation that if Xis a +C∞-scheme then ¯Xis the associated C∞-stack, and if f:X→Yis a mor- +phism ofC∞-schemes then ¯f:¯X→¯Yis the associated 1-morphism of C∞- +stacks. Write ¯C∞Schlfp,¯C∞Schlf,¯C∞Schfor the full 2-subcategories of C∞- +stacksXinC∞Stawhich are equivalent to ¯XforXinC∞Schlfp,C∞Schlf +orC∞Sch, respectively. When we say that a C∞-stackXis aC∞-scheme, we +mean thatX∈¯C∞Sch. +Since(C∞Sch,J)isasubcanonicalsite, theembedding C∞Sch→C∞Sta +takingX/ma√sto→¯X,f/ma√sto→¯fis fully faithful. We write this as a full and faithful +functorFC∞Sta +C∞Sch:C∞Sch→C∞StamappingFC∞Sta +C∞Sch:X/ma√sto→¯Xon objects +andFC∞Sta +C∞Sch:f/ma√sto→¯fon (1-)morphisms. Hence ¯C∞Schlfp,¯C∞Schlf,¯C∞Sch +areequivalentto C∞Schlfp,C∞Schlf,C∞Sch, consideredas 2-categorieswith +only identity 2-morphisms. In practice one often does not distinguis h between +schemes and stacks which are equivalent to schemes, that is, one id entifies +C∞Schlfp,...,C∞Schand¯C∞Schlfp,...,¯C∞Sch. +Remark 6.4. Behrend and Xu [5, Def. 2.15] use ‘ C∞-stack’to mean something +different, a stack Xover the site ( Man,JMan) of manifolds with Grothendieck +topologyJManassociated to the Grothendieck pretopology PJMangiven by +opencovers,suchthatthereexistsasurjectiverepresentable submersion π:¯U→ +Xfrom some manifold U. These arealsocalled ‘smooth stacks’or‘differentiable +stacks’in[5,32,49,55]. Thequotient[ V/G]ofamanifold VbyaLiegroup Gisan +example of a differentiable stack. By Zung’s linearization theorem [71 , Th. 2.3], +a differentiable stack Xwith proper diagonal is Zariski locally equivalent to +such a quotient [ V/G] withGcompact. Our C∞-stacks are a far larger class of +more singular objects than the differentiable stacks of [5,32,49, 55]. +Theorems 4.25(b) and A.23, Corollary A.26 and Proposition 6.2 imply: +Theorem 6.5. LetXbe aC∞-stack. ThenXis equivalent to the stack +[V⇒U]associated to a groupoid (U,V,s,t,u,i,m)inC∞Sch. Conversely, +any groupoid in C∞Schdefines aC∞-stack[V⇒U]. All fibre products exist +in the2-category C∞Sta. +QuotientC∞-stacks[X/G] are a special class of C∞-stacks. +62Definition 6.6. AC∞-groupGis a group object in C∞Sch, that is, a C∞- +schemeG= (G,OG) equipped with an identity element 1 ∈Gand multiplica- +tion and inverse morphisms m:G×G→G,i:G→GinC∞Schsuch that +(∗,G,π,π,1,i,m) is a groupoid in C∞Sch. Here∗= SpecRis a point, and +π:G→∗is the projection, and we regard 1 ∈Gas a morphism 1 : ∗→G. +LetGbe aC∞-group, and XaC∞-scheme. A ( left)actionofGonXis a +morphismµ:G×X→Xsuch that +/parenleftbig +X,G×X,πX,µ,1×idX,(i◦πG)×µ,(m◦((πG◦π1)×(πG◦π2)))×(πX◦π2)/parenrightbig +(6.1) +is a groupoid object in C∞Sch, where in the final morphism π1,π2are the +projections from ( G×X)×πX,X,µ(G×X) to the first and second factors +G×X. Then define the quotientC∞-stack[X/G] to be the stack [ G×X⇒X] +associated to the groupoid (6.1). It is a C∞-stack. +IfG= (G,OG) is aC∞-group then the underlying space Gis a topological +group, and is in particular a group, and if G= (G,OG) acts onX= (X,OX) +thenGacts continuously on X. +IfGis a Lie group then G=FC∞Sch +Man(G) is aC∞-group in a natural way, by +applyingFC∞Sch +Mantothesmoothmultiplicationandinversemaps m:G×G→G +andi:G→G. If a Lie group Gacts smoothly on a manifold Xwith action +µ:G×X→Xthen theC∞-groupG=FC∞Sch +Man(G) acts on the C∞-scheme +X=FC∞Sch +Man(X) with action µ=FC∞Sch +Man(µ) :G×X→X, so we can form +the quotient C∞-stack [X/G]. +Example 6.7. LetGbe aC∞-group, and X=∗be the point in C∞Sch, with +trivialG-action. The quotient C∞-stack [∗/G] is known as BG, the classifying +stack for principal G-bundles on C∞-schemes. +IfSis aC∞-scheme, a principalG-bundle(P,π,µ) overSis aC∞-scheme +P, a morphism π:P→S, and aG-actionµ:G×P→PofGonP, such +thatπisG-invariant, and Smay be covered by open C∞-subschemes U⊆S +such that there exists an isomorphism π−1(U)∼=G×Uwhich identifies the +G-action onπ−1(U)⊆Pwith the product of the left G-action onGand the +trivialG-action onU, and identifies π|···:π−1(U)→UwithπU:G×U→U. +Often we write Pas the principal bundle, leaving π,µimplicit. +One well known way to write BGexplicitly as a category fibred in groupoids +pX:X→C∞Sch, as in§A.2, is to defineXto be the category with objects +pairs (S,P) of aC∞-schemeSandPa principal G-bundle over S, and mor- +phisms (f,u) : (S,P)→(T,Q) consisting of C∞-scheme morphisms f:S→T +andu:P→Q, such that uisG-equivariant and +P +π/d15/d15u/d47/d47Q +π/d15/d15 +Sf/d47/d47T(6.2) +is a Cartesian square in C∞Sch, which implies that Pis canonicallyisomorphic +to the pullback principal G-bundlef∗(Q). Composition of morphisms is ( g,v)◦ +63(f,u) = (g◦f,v◦u), and identity morphisms are id (S,P)= (idS,idP). The +functorpX:X→C∞SchmapspX: (S,P)/ma√sto→Sonobjectsand pX: (f,u)/ma√sto→f +on morphisms. +In§7.1 we will give a more detailed treatment of quotient C∞-stacks [X/G] +of aC∞-schemeXby a finite group G. +6.2 Properties of 1-morphisms of C∞-stacks +We use the material of §A.4. We define some classes of C∞-scheme morphisms. +Definition 6.8. Letf= (f,f♯) :X= (X,OX)→Y= (Y,OY) be a morphism +inC∞Sch. Then: +•We callfanopen embedding ifV=f(X) is an open subset in Yand +(f,f♯) : (X,OX)→(V,OY|V) is an isomorphism. +•We callfaclosed embedding iff:X→Yis a homeomorphism with +a closed subset of Y, andf♯:f−1(OY)→OXis a surjective morphism +of sheaves of C∞-rings. Equivalently, fis an isomorphism with a closed +C∞-subscheme of Y. Over affine open subsets U∼=SpecCinY,fis +modelled on the natural morphism Spec( C/I)֒→SpecCfor some ideal I +inC. +•We callfanembedding if we may write f=g◦hwherehis an open +embedding and gis a closed embedding. +•We callf´ etaleif eachx∈Xhas an open neighbourhood UinXsuch +thatV=f(U) is open in Yand (f|U,f♯|U) : (U,OX|U)→(V,OY|V) is +an isomorphism. That is, fis a local isomorphism. +•We callfproperiff:X→Yis a proper map of topological spaces, that +is, ifS⊆Yis compact then f−1(S)⊆Xis compact. +•We say that fhas finite fibres iff:X→Yis a finite map, that is, f−1(y) +is a finite subset of Xfor ally∈Y. +•We callfseparated iff:X→Yis a separated map of topological spaces, +that is, ∆ X=/braceleftbig +(x,x) :x∈X/bracerightbig +is a closed subset of the topological fibre +productX×f,Y,fX=/braceleftbig +(x,x′)∈X×X:f(x) =f(x′)/bracerightbig +. +•We callfclosediff:X→Yis a closed map of topological spaces, that +is,S⊆Xclosed implies f(S)⊆Yclosed. +•We callfuniversally closed if whenever g:W→Yis a morphism then +πW:X×f,Y,gW→Wis closed. +•We callfasubmersion if for allx∈Xwithf(x) =y, there exists an open +neighbourhood UofyinYand a morphism g= (g,g♯) : (U,OY|U)→ +(X,OX) withg(y) =xandf◦g= id(U,OY|U). +64•We callflocally fair , orlocally finitely presented , if whenever Uis a locally +fair, or locally finitely presented C∞-scheme, respectively, and g:U→Y +is a morphism then X×f,Y,gUis locally fair, or locally finitely presented, +respectively. +Remark 6.9. These are mostly analogues of standard concepts in algebraic +geometry, as in Hartshorne [31] for instance. But because the to pology onC∞- +schemes is finer than the Zariski topology in algebraic geometry — fo r example, +affineC∞-schemes are Hausdorff — our definitions of ´ etale and proper are s im- +pler than in algebraic geometry. (Open or closed) embeddings corre spond to +(open or closed) immersions in algebraic geometry, but we prefer th e word ‘em- +bedding’, as immersion has a different meaning in differential geometry . Closed +morphisms are not invariant under base change, which is why we defin e univer- +sally closed. If X,Yare manifolds and X,Y=FC∞Sch +Man(X,Y), thenf:X→Y +is a submersion of C∞-schemes if and only if f=FC∞Sch +Man(f) forf:X→Ya +submersion of manifolds. +Definition 6.10. LetPbe a property of morphisms in C∞Sch. We say that +Pis stable under open embedding if whenever f:U→VisPandi:V→W +is an open embedding, then i◦f:U→WisP. +The next proposition is elementary. See Laumon and Bailly [46, §3.10] and +Noohi [55, Ex. 4.6] for similar lists for the ´ etale and topological sites . +Proposition 6.11. The following properties of morphisms in C∞Schare in- +variant under base change and local in the target in the site (C∞Sch,J),in +the sense of§A.4:open embedding, closed embedding, embedding, ´ etale, prop er, +has finite fibres, separated, universally closed, submersio n, locally fair, locally +finitely presented. The following properties are also stabl e under open embed- +ding, in the sense of Definition 6.10:open embedding, embedding, ´ etale, has +finite fibres, separated, submersion, locally fair, locally finitely presented. +As in§A.4, this implies that these properties are also defined for repre- +sentable 1-morphisms in C∞Sta. In particular, if Xis aC∞-stack then ∆ X: +X →X×X is representable, and if Π : ¯U→Xis an atlas then Π is repre- +sentable, so we can require that ∆ Xor Π has some of these properties. +Definition 6.12. LetXbe aC∞-stack. Following [46, Def. 7.6], we say that X +isseparated if the diagonal 1-morphism ∆ X:X→X×X is universally closed. +IfX=¯Xfor someC∞-schemeX= (X,OX) thenXis separated if and only if +∆X:X→X×Xis closed, that is, if and only if Xis Hausdorff. +Proposition 6.13. LetW=X×f,Z,gYbe a fibre product of C∞-stacks with +X,Yseparated. ThenWis separated. +65Proof.We have a 2-commutative diagram with both squares 2-Cartesian: +W∆W/d47/d47 +/d117/d117❦❦❦❦❦❦❦❦❦❦❦π1/d41/d41❙❙❙❙❙❙❙❙❙ W×W +/d41/d41❙❙❙❙❙❙❙ +Z +Z /d41/d41❙❙❙❙❙❙❙❙❙❙❙ X×f◦∆Z,Z×Z,g◦ZY +/d117/d117❦❦❦❦❦❦❦❦ +/d41/d41❙❙❙❙❙❙π2/d53/d53❦❦❦❦❦❦❦ +X×X×Y×Y . +IZ X×Y∆X×∆Y/d53/d53❦❦❦❦❦❦❦(6.3) +Let [V⇒U] be a groupoid presentation of Z, and consider the fourth 2- +Cartesian diagram of (A.12), with surjective rows. The left hand mo rphism +¯uׯidUhas a left inverse πU, and so is automatically universally closed. Hence +Zis universally closed by Propositions A.18(c) and 6.11, so π1in (6.3) is uni- +versally closed by Propositions A.18(a) and 6.11. Also ∆ X,∆Yare universally +closed asX,Yare separated, so ∆ X×∆Yin (6.3) is universally closed, and +π2is universally closed. Thus ∆ W∼=π2◦π1is universally closed, and Wis +separated. +6.3 Open C∞-substacks and open covers +Definition 6.14. LetXbe aC∞-stack. AC∞-substackYinXis a substack +ofX, in the sense of Definition A.7, which is also a C∞-stack. It has a natural +inclusion 1-morphism iY:Y֒→X. We callYanopenC∞-substack ofXif +iYis a representable open embedding, a closedC∞-substack ofXifiYis a +representable closed embedding, and a locally closed C∞-substack ofXifiYis +a representable embedding. +Anopen cover{Ua:a∈A}ofXis a family of open C∞-substacksUainX +with/coproducttext +a∈AiUa:/coproducttext +a∈AUa→Xsurjective. We write U⊆XwhenUis an open +C∞-substack ofX, and/uniontext +a∈AU=Xto mean that/coproducttext +a∈AiUais surjective. +Somepropertiesof∆ X,ιX,XandatlasesforXcanbetestedontheelements +of an open cover. The proof is elementary. +Proposition 6.15. LetXbe aC∞-stack, and{Ua:a∈A}an open cover +ofX. Suppose PandQare properties of morphisms in C∞Schwhich are +invariant under base change and local in the target in (C∞Sch,J),and that P +is stable under open embedding. Then: +(a)LetΠa:¯Ua→Uabe an atlas forUafora∈A. SetU=/coproducttext +a∈AUaand +Π =/coproducttext +a∈AiUa◦Πa:¯U→X. ThenΠis an atlas forX,andΠisPif +and only if ΠaisPfor alla∈A. +(b)∆X:X→X×X isPif and only if ∆Ua:Ua→Ua×UaisPfor alla∈A. +(c)ιX:IX→XisQif and only if ιUa:IUa→UaisQfor alla∈A. +(d)X:X→IXisQif and only if Ua:Ua→IUaisQfor alla∈A. +IfX=¯Ufor someC∞-schemeU= (U,OU), then the open C∞-substacks +ofXare precisely those subsheaves of the form (V,OU|V) for all open V⊆U, +that is, they are the images in C∞Staof the open C∞-subschemes of U. We +can also describe the open substacks of stacks [ V⇒U] associated to groupoids: +66Proposition 6.16. Let(U,V,s,t,u,i,m)be a groupoid in C∞SchandX= +[V⇒U]the associated C∞-stack, and write U= (U,OU),and so on. Then open +C∞-substacksX′ofXare naturally in 1-1correspondence with open subsets +U′⊆Uwiths−1(U′) =t−1(U′),whereX′= [V′⇒U′]forU′= (U′,OU|U′) +andV′= (s−1(U′),OV|s−1(U′)). If(U,V,s,t,u,i,m)is as in(6.1),so thatX +is a quotient C∞-stack[U/G],then openC∞-substacksX′ofXcorrespond to +G-invariant open subsets U′⊆U. +Proof.From Theorem A.23, as X= [V⇒U] we have a natural surjective, +representable 1-morphism Π : ¯U→X. IfX′is an openC∞-substack ofXthen +¯U×Π,X,iX′X′is an open C∞-substack of ¯U, and so is of the form (U′,OU|U′) +for some open U′⊆U. We have natural equivalences +(s−1(U′),OV|s−1(U′))≃¯U′×i¯U′,¯U,¯s¯V≃X′×X(¯U×id¯U,¯U,¯s¯V)≃X′×i′ +X,X,πX¯V +≃X′×X(¯U×id¯U,¯U,¯t¯V)≃¯U′×i¯U′,¯U,¯t¯V≃(t−1(U′),OV|t−1(U′)), +by associativity properties of fibre products in 2-categories, whic h implies that +s−1(U′) =t−1(U′). Conversely, if s−1(U′) =t−1(U′) then defining U′,V′as in +the proposition, we get a C∞-stackX′= [V′⇒U′] which is naturally an open +C∞-substack ofX. WhenX= [U/G], we see that s−1(U′) =t−1(U′) if and +only ifU′isG-invariant. +6.4 The underlying topological space of a C∞-stack +Following Noohi [55, §4.3,§11] in the case of topological stacks, we associate a +topological space Xtopto aC∞-stackX. In§7.4, ifXis a Deligne–Mumford +C∞-stack, we will also give Xtopthe structure of a C∞-scheme. +Definition 6.17. LetXbe aC∞-stack. Write∗for the point Spec Rin +C∞Sch, and¯∗for the associated point in C∞Sta. DefineXtopto be the +set of 2-isomorphism classes [ x] of 1-morphisms x:¯∗→X. +SupposeU⊆Xis an openC∞-substack. SinceUis a subcategory of X, any +1-morphism u:¯∗→U, regardedasafunctorfromthecategory ¯∗tothe category +U, is also a 1-morphism u:¯∗→X. Also, asUis a strictly full subcategory of +X, ifx:¯∗→Xis a 1-morphism and η:u⇒xa 2-morphism of 1-morphisms +¯∗→X, thenxis also a 1-morphism u:¯∗→U, andηis also a 2-morphism of +1-morphisms ¯∗→U. This implies that Utopis a subset ofXtop. +DefineTXtop=/braceleftbig +Utop:U⊆XisanopenC∞-substackinX/bracerightbig +, asetofsubsets +ofXtop. We claimthatTXtopisatopologyonXtop. Toseethis, notethattaking +Uto beXor the empty C∞-substack givesXtop,∅∈TXtop. IfU,V⊆Xare +openC∞-substacks ofXthen the intersection of subcategories W=U∩Vis +an openC∞-substack ofXequivalent to the fibre product U×iU,X,iVV, with +Wtop=Utop∩Vtop, soTXtopis closed under finite intersections. +If{Ua:a∈A}is a family of open C∞-substacks inX, defineVto be the +unique smallest strictly full subcategory of Xwhich containsUafor eacha∈A +and is closed under the stack axiom (A.9) in Definition A.6. Then Vis an open +67C∞-substack ofX, which we write as V=/uniontext +a∈AUa, andVtop=/uniontext +a∈AUatop. +SoTXtopis closed under arbitrary unions. +Thus (Xtop,TXtop) is a topological space, which we call the underlying topo- +logical space ofX, and usually write as Xtop. It has the following properties. +Iff:X →Y is a 1-morphism of C∞-stacks then there is a natural continu- +ous mapftop:Xtop→Ytopdefined by ftop([x]) = [f◦x]. Iff,g:X →Y +are 1-morphisms and η:f⇒gis a 2-isomorphism then ftop=gtop. Map- +pingX /ma√sto→X top,f/ma√sto→ftopand 2-morphisms to identities defines a 2-functor +FTop +C∞Sta:C∞Sta→Top, where the category of topological spaces Topis +regarded as a 2-category with only identity 2-morphisms. +IfX= (X,OX) is aC∞-scheme, so that ¯Xis aC∞-stack, then ¯Xtopis +naturallyhomeomorphicto X, and we will identify ¯XtopwithX. Iff= (f,f♯) : +X= (X,OX)→Y= (Y,OY) is a morphism of C∞-schemes, so that ¯f:¯X→¯Y +is a 1-morphism of C∞-stacks, then ¯ftop:¯Xtop→¯Ytopisf:X→Y. +For aC∞-stackX, we can characterize Xtopby the following universal +property. We are given a topological space Xtopand for every 1-morphism +f:¯U→Xfor aC∞-schemeU= (U,OU) we are given a continuous map +ftop:U→Xtop, such that if fis 2-isomorphic to h◦¯gfor some morphism +g= (g,g♯) :U→Vand 1-morphism h:V→Xthenftop=htop◦g. IfX′ +top, +f′ +topare alternative choices of data with these properties then there is a unique +continuous map j:Xtop→X′ +topwithf′ +top=j◦ftopfor allf. +We can also make Xtopinto aC∞-ringed spaceXtop: +Definition 6.18. LetXbe aC∞-stack. Define a sheaf of C∞-ringsOXtop +onXtopas follows: each open set in XtopisUtopfor some unique open C∞- +substackU ⊆X. DefineOXtop(Utop) to be the set of 2-isomorphism classes +[c] of 1-morphisms c:U →¯R. Iff:Rn→Ris smooth and [ c1],...,[cn]∈ +OXtop(Utop), define Φ f/parenleftbig +[c1],...,[cn]/parenrightbig +=/bracketleftbig¯f◦(c1×···×cn)/bracketrightbig +, using the compo- +sitionUc1×···×cn−→¯R×···×¯R¯f−→¯R. ThenOXtop(Utop) is aC∞-ring. +IfVtop⊆Utop⊆Xtopare open, so that V ⊆U ⊆X , define aC∞-ring +morphismρUV:OXtop(Utop)→OXtop(Vtop) byρUV: [c]/ma√sto→[c|V]. It is now +easy to check that OXtopis a presheaf of C∞-rings onXtop, but it is less +obvious that it is a sheaf. To see this, note that by general proper ties of stacks, +U /ma√sto→Hom(U,¯R) is a 2-sheaf (stack) of groupoids on the topological space +Xtop, whereHom(U,¯R) is the groupoid of 1- and 2-morphisms U →¯R, and +OXtop(Utop) is its set of isomorphism classes. +Starting with a 2-sheaf and taking sets of isomorphism classes gene rally +yields only a presheaf of sets, not a sheaf. But as ¯Ris aC∞-scheme the +groupoids Hom(U,¯R) are discrete (have no nontrivial automorphisms), so tak- +ing isomorphism classes loses no information, and the 2-sheaf prope rty implies +thatOXtopis a sheaf of sets, and so of C∞-rings. ThusXtop= (Xtop,OXtop) is +aC∞-ringed space, the underlying C∞-ringed space ofX. +For generalXthisXtopneed not be a C∞-scheme. If it is, we call Xtopthe +coarse moduli C∞-scheme ofX. Coarse moduli C∞-schemes have the following +universal property: there is a 1-morphism π:X →¯Xtopcalled the structural +68morphism , such that if f:X→¯Yis a 1-morphism for any C∞-schemeYthen +fis 2-isomorphic to ¯ g◦πfor some unique C∞-scheme morphism g:Xtop→Y. +We can think of a C∞-stackXas being a topological space Xtopequipped +with some complicated extra geometrical structure, just as manif olds and orb- +ifolds are usually thought of as topological spaces equipped with ext ra structure +coming from an atlas of charts. As in Noohi [55, Ex. 4.13], it is easy to d escribe +Xtopusing a groupoid presentation [ V⇒U] ofX: +Proposition 6.19. LetXbe equivalent to the C∞-stack[V⇒U]associated to +a groupoid (U,V,s,t,u,i,m)inC∞Sch,whereU= (U,OU),s= (s,s♯),and so +on. Define∼onUbyp∼p′if there exists q∈Vwiths(q) =pandt(q) =p′. +Then∼is an equivalence relation on U,so we can form the quotient U/∼,with +the quotient topology. There is a natural homeomorphism Xtop∼=U/∼. +For a quotient C∞-stackX≃[U/G]we haveXtop∼=U/G. +Using this we can deduce properties of Xtopfrom properties of Xexpressed +interms ofV⇒U. Forinstance, ifXis separatedthen s×t:V→U×Uis (uni- +versally) closed, and we can take UHausdorff. But the quotient of a Hausdorff +topological space by a closed equivalence relation is Hausdorff, yieldin g: +Lemma 6.20. LetXbe a separated C∞-stack. Then the underlying topological +spaceXtopis Hausdorff. +Next we discuss isotropy groups ofC∞-stacks. +Definition 6.21. LetXbe aC∞-stack, and [ x]∈Xtop. Pick a representative +xfor [x], so thatx:¯∗→Xis a 1-morphism. Then there exists a C∞-scheme +G= (G,OG), unique up to isomorphism, with ¯G=¯∗×x,X,x¯∗. Applying the +construction of the groupoid in Definition A.21 with Π : U→Xreplaced by +x:¯∗→X, we giveGthe structure of a C∞-group. The underlying group Gis +canonically isomorphic to the group of 2-morphisms η:x⇒x. +With [x]fixed, this C∞-groupGisindependent ofchoicesuptononcanonical +isomorphism; roughly, Gis canonical up to conjugation in G. We define the +isotropy group (ororbifold group , orstabilizer group )IsoX([x])orIso([x])of[x]to +be thisC∞-groupG, regarded as a C∞-group up to noncanonical isomorphism. +IfX= [V⇒U] is associatedto a groupoid( U,V,s,t,u,i,m) thenx:¯∗→X +factors through ¯ w:¯∗→¯Uup to 2-isomorphism for some point w∈U, and then +Gis isomorphic to the C∞-subscheme G′=s−1(w)∩t−1(w) inV, with identity +u|w:∗→G′, inversei|G′:G′→G′, and multiplication m|G′×G′:G′×G′→G′. +Iff:X→Yis a 1-morphism of C∞-stacks and [ x]∈Xtopwithftop([x]) = +[y]∈Ytop, fory=f◦x, then at the level of sets we define f∗: IsoX([x])→ +IsoY([y]) byf∗(η) = idf∗η. This is a group morphism, by compatibility of +horizontal and vertical composition in 2-categories. We can exten df∗naturally +to a morphism f∗: IsoX([x])→IsoY([y]) ofC∞-groups, such that +¯f∗:IsoX([x]) =¯∗×x,X,x¯∗−→¯∗×f◦x,Y,f◦x¯∗=IsoY([y]) +is induced from f:X→Yby the universal property of fibre products. Then +f∗,f∗are independent of the choice of x∈[x] up to conjugation in Iso Y([y]). +696.5 Gluing C∞-stacks by equivalences +Here are two propositions on gluing C∞-stacks by equivalences. They are exer- +cises in stack theory, with no special C∞issues, and also hold for other classes +of stacks. See Rydh [61, Th. C] for stronger results for algebraic stacks. +Proposition 6.22. SupposeX,YareC∞-stacks,U ⊆X,V ⊆Yare open +C∞-substacks, and f:U→Vis an equivalence in C∞Sta. Then there exist +aC∞-stackZ,openC∞-substacks ˆX,ˆYinZwithZ=ˆX∪ˆY,equivalences +g:X →ˆXandh:Y→ˆYsuch thatg|Uandh|Vare both equivalences with +ˆX∩ˆY,and a2-morphism η:g|U⇒h◦f:U→ˆX∩ˆYinC∞Sta. Furthermore, +Zis independent of choices up to equivalence. +Proposition 6.23. SupposeX,YareC∞-stacks,U,V ⊆ X are openC∞- +substacks withX=U∪V, f:U → Y andg:V → Y are1-morphisms, +andη:f|U∩V⇒g|U∩Vis a2-morphism in C∞Sta. Then there exists a 1- +morphismh:X →Y and2-morphisms ζ:h|U⇒f, θ:h|V⇒gsuch that +θ|U∩V=η⊙ζ|U∩V:h|U∩V⇒g|U∩V. Thishis unique up to 2-isomorphism. +In general, hisnotindependent up to 2-isomorphism of the choice of η. +Here is an example in which his not independent of ηup to 2-isomorphism +in the last part of Proposition 6.23. +Example 6.24. LetXbe theC∞-stack associated to the circle X=/braceleftbig +(x,y)∈ +R2:x2+y2= 1/bracerightbig +, andU,V⊆Xthe substacks associated to the open sets U=/braceleftbig +(x,y)∈X:x >−1 +2/bracerightbig +andV=/braceleftbig +(x,y)∈X:x <1 +2/bracerightbig +. LetYbe the quotient +C∞-stack [∗/Z2]. Then 1-morphisms f:X →Y correspond to principal Z2- +bundlesPf→X, and for 1-morphisms f,g:X→Ywith principal Z2-bundles +Pf,Pg→X,a2-morphism η:f⇒gcorrespondstoanisomorphismofprincipal +Z2-bundlesPf∼=Pg. The same holds for 1-morphisms U,V,U∪V→Y and +their 2-morphisms. +Letf:U → Y andg:V → Y be the 1-morphisms corresponding to +the trivial Z2-bundlesPf=Z2×U→U,Pg=Z2×V→V. Then 2- +morphisms η:f|U∩V⇒g|U∩Vcorrespond to automorphisms of the trivial +Z2-bundleZ2×(U∩V)→U∩V, that is, to continuous maps U∩V→Z2. +Note thatU∩Vhas two connected components/braceleftbig +(x,y)∈X:−1 +2< x <1 +2, +y>0/bracerightbig +and/braceleftbig +(x,y)∈X:−1 +2