# Differential forms on $C^{\infty}$-ringed spaces.

Eugene Lerman

###### Abstract.

We construct a complex of differential forms on a local $C^{\infty}$-ringed
space. The two main classes of spaces we have in mind are differential spaces
in the sense of Sikorski and $C^{\infty}$-schemes. Just as in the case of
manifolds the construction is functorial. Consequently forms can be integrated
over simplices and Stokes’ theorem holds.

###### Contents

  1. 1 Introduction
  2. 2 $C^{\infty}$-rings
  3. 3 Differential spaces
  4. 4 Local $C^{\infty}$-ringed spaces
  5. 5 $C^{\infty}$-schemes
  6. 6 Kähler differentials of $C^{\infty}$-rings
  7. 7 $C^{\infty}$-algebraic de Rham complex
  8. 8 Differential forms on $C^{\infty}$-ringed spaces, integration and Stokes’ theorem
  9. A Kähler differentials on concrete finitely presented differential spaces

##  1\. Introduction

The goal of this paper is to construct a complex of sheaves of differential
forms on local $C^{\infty}$-ringed spaces. The category
$\mathsf{LC^{\infty}RS}$ of local $C^{\infty}$-ringed spaces includes
manifolds, manifolds with corners, differential spaces in the sense of
Sikorski [Sn], $C^{\infty}$-differentiable space of Navarro González and
Sancho de Salas [NGSS], affine $C^{\infty}$-schemes of Dubuc [D] and Kreck’s
stratifolds [Kr].

The sheaves of commutative differential graded algebras (CDGAs) that we
construct agree with the ordinary CDGA of de Rham differential forms on the
manifolds with corners. Our construction of differential forms is functorial,
hence they can be integrated over simplexes and ($C^{\infty}$-)singular
chains. As a result a version of Stokes’s theorem holds.

On the level of 0- and 1-forms our sheaves are cotangent complexes of
$C^{\infty}$-Kähler differentials studies by Joyce [J]. In the case of
differential spaces our CDGAs appear to agree with the differential forms
defined by Mostow [Mos]. In the case of stratifolds our differential forms
include Kreck’s.

In an attempt to make this paper broadly accessible, the paper includes a fair
amount of expository material. I start by reviewing $C^{\infty}$-rings.
Differential spaces in the sense of Sikorski are reviewed next. The fact that
Sikorski differential spaces carry sheaves of $C^{\infty}$-rings is made
explicit. Next local $C^{\infty}$-ringed spaces are introduced. I prove that
there is a full and faithful embedding of the category of differential spaces
into the category $\mathsf{LC^{\infty}RS}$ of local $C^{\infty}$-ringed
spaces.

There is a global sections functor $\Gamma$ from $\mathsf{LC^{\infty}RS}$ to
the opposite of the category $C^{\infty}\mathsf{Ring}$ of $C^{\infty}$-rings.
I review a construction of its right adjoint
$\operatorname{Spec}:{(C^{\infty}\mathsf{Ring})}^{\mbox{\sf{\tiny{op}}}}\to\mathsf{LC^{\infty}RS}$.
The existence of the functor $\operatorname{Spec}$ was shown by Dubuc in 1981
[D]. Our proof is a simplified version of the proof in [J]. We use the fact
that it’s enough to construct $\operatorname{Spec}$ on objects together with
the components of the counit of the desired adjunction; there is no need to
explicitly construct $\operatorname{Spec}$ on morphisms. In Section 6 we
discuss Kähler differentials of $C^{\infty}$-rings and some of their
properties. The Kähler differentials for Fermat theories (which include
$C^{\infty}$-rings) were constructed by Dubuc and Kock [DK]. The construction
I present follows Joyce [J]. I compute a few examples. Note that by definition
of Kähler differentials, given a $C^{\infty}$-ring $\mathscr{A}$, the module
$\operatorname{Hom}(\Omega^{1}_{\mathscr{A}},\mathscr{A})$ dual to the module
$\Omega^{1}_{\mathscr{A}}$ of Kähler differentials of $\mathscr{A}$ is
isomorphic to the module $\operatorname{Der}(\mathscr{A})$ of
$C^{\infty}$-ring derivations of $\mathscr{A}$. However, for a general ring
$\mathscr{A}$, the double dual map

$\Omega^{1}_{\mathscr{A}}\to\operatorname{Hom}(\operatorname{Der}(\mathscr{A}),\mathscr{A})$

is neither injective nor surjective. This may be a reason why defining
differential forms on differential spaces by taking duals of derivations lead
to paradoxical results.

In Section 7 I construct the $C^{\infty}$-ring analogue of the algebraic de
Rham complex. This amounts to extending the universal differential
$d:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ to all of the exterior powers of
the module $\Omega^{1}_{\mathscr{A}}$. The issue is that
$\Omega^{1}_{\mathscr{A}}$ is not a free module over $\mathscr{A}$. However a
modification of the construction to extend $d$ used in algebraic geometry does
work which is a pleasant surprise.

In Section 8 I use the naturality of the construction of the previous section
to attach to very $C^{\infty}$-ringed space $(M,\mathscr{A})$ a presheaf
$(\Lambda^{\bullet}\Omega_{\mathscr{A}}^{1},\wedge,d)$ of CDGA’s. We check
that in the case where $M$ is a manifold with corners and
$\mathscr{A}=C^{\infty}_{M}$, the sheaf of ordinary smooth functions, the
presheaf $\Lambda^{\bullet}\Omega_{\mathscr{A}}$ is isomorphic to the sheaf
$\bm{\Upomega}^{\bullet}_{dR,M}$ of ordinary differential forms on the
manifold $M$. I show that a map
$(f,f_{\\#}):(M,\mathscr{A})\to(N,\mathscr{B})$ of local $C^{\infty}$-ringed
spaces induces a map $\tilde{f}:\Lambda^{\bullet}\Omega^{1}_{\mathscr{B}}\to
f_{*}(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}})$ of presheaves of CDGAs. I
then sheafify $(\Lambda^{\bullet}\Omega_{\mathscr{A}}^{1},\wedge,d)$ to obtain
a sheaf of CDGAs on a $C^{\infty}$-ringed space. I think of it as a sheaf of
differential forms. Using the fact that the sheafification functor is a left
adjoint to inclusion I show that maps of local $C^{\infty}$-ringed spaces give
rise to maps between sheaves of CDGAs of differential forms (Theorem 8.13).
The section ends with integration of differential forms over singular chains
and a version of Stokes’s theorem. In the appendix I describe a class of
differential spaces (which also happen to be affine $C^{\infty}$-schemes) for
which the presheaf of Kähler differentials is automatically a sheaf. The class
includes manifolds (without boundary) and some singular spaces as well, such
as $M:=\\{(x,y)\in\mathbb{R}^{2}\mid xy=0\\}$.

### Acknowledgements

I am grateful to Yael Karshon for years of conversations on the topic of this
paper and many related subjects. I thank Reyer Sjamaar, Rui Loja Fernandes and
Casey Blacker for an number of useful discussions.

##  2\. $C^{\infty}$-rings

In this section we recall the notion of a $C^{\infty}$-ring and collect a
number of useful facts. I couldn’t resist including most proofs. Our main
references are [MR] and [J]. For many practical purposes a $C^{\infty}$-ring
can be thought of a (nonempty) set $\mathscr{C}$ with an infinite collection
of $m$-ary operations $g_{\mathscr{C}}:\mathscr{C}^{m}\to\mathscr{C}$ (for all
$m\in\mathbb{Z}_{\geq 0}$), one for every smooth function $g\in
C^{\infty}(\mathbb{R}^{m})$. The composition of operations should correspond
to the composition of functions that define these operations, and so we end up
with the following “definition”:

######  Definition 2.1 (preliminary).

A $C^{\infty}$-ring is a (nonempty) set $\mathscr{C}$ together with operations

$g_{\mathscr{C}}:\mathscr{C}^{m}\to\mathscr{C}$

for all $m$ and all $g\in C^{\infty}(\mathbb{R}^{m})$ such that for all
$n,m\geq 0$, all $g\in C^{\infty}(\mathbb{R}^{m})$ and all
$f_{1},\ldots,f_{m}\in C^{\infty}(\mathbb{R}^{n})$

(2.1)
$({g\circ(f_{1},\ldots,f_{m})})_{\mathscr{C}}(c_{1},\ldots,c_{n})=g_{\mathscr{C}}({(f_{1})}_{\mathscr{C}}(c_{1},\ldots,c_{n}),\ldots,{(f_{m})}_{\mathscr{C}}(c_{1},\ldots,c_{n}))$

for all $(c_{1},\ldots,c_{n})\in\mathscr{C}^{n}$. Additionally we require that
for every $m>0$ and for every coordinate function
$x_{j}:\mathbb{R}^{m}\to\mathbb{R}$, $1\leq j\leq m$,

(2.2) $(x_{j})_{\mathscr{C}}(c_{1},\ldots,c_{m})=c_{j}.$

######  Example 2.2.

Let $M$ be a $C^{\infty}$-manifold and $\mathscr{A}=C^{\infty}(M)$ the set of
smooth functions. Then $\mathscr{A}$ is a $C^{\infty}$ ring: for any $n$, any
$a_{1},\ldots,a_{n}\in C^{\infty}(M)$ and $f\in C^{\infty}(\mathbb{R}^{n})$
the composite $f\circ(a_{1},\ldots,a_{n}):X\to\mathbb{R}$ is smooth. Therefore
we may define

$f_{\mathscr{A}}(a_{1},\ldots,a_{n}):=f\circ(a_{1},\ldots,a_{n}).$

We leave it to the reader to check that (2.1) and (2.2) hold.

######  Example 2.3.

If the manifold $M$ in the example above is a single point, then
$C^{\infty}(M)$ “is” the real line $\mathbb{R}$. The operation of $f\in
C^{\infty}(\mathbb{R}^{n})$ on $(a_{1},\ldots,a_{n})\in\mathbb{R}^{n}$ is then
the evaluation of the function $f$ on the $n$-tuple of real numbers:

$f_{\mathbb{R}}(a_{1},\ldots,a_{n}):=f(a_{1},\ldots,a_{n}).$

######  Example 2.4.

Let $X$ be a topological space and $\mathscr{A}=C^{0}(X)$ the set of
continuous functions. Then $\mathscr{A}$ is a $C^{\infty}$ ring: for any $n$,
any $a_{1},\ldots,a_{n}\in C^{0}(X)$ and $f\in C^{\infty}(\mathbb{R}^{n})$ the
composite $f\circ(a_{1},\ldots,a_{n}):X\to\mathbb{R}$ is again continuous. We
then may define

$f_{\mathscr{A}}(a_{1},\ldots,a_{n}):=f\circ(a_{1},\ldots,a_{n})$

which gives the set $C^{0}(X)$ the structure of a $C^{\infty}$-ring.

######  Remark 2.5.

It is a little hard to explain why we require (2.2) to hold without giving the
actual definition, Definition 2.11 below.

######  Remark 2.6.

Note a subtlety: if $m=0$, $\mathscr{C}^{0}$ is a singleton $\\{*\\}$,
$\mathbb{R}^{0}=\\{0\\}$ is the zero dimensional vector space, and
$C^{\infty}(\mathbb{R}^{0})=\mathbb{R}$. Hence the nullery operations are
functions $c_{\mathscr{C}}:\\{*\\}\to\mathscr{C}$, one for every real number
$c\in\mathbb{R}=C^{\infty}(\mathbb{R}^{0})$. We identify $c_{\mathscr{C}}$
with the element $c_{\mathscr{C}}(*)\in\mathscr{C}$.

Since $g(x,y)=xy\in C^{\infty}(\mathbb{R}^{2})$, we have a multiplication map
$g_{\mathscr{C}}:\mathscr{C}\times\mathscr{C}\to\mathscr{C}$. We write
$a_{1}a_{2}$ for $g_{\mathscr{C}}(a_{1},a_{2})$. With this notation it is not
hard to show that

$0_{\mathscr{C}}a=0_{\mathscr{C}}\qquad\textrm{ and }1_{\mathscr{C}}a=a$

for all $a\in\mathscr{C}$. Consequently if $0_{\mathscr{C}}=1_{\mathscr{C}}$
then $\mathscr{C}=\\{0_{\mathscr{C}}\\}$. Otherwise the $C^{\infty}$ ring
$\mathscr{C}$ contains a copy of the reals $\mathbb{R}$, which we think of as
the set of scalars.

######  Remark 2.7.

Any nonzero $C^{\infty}$-ring $\mathscr{A}$ has an underlying unital
$\mathbb{R}$-algebra. This is because to define an $\mathbb{R}$-algebra
structure we need scalars, addition and multiplication (plus various
compatibility and associativity conditions). In the previous remark we
explained where the multiplication and the scalars come from. The addition is
the operation defined by the function $f(x,y)=x+y\in
C^{\infty}(\mathbb{R}^{2})$.

We will not notationally distinguish between $C^{\infty}$-rings and their
underlying $\mathbb{R}$-algebras.

######  Definition 2.8.

A morphism of $\mathscr{A}\to\mathscr{B}$ of $C^{\infty}$-rings is a function
$\varphi:\mathscr{A}\to\mathscr{B}$ which preserves all the operations: for
any $n$, any $a_{1},\ldots,a_{n}\in\mathscr{A}$ and any $f\in
C^{\infty}(\mathbb{R}^{n})$

$\varphi(f_{\mathscr{A}}(a_{1},\ldots,a_{n}))=f_{\mathscr{B}}(\varphi(a_{1}),\ldots,\varphi(a_{n})).$

######  Notation 2.9 (The category $C^{\infty}\mathsf{Ring}$ of
$C^{\infty}$-rings).

It is easy to see that the composite of two morphisms of $C^{\infty}$-rings is
again a morphism of $C^{\infty}$-rings. Consequently $C^{\infty}$-rings form a
category that we denote by $C^{\infty}\mathsf{Ring}$.

The “official” definition of a $C^{\infty}$-ring proceeds as follows. We need
to first define the category $\mathsf{Euc}$ of Euclidean (a.k.a. Cartesian)
spaces.

######  Definition 2.10 (The category $\mathsf{Euc}$ of Euclidean spaces).

The objects of the category $\mathsf{Euc}$ are the coordinate vector spaces
$\mathbb{R}^{n}$, $n\geq 0$, thought of as $C^{\infty}$-manifolds. The
morphisms are $C^{\infty}$ maps.

We note that all objects of $\mathsf{Euc}$ are finite powers of one object:

$\mathbb{R}^{n}=(\mathbb{R}^{1})^{n}.$

######  Definition 2.11.

A $C^{\infty}$-ring ${\mathcal{C}}$ is a functor

${\mathcal{C}}:\mathsf{Euc}\to\mathsf{Set}$

from the category $\mathsf{Euc}$ of Euclidean spaces to the category
$\mathsf{Set}$ of sets that preserves finite products.

######  Remark 2.12.

Definition 2.11 says that given a $C^{\infty}$ ring ${\mathcal{C}}$ we have:

  1. (2.12.i)

For any $n\geq 0$, ${\mathcal{C}}(\mathbb{R}^{n})$ is an $n$-fold product of
the set $\mathscr{C}:={\mathcal{C}}(\mathbb{R}^{1})$:

${\mathcal{C}}(\mathbb{R}^{n}\xrightarrow{x_{i}}\mathbb{R})=\mathscr{C}\,^{n}\xrightarrow{pr_{i}}\mathscr{C},$

where $x_{i}$ is the $i$th coordinate function and $pr_{i}$ is the projection
on the $i$th factor.

  2. (2.12.ii)

For any smooth function
$g=(g_{1},\ldots,g_{m}):\mathbb{R}^{n}\to\mathbb{R}^{m}$ we have a map of sets

${\mathcal{C}}(g):{\mathcal{C}}(\mathbb{R}^{n})=\mathscr{C}^{n}\to\mathscr{C}^{m}={\mathcal{C}}(\mathbb{R}^{m})$

with

$pr_{j}\circ{\mathcal{C}}(g)={\mathcal{C}}(g_{j}).$

  3. (2.12.iii)

For any triple of smooth functions
$\mathbb{R}^{n}\xrightarrow{g}\mathbb{R}^{m}\xrightarrow{f}\mathbb{R}$ we have
three maps of sets ${\mathcal{C}}(g)$, ${\mathcal{C}}(f)$,
${\mathcal{C}}(f\circ g)$ with

${\mathcal{C}}(f)\circ{\mathcal{C}}(g)={\mathcal{C}}(f\circ g).$

  4. (2.12.iv)

${\mathcal{C}}(\mathbb{R}^{0})$ is a single point set $\\{*\\}$. This is
because by definition product-preserving functors take terminal objects to
terminal objects.

It follows that $\mathscr{C}={\mathcal{C}}(\mathbb{R}^{1})$ is a
$C^{\infty}$-ring in the sense of Definition 2.1.

######  Remark 2.13.

Conversely, given a $C^{\infty}$-ring $\mathscr{C}$ in the senes of Definition
2.1 (i.e., a set $\mathscr{C}$ with operations) there is a product-preserving
functor ${\mathcal{C}}:\mathsf{Euc}\to\mathsf{Set}$ with
${\mathcal{C}}(\mathbb{R}^{1})=\mathscr{C}$. For a proof see for example [J].

######  Remark 2.14.

It is not hard to see that if
${\mathcal{C}},{\mathcal{B}}:\mathsf{Euc}\to\mathsf{Set}$ are two product
preserving functors and $\alpha:{\mathcal{C}}\Rightarrow{\mathcal{B}}$ is a
natural transformation, then
$\alpha_{\mathbb{R}^{1}}:{\mathcal{C}}(\mathbb{R}^{1})\to{\mathcal{B}}(\mathbb{R}^{1})$
is a map of $C^{\infty}$-rings in the sense of Definition 2.8. Consequently we
can equivalently define the category $C^{\infty}\mathsf{Ring}$ as the category
of product-preserving functors from $\mathsf{Euc}$ to $\mathsf{Set}$.

From now on we will not distinguish between the two definitions.

The category of $C^{\infty}$-rings is complete and co-complete for category
theoretic reasons [Bor, Theorem 3.4.5]. Here are, roughly, the reasons:
Product preserving functors commute with limits and the category
$\mathsf{Set}$ of sets is complete. Consequently $C^{\infty}\mathsf{Ring}$ is
complete and the limits are computed “objectwise.” The co-completeness follows
from the fact that $C^{\infty}\mathsf{Ring}$ is a reflective subcategory of
the functor category $[\mathsf{Euc},\mathsf{Set}]$.
The following well-known lemma is very useful for dealing with
$C^{\infty}$-rings. I can’t resist including its proof.

######  Lemma 2.15 (Hadamard’s lemma).

For any smooth function $f:\mathbb{R}^{n}\to\mathbb{R}$ there exist smooth
functions $g_{1},\ldots,g_{n}\in C^{\infty}(\mathbb{R}^{2n})$ so that

(2.3) $f(x)-f(y)=\sum_{i=1}^{n}(x_{i}-y_{i})g_{i}(x,y)$

for any pair of points $x,y\in\mathbb{R}^{n}$.

###### Proof.

$\displaystyle f(x)-f(y)=$
$\displaystyle\int_{0}^{1}\frac{d}{dt}\,f(tx+(1-t)y)\,dt$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\sum_{i=1}^{n}\partial_{i}f(tx+(1-t)y)(x_{i}-y_{i})\,dt$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(x_{i}-y_{i})\int_{0}^{1}\partial_{i}f(tx+(1-t)y)(x_{i}-y_{i})\,dt.$

Define

$g_{i}(x,y)=\int_{0}^{1}\partial_{i}f(tx+(1-t)y)\,dt\,dt.$

∎

######  Remark 2.16.

Note that for any smooth function $f\in C^{\infty}(\mathbb{R}^{n})$ and any
functions $g_{1},\ldots,g_{n}\in C^{\infty}(\mathbb{R}^{2n})$ so that (2.3)
holds, we must have $g_{i}(x,x)=(\partial_{i}f)\,(x)$ for $i=1,\ldots,n$.

Here is another well-known result.

######  Lemma 2.17.

For any ideal $I$ in a $C^{\infty}$-ring $\mathscr{C}$ the quotient
$\mathbb{R}$-algebra $\mathscr{C}/I$ is naturally a $C^{\infty}$-ring.

###### Proof.

We prove that for any $n>0$ and any function $f\in C^{\infty}(\mathbb{R}^{n})$
the map

$f_{\mathscr{C}/I}:(\mathscr{C}/I)^{n}\to\mathscr{C}/I,\qquad
f_{\mathscr{C}/I}(c_{1}+I,\ldots,c_{n}+I):=f_{\mathscr{C}}(c_{1},\ldots,c_{n})+I$

is well-defined for all $(c_{1}+I,\ldots,c_{n}+I)\in(\mathscr{C}/I)^{n}$.
Consider $h(x,y)=f(x)-f(y)\in C^{\infty}(\mathbb{R}^{2n})$. By Hadamard’s
lemma

$h(x,y)=\sum_{i=1}^{n}(x_{i}-y_{i})g_{i}(x,y)$

for some $g_{1},\ldots,g_{n}\in C^{\infty}(\mathbb{R}^{2n})$. Hence for any
$(c_{1},\ldots,c_{n}),(c_{1}^{\prime},\ldots,c_{n}^{\prime})\in(\mathscr{C})^{n}$

$f_{\mathscr{C}}(c_{1},\ldots,c_{n})-f_{\mathscr{C}}(c_{1}^{\prime},\ldots,c_{n}^{\prime})=h_{\mathscr{C}}(c_{1},\ldots,c_{n},c_{1}^{\prime},\ldots,c_{n}^{\prime})=\sum_{i=1}^{n}(c_{i}-c_{i}^{\prime}){g_{i}}_{\mathscr{C}}(c,c^{\prime}).$

Hence if $c_{i}-c^{\prime}_{i}\in I$ for all $i$, then
$f_{\mathscr{C}}(c_{1},\ldots,c_{n})-f_{\mathscr{C}}(c_{1}^{\prime},\ldots,c_{n}^{\prime})\in
I$. It follows that the function $f_{\mathscr{C}/I}$ is well-defined. ∎

######  Example 2.18.

Let $M$ be a $C^{\infty}$-manifold, $p\in M$ a point. Recall that a germ of a
smooth function at $p$ is the equivalence class of a pair $(U,g)$ where $U$ is
an open neighborhood of $p$ and $g\in C^{\infty}(U)$. Two pairs $(U,g)$,
$(V,h)$ are equivalent if the functions $g,h$ agree on a neighborhood $W$ of
$p$ contained in $U\cap V$. We denote the class of $(U,g)$ by $[(U,g)]$ and by
$g_{p}$.

The set $C^{\infty}(M)_{p}$ of all germs of smooth functions at $p$ is an
$\mathbb{R}$ algebra and the map

$\pi_{p}:C^{\infty}(M)\to C^{\infty}(M)_{p},\quad\pi_{p}(f)=[(M,f)]\equiv
f_{p}$

is an $\mathbb{R}$-algebra map. Note that $\pi_{p}$ is surjective. This is
because for any germ $[(U,g)]$ there is a smooth function $\rho\in
C^{\infty}(M)$ with $\operatorname{supp}\rho\subset U$ and $\rho$ identically
1 on a smaller neighborhood of $p$. And then $\rho g\in C^{\infty}(M)$ and
$[(U,g)]=[M,\rho g)]=\pi_{p}(\rho g)$.

It follows that the algebra of germs $C^{\infty}(M)_{p}$ is isomorphic to the
quotient $C^{\infty}(M)/\ker\pi_{p}$ hence acquires a $C^{\infty}$-algebra
structure. Explicitly for any $k>0$, $g_{1},\ldots,g_{k}\in C^{\infty}(M)$ and
$\varphi\in C^{\infty}(\mathbb{R}^{k})$

$\varphi_{C^{\infty}(M)_{p}}((g_{1})_{p},\ldots,(g_{k})_{p})=(\varphi\circ(g_{1},\ldots,g_{k}))_{p}=(\varphi_{C^{\infty}(M)}(g_{1},\ldots,g_{k}))_{p}.$

###  $\mathbb{R}$-points of $C^{\infty}$-rings

Recall that the real line has the structure of a $C^{\infty}$-ring; see
Example 2.3 above.

######  Definition 2.19.

An $\mathbb{R}$-point of a $C^{\infty}$-ring $\mathscr{A}$ is a map of
$C^{\infty}$-rings $p:\mathscr{A}\to\mathbb{R}$.

######  Example 2.20.

Let $M$ be a smooth manifold and $C^{\infty}(M)$ the $C^{\infty}$-ring of
smooth functions on $M$. Then for any point $x\in M$ the the evaluation map

$\operatorname{\mathsf{ev}}_{x}:C^{\infty}(M)\to\mathbb{R},\qquad\operatorname{\mathsf{ev}}_{x}(f)=f(x)$

is an $\mathbb{R}$-point of the $C^{\infty}$-ring $C^{\infty}(M)$.

More generally if $(M,C^{\infty}(M))$ be a differential space (see Definition
3.1 in the introduction) and $x\in M$ a point then the evaluation map
$\operatorname{\mathsf{ev}}_{x}:C^{\infty}(M)\to\mathbb{R}$, $f\mapsto f(x)$
is an $\mathbb{R}$-point in the $C^{\infty}$-ring $C^{\infty}(M)$.

######  Remark 2.21.

An $\mathbb{R}$-point of a $C^{\infty}$-ring is also a map of
$\mathbb{R}$-algebras. The converse is true as well: a map of
$\mathbb{R}$-algebras $p:\mathscr{A}\to\mathbb{R}$ is a map of
$C^{\infty}$-rings. See [MR, Proposition 3.6 (b), p. 33]. The proof relies on
the so called Milnor’s exercise (which appears to be a result of Pursell — see
[Pu, Chapter 8]): for any (second countable) manifold $M$ and any
$\mathbb{R}$-algebra map $\varphi:C^{\infty}(M)\to\mathbb{R}$ there is a point
$p\in M$ so that $\varphi(f)=f(p)$ for all $f\in C^{\infty}(M)$. It will be
useful for us to have available the following version of Milnor’s exercise.

######  Lemma 2.22.

Let $M$ be a topological space, $\mathscr{A}\subset C^{0}(M)$ a
$C^{\infty}$-subring. Suppose $\mathscr{A}$ contains a proper function. Then
any $\mathbb{R}$-algebra map $\varphi:\mathscr{A}\to\mathbb{R}$ is an
evaluation on a point $p\in M$:

$\varphi(g)=\operatorname{\mathsf{ev}}_{p}(g)$

for all $g\in\mathscr{A}$. Here and elsewhere
$\operatorname{\mathsf{ev}}_{p}(g)=g(p)$, the value of the function $g$ on a
point $p\in M$.

Consequently all $\mathbb{R}$ algebra maps from $\mathscr{A}$ to $\mathbb{R}$
are maps of $C^{\infty}$-rings.

###### Proof.

By assumption there is a proper function $f\in\mathscr{A}\subset C^{0}(M)$.
Let $\varphi:\mathscr{A}\to\mathbb{R}$ be an $\mathbb{R}$-algebra map. Note
that by convention $\varphi(1_{\mathscr{A}})=1$, where $1_{\mathscr{A}}$ is
the constant function that assigns 1 to every point of $M$. Let
$r=\varphi(f)$.

If the set $\\{f=r\\}$ is empty, then $f-r\in\mathscr{A}$ is a nowhere zero
function with $\varphi(f-r)=0$. Since $f-r$ is also proper it is bounded away
from zero. There is a smooth function $k:\mathbb{R}\to\mathbb{R}$ so that
$k(x)=1/x$ for $x$ sufficiently far away from 0, and then $k\circ
f=\frac{1}{f}$. Since $\mathscr{A}$ is a $C^{\infty}$-subring of $C^{0}(M)$ it
follows that $\frac{1}{f}\in\mathscr{A}$. Hence $f$ is invertible in the
$\mathbb{R}$-algebra $\mathscr{A}$ and therefore the ideal $\ker\varphi$ is
all of $\mathscr{A}$. This contradicts that $\varphi(1)=1$. We conclude that
the set $\\{f=r\\}$ is nonempty.

Since $f$ is proper, the set $\\{f=r\\}$ is compact. Let
$\operatorname{\mathsf{ev}}_{p}:\mathscr{A}\to\mathbb{R}$ denote the
evaluation at $p\in M$. If $\varphi=\operatorname{\mathsf{ev}}_{p}$ for some
$p\in M$ then $g(p)=0$ for any $g\in\ker\varphi$. So if
$\varphi\not=\operatorname{\mathsf{ev}}_{p}$ for any $p\in M$ then the
intersection

$\bigcap_{g\in\ker\varphi}g^{-1}(0)$

is empty. That is, $\\{M\smallsetminus g^{-1}(0)\\}_{g\in\ker\varphi}$ is an
open cover of $M$ and of the set $\\{f=r\\}$. Since $\\{f=r\\}$ is compact
(and nonempty) there is $n>0$, $g_{1},\ldots,g_{n}\in\ker\varphi$ so that
$\\{M\smallsetminus g_{i}^{-1}(0)\\}_{i=1}^{n}$ covers $\\{f=r\\}$. Then the
function

$h:=(f-r)^{2}+g_{1}^{2}+\cdots+g_{n}^{2}$

is bounded away from 0 as well hence invertible. Since

$\varphi((f-r)^{2}+g_{1}^{2}+\cdots+g_{n}^{2})=(\varphi(f)-r)^{2}+\varphi(g_{1})^{2}+\cdots+\varphi(g_{n})^{2}=0$

$h$ is in the kernel of $\varphi$ and therefore the ideal $\ker\varphi$ is all
of $\mathscr{A}$. Again this contradicts the fact that $\varphi$ is not
identically zero. We conclude there is $p\in M$ so that for any
$g\in\ker\varphi$ , $g(p)=0$.

It remains to show that $\varphi=\operatorname{\mathsf{ev}}_{p}$. For any
function $g\in\mathscr{A}$, $g-\varphi(g)\in\ker\varphi$. Hence

$0=(g-\varphi(g))(p)=g(p)-\varphi(g)=\operatorname{\mathsf{ev}}_{p}(g)-\varphi(g).$

Thus $\varphi=\operatorname{\mathsf{ev}}_{p}$ and we are done. ∎

######  Remark 2.23.

Lemma 2.22 may not be optimal. According to Cukrowski and Stronowski [CS] if
$(M,\mathscr{F})$ is a differential space and $\mathscr{F}$ is generated, in
the sense defined in [CS], by a countable set, then every $\mathbb{R}$-point
of $\mathscr{F}$ is an evaluation at a point of $M$.

They also give an example ([CS, Example I] ) of a differential space
$(M,\mathscr{F})$ and an $\mathbb{R}$-point $\chi:\mathscr{F}\to\mathbb{R}$
which is not an evaluation at any point of $M$.

We will come back to this when we discuss the relation between differential
spaces and affine $C^{\infty}$-schemes. See Remark 5.8.

######  Example 2.24.

The $C^{\infty}$-ring $C^{\infty}(\mathbb{R})/\langle x^{2}\rangle$, where
$\langle x^{2}\rangle$ denotes the ideal generated by $x^{2}$, has only one
$\mathbb{R}$-point: the point $p$ is given by $p(f+\langle x^{2}\rangle)=f(0)$
for all $f\in C^{\infty}(\mathbb{R})$.

###### Proof.

Let $\varphi:C^{\infty}(\mathbb{R})/\langle x^{2}\rangle\to\mathbb{R}$ be an
$\mathbb{R}$-point and $\pi:C^{\infty}(\mathbb{R})\to
C^{\infty}(\mathbb{R})/\langle x^{2}\rangle$ the projection map. Then
$\varphi\circ\pi:C^{\infty}(\mathbb{R})\to\mathbb{R}$ is also an
$\mathbb{R}$-point. Note that $f(x)=x^{2}\in C^{\infty}(\mathbb{R})$ is a
proper function. By Lemma 2.22 there is $q\in\mathbb{R}$ so that
$\varphi\circ\pi=\operatorname{\mathsf{ev}}_{q}$. Since $\pi(x^{2})=0$,

$0=\varphi\circ\pi(x^{2})=\operatorname{\mathsf{ev}}_{q}(x^{2}).$

Thus $q=0$ and $\varphi=p$. ∎

######  Definition 2.25.

A $C^{\infty}$-ring $\mathscr{A}$ is point determined if $\mathbb{R}$-points
separate elements of $\mathscr{A}$. That is, if $a\in\mathscr{A}$ and
$a\not=0$ then there is an $\mathbb{R}$-point $p:\mathscr{A}\to\mathbb{R}$ so
that $p(a)\not=0$.

######  Example 2.26.

If $(M,\mathscr{F})$ is a differential space then $\mathscr{F}$ is point-
determined: a function on $M$ is non-zero only if there is a point at which it
takes a non-zero value.

######  Remark 2.27.

Not all $C^{\infty}$-rings are point determined. The simplest example is
probably the $C^{\infty}$-ring of $\mathscr{A}=C^{\infty}(\mathbb{R})/\langle
x^{2}\rangle$ of Example 2.24: the value of the one $\mathbb{R}$-point $p$ of
$\mathscr{A}$ on the nonzero element $x+\langle x^{2}\rangle\in\mathscr{A}$ is
zero.

### Free and finitely generated $C^{\infty}$-rings

######  Definition 2.28.

A $C^{\infty}$-ring $\mathscr{A}$ is free on a subset $X\subset\mathscr{A}$ or
is freely generated by a subset $X$ if for any $C^{\infty}$-ring $\mathscr{B}$
and any map of sets $f:X\to\mathscr{B}$ there is a unique map of
$C^{\infty}$-rings $\tilde{f}:\mathscr{A}\to\mathscr{B}$ so that
$\tilde{f}|_{X}=f$.

######  Remark 2.29.

In the definition above we are deliberately suppressing the underlying set
functor $U:C^{\infty}\mathsf{Ring}\to\mathsf{Set}$.

######  Lemma 2.30.

The $C^{\infty}$-ring $C^{\infty}(\mathbb{R}^{n})$ is a free ring on $n$
generators. The generators are the coordinate functions
$x_{1},\ldots,x_{n}:\mathbb{R}^{n}\to\mathbb{R}$.

###### Proof.

For any $C^{\infty}$-ring $\mathscr{A}$, any map
$\varphi:C^{\infty}(\mathbb{R}^{n})\to\mathscr{A}$ and any $f\in
C^{\infty}(\mathbb{R}^{n})$

$\varphi(f)=\varphi(f_{C^{\infty}(\mathbb{R}^{n})}(x_{1},\ldots,x_{n}))=f_{\mathscr{A}}(\varphi(x_{1}),\ldots,\varphi(x_{n})).$

∎

######  Definition 2.31.

A $C^{\infty}$-ring $\mathscr{A}$ is finitely generated if there is a
surjective $C^{\infty}$-ring map
$\varphi:C^{\infty}(\mathbb{R}^{n})\to\mathscr{A}$ for some $n>0$.
Equivalently $\mathscr{A}$ is finitely generated if $\mathscr{A}$ is
isomorphic to $C^{\infty}(\mathbb{R}^{n})/I$ for some ideal $I\subset
C^{\infty}(\mathbb{R}^{n})$.

######  Remark 2.32.

There are many $C^{\infty}$-rings that are not finitely generated. See Remark
3.21 below for a simple example.

### Derivations and $C^{\infty}$-derivations.

A vector field $v$ on a manifold $M$ can be defined as a derivation
$v:C^{\infty}(M)\to C^{\infty}(M)$ of the $\mathbb{R}$-algebra of smooth
functions on $M$: $v$ is $\mathbb{R}$-linear and for any two functions $f,g\in
C^{\infty}(M)$

$v(fg)=v(f)g+fv(g).$

It is well-known that vector fields on manifolds satisfy a version of a chain
rule: for any $n>0$, any $f\in C^{\infty}(\mathbb{R}^{n})$ and any
$a_{1},\ldots,a_{n}\in C^{\infty}(M)$

$v(f\circ(a_{1},\ldots,a_{n}))=\sum_{i=1}^{n}(\partial_{i}f)\circ(a_{1},\ldots,a_{n})\,v(a_{i}).$

This fact motivates the following notion of a derivation of a
$C^{\infty}$-ring $\mathscr{A}$ (we will come back to Definition 2.33 and
generalize it after we define modules over $C^{\infty}$-rings; see Definition
6.4).

######  Definition 2.33.

A $C^{\infty}$-derivation of a $C^{\infty}$ -ring $\mathscr{A}$ is a map
$X:\mathscr{A}\to\mathscr{A}$ so that for any $n>0$, any $f\in
C^{\infty}(\mathbb{R}^{n})$ and any $a_{1},\ldots,a_{n}\in\mathscr{A}$

$X(f_{\mathscr{A}}(a_{1},\ldots,a_{n}))=\sum_{i=1}^{n}(\partial_{i}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\,X(a_{i}).$

######  Notation 2.34.

The set of all $C^{\infty}$-derivations of a $C^{\infty}$-ring $\mathscr{A}$
forms a real vector space. We denote this vector space by
$\operatorname{Der}(\mathscr{A})$.

######  Remark 2.35.

One can show that if a $C^{\infty}$-ring $\mathscr{A}$ is finitely generated
and point determined then any $\mathbb{R}$-algebra derivation
$v:\mathscr{A}\to\mathscr{A}$ is automatically a $C^{\infty}$-derivation [KL].
In fact more is true; see [Ya].

######  Lemma 2.36.

Let $\mathscr{A}$ be a $C^{\infty}$-ring, $w:\mathscr{A}\to\mathscr{A}$ a
derivation and $a,b\in\mathscr{A}$ two elements with $ab=1$ (i.e., $a$ is
invertible and $b$ is the inverse of $a$). Then

$w(b)=-b^{2}w(a).$

###### Proof.

Since $w$ is a derivation, $w(1)=0$. Hence $0=w(ab)=w(a)b+a\,w(b)$ and the
result follows. ∎

### Localization of $C^{\infty}$-rings

######  Definition 2.37.

Let $S$ be a subset of a $C^{\infty}$-ring $\mathscr{A}$. A localization of
$\mathscr{A}$ at $S$ is a $C^{\infty}$-ring $\mathscr{A}\\{S^{-1}\\}$ together
with a map $\nu:\mathscr{A}\to\mathscr{A}\\{S^{-1}\\}$ which has the following
universal property:

For any $C^{\infty}$-ring $\mathscr{B}$ and any map
$f:\mathscr{A}\to\mathscr{B}$ so that $f(s)\in\mathscr{B}$ is invertible in
$\mathscr{B}$ there is a unique map
$\bar{f}:\mathscr{A}\\{S^{-1}\\}\to\mathscr{B}$ so that the diagram

$\textstyle{\mathscr{A}}$$\textstyle{\mathscr{B}}$$\textstyle{\mathscr{A}\\{S^{-1}\\}}$$\scriptstyle{\nu}$$\scriptstyle{f}$$\scriptstyle{\bar{f}}$

commutes.

Localizations always exist. See [MR, pp. 23] for a sketch of a proof. We next
define a local $C^{\infty}$-ring.

######  Definition 2.38.

A $C^{\infty}$-ring $\mathscr{A}$ is local if it has exactly one maximal ideal
$\mathfrak{m}$ and if the quotient $\mathscr{A}/\mathfrak{m}$ is isomorphic to
$\mathbb{R}$.

######  Remark 2.39.

Moerdijk and Reyes call our local $C^{\infty}$-rings pointed local
$C^{\infty}$-rings. This is because a local $C^{\infty}$-ring $\mathscr{A}$ in
our sense has a unique $\mathbb{R}$-point: it’s the quotient map
$\mathscr{A}\to\mathscr{A}/\mathfrak{m}$ followed by the isomorphism
$\mathscr{A}/\mathfrak{m}\simeq\mathbb{R}$, where $\mathfrak{m}$ is the
maximal ideal.

##  3\. Differential spaces

In this section we define Sikorski differential spaces smooth maps between
differential spaces, officially define the category $\mathsf{DiffSpace}$ of
differential spaces and smooth maps, and talk about induced differential
spaces structure on subsets.

Recall a common definition of a differential space.

######  Definition 3.1 (Differential space in the sense of Sikorski).

A differential space is a pair $(M,\mathscr{F})$ where $M$ is a topological
space and $\mathscr{F}$ is a set of real valued functions on $M$ so that

  1. (3.1.i)

For any natural number $k$, for any $a_{1},\ldots,a_{k}$ and any smooth
function $f\in C^{\infty}(\mathbb{R}^{k})$ the composite
$f\circ(a_{1},\ldots,a_{k})$ is in $\mathscr{F}$;

  2. (3.1.ii)

If $g:M\to\mathbb{R}$ is a function with the property that for any $x\in M$
there is an open neighborhood $U$ of $x$ and $f\in\mathscr{F}$ ($f$ depends on
$x$) so that $g|_{U}=f|_{U}$ then $g\in\mathscr{F}$.

Additionally we require that the topology on $M$ is the smallest topology for
which all the functions in $\mathscr{F}$ are continuous.

One refers to elements of $\mathscr{F}$ as smooth functions on the space $M$.
The whole set $\mathscr{F}$ is called a smooth structure on $M$.

It will be convenient at this point to change our notation.

######  Notation 3.2.

We denote a differentials space as a pair $(M,C^{\infty}(M))$, where $M$ is a
space and $C^{\infty}(M)$ is set of (continuous) functions on $M$ satisfying
the conditions 3.1.i and 3.1.ii of Definition 3.1. In other words we denote
the set $\mathscr{F}$ by $C^{\infty}(M)$.

######  Remark 3.3.

  * •

One thinks of the family of functions $C^{\infty}(M)$ on a differential space
$(M,C^{\infty}(M))$ as the set of all abstract “smooth” functions on $M$.

  * •

Condition (3.1.i) of Definition 3.1 says that the set $C^{\infty}(M)$ is a
$C^{\infty}$-ring: for any $n>0$, $a_{1},\ldots,a_{n}\in C^{\infty}(M)$ and
$f\in C^{\infty}(\mathbb{R}^{n})$ the composite
$f\circ(a_{1},\ldots,a_{n}):M\to\mathbb{R}$ is again in $C^{\infty}(M)$ by
(3.1.i). Thus setting

$f_{C^{\infty}(M)}(a_{1},\ldots,a_{n}):=f\circ(a_{1},\ldots,a_{n})$

for $f\in C^{\infty}(\mathbb{R}^{n})$ and $a_{1},\ldots,a_{n}\in
C^{\infty}(M)$) makes $C^{\infty}(M)$ into a $C^{\infty}$-ring. (The standard
definitions of differential spaces are usually silent about the case $n=0$.)

  * •

Condition (3.1.ii) of Definition 3.1 is a condition on the sheaf of functions
on $M$ generated by $C^{\infty}(M)$. See Remark 3.11.

Before we give our first example of a differential space we record a fact that
is equivalent to a theorem of Whitney.

######  Lemma 3.4.

Let $M$ be a manifold and $U\subset M$ an open set. Then there is a
$C^{\infty}$-function $f:M\to[0,1]$ (in the ordinary sense) so that $U=\\{x\in
M\mid f(x)\not=0\\}$.

###### Proof.

By a theorem of Whitney any closed subset of $M$ is the zero set of some
smooth function (remember that all our manifolds are Hausdorff and second
countable). See, for example, [Lee, Theorem 2.29]. Hence if $U\subset M$ is
open then there is a smooth function $f:M\to\mathbb{R}$ so that $U=\\{x\in
M\mid f(x)\not=0\\}$. ∎

######  Example 3.5.

Let $M$ be a smooth manifold Then the pair $(M,C^{\infty}(M))$, where
$C^{\infty}(M)$ is the algebra of $C^{\infty}$ functions, is a differential
space in the sense of Definition 3.1. The only hard thing to check is that the
topology on $M$ is the smallest among the topologies for which smooth
functions are continuous. But this follows from Lemma 3.4 above.

######  Definition 3.6.

A map or a morphism or a smooth function from a differential space
$(M,C^{\infty}(M))$ to a differential space $(N,C^{\infty}(N))$ is a function
$\varphi:M\to N$ so that for any $f\in C^{\infty}(N)$ the composite
$f\circ\varphi$ is in $C^{\infty}(M)$.

######  Remark 3.7.

If $\varphi:(M,C^{\infty}(M))\to(N,C^{\infty}(N))$ is a map of differential
spaces then $\varphi$ is automatically continuous. This is because the
topology on a differential space is the smallest topology making the smooth
real-valued functions continuous.

######  Remark 3.8.

It is easy to see that differential spaces and smooth maps form a category,
which we refer to as the category of differential spaces  and denote it by
$\mathsf{DiffSpace}$.

######  Definition 3.9 (The structure sheaf $C^{\infty}_{M}$ of a differential
space $(M,C^{\infty}(M))$).

Let $(M,C^{\infty}(M))$ be a differential spaces. Define a presheaf
${\mathcal{P}}$ of $C^{\infty}$-rings on $M$ as follows: for an open set
$U\subset M$

${\mathcal{P}}(U):=\\{f|_{U}\mid f\in C^{\infty}(M)\\}$

and let the structure maps of ${\mathcal{P}}$ to be the restrictions of
functions.

Define the structure sheaf $C^{\infty}_{M}$ of a differential space
$(M,C^{\infty}(M))$ to be the sheafification of presheaf ${\mathcal{P}}$.

The structure sheaf $C^{\infty}_{M}$ of a differential space has a concrete
description: see Lemma 3.10 below.

######  Lemma 3.10.

Let $(M,C^{\infty}(M))$ be a differential space, ${\mathcal{P}}$ the presheaf
on $M$ defined by

${\mathcal{P}}(U):=\\{f|_{U}\mid f\in C^{\infty}(M)\\}$

as in Definition 3.9. For an open subset $U\subseteq M$ let
$C^{\infty}_{M}(U)$ be the ring of all function $f:U\to\mathbb{R}$ with the
following property: for any point $p\in U$ there is an open neighborhood $V$
of $p$ in $U$ and $g\in C^{\infty}(M)$ with $g|_{V}=f|_{V}$. Then
$C^{\infty}_{M}$ is a sheaf and is a sheafification of the presheaf
${\mathcal{P}}$.

###### Proof.

For every open set $U\subset M$ define

$\begin{split}C^{\infty}_{M}(U):=&\\{f:U\to\mathbb{R}\mid\textrm{ for any
}x\in U\textrm{ there is an open neighborhood }V\\\ &\textrm{ of }x\textrm{ in
}U\textrm{ and }g\in C^{\infty}(M)\textrm{ with }g|_{V}=f|_{V}\\}.\end{split}$

It is easy to see that $C^{\infty}_{M}$ is a presheaf and that $P(U)\subset
C^{\infty}_{M}(U)$ for all opens $U$. Moreover the presheaves $C^{\infty}_{M}$
and $P$ have the same stalks. So to prove that $C^{\infty}_{M}$ is a
sheafification of $P$ it is enough to check that $C^{\infty}_{M}$ is a sheaf.
This is not hard. ∎

######  Remark 3.11.

We now see what condition (3.1.ii) of Definition 3.1 is saying. It says that
the $C^{\infty}$-ring of functions $C^{\infty}(M)$ on a differential space
$(M,C^{\infty}(M))$ is the $C^{\infty}$-ring of global sections of the sheaf
$C^{\infty}_{M}$ induced on $M$ by $C^{\infty}(M)$ and that we don’t get any
more (or fewer) global sections.

Given a differential space $(N,C^{\infty}(N))$ there are various ways to
define a differential space structure on a subset $M$ of $N$. See, for
example, [Sn, pp. 17-18].

######  Definition 3.12 (smooth functions on subsets of differential spaces).

Let $(N,C^{\infty}(N))$ a differential space and $M\subset N$ a subset. A
function $f:M\to\mathbb{R}$ is smooth if for every point $x\in M$ there is a
smooth function $g_{x}\in C^{\infty}(N)$ and an open neighborhood $U_{x}$ of
$x$ in $N$ so that $g_{x}|_{U_{x}\cap X}=f|_{U_{x}\cap M}$.

######  Notation 3.13.

Let $M$ be a subset of a differential space $(N,C^{\infty}(N))$, as in
Definition 3.12 above. We set

$C^{\infty}(M):=\\{f:M\to\mathbb{R}\mid f\textrm{ is smooth in the sense of
Definition~{}\ref{def:01}}\\}.$

We will see shortly in Lemma 3.15 below that $C^{\infty}(M)$, as defined
above, is a $C^{\infty}$-ring. We’ll prove it in the case when $N$ is a
manifold. For a more general case see [Sn].

######  Lemma 3.14.

Let $N$ be a second countable Hausdorff $C^{\infty}$ manifold and $M\subset N$
a subset as in Definition 3.12 above. Then $f:M\to\mathbb{R}$ is smooth if and
only if there is an open neighborhood $U$ of $M$ in $N$ and a smooth function
$g:U\to\mathbb{R}$ so that $f=g|_{M}$ (both $U$ and $g$ depend on $f$).
Moreover if $M$ is closed in $N$ we may take $U=N$.

###### Proof.

Let $U=\bigcup_{x\in M}U_{x}$ where $U_{x}$’s are the open sets of Definition
3.12. Then $\\{U_{x}\\}_{x\in M}$ is an open cover of $U$. Since $N$ is second
countable, $U$ is a second countable Hausdorff manifold hence paracompact.
Consequently there is a partition of unity $\\{\rho_{x}\\}$ subordinate to
$\\{U_{x}\\}_{x\in M}$. The function $g=\sum\rho_{x}g_{x}\in C^{\infty}(U)$
then has the property that $g|_{M}=f$.

If $M$ is closed, we may consider an open cover of $N$ consisting of all the
sets $U_{x}$, $x\in M$ and $U_{0}:=N\smallsetminus M$. Choose a partition of
unity subordinate to this cover of $X$ and again set $g=\sum_{x\in
M}\rho_{x}g_{x}$ (note that we are omitting the function $\rho_{0}$ with
$\operatorname{supp}\rho_{0}\subseteq U_{0}$ from the sum). Then $g$ is a
smooth function on all of $N$ and $g|_{M}=f$. ∎

######  Lemma 3.15.

Let $N$ be a $C^{\infty}$ second countable Hausdorff manifold and $M\subset N$
a subset. The set $C^{\infty}(M)$ of smooth functions on $M$ (Definition 3.12,
Notation 3.13) is a $C^{\infty}$-ring.

###### Proof.

Let $U\subset M$ be an open neighborhood of $M$ in $N$. It’s easy to see that
$C^{\infty}(U)|_{M}$ is a $C^{\infty}$ ring. First of all
$C^{\infty}(U)|_{M}=C^{\infty}(U)/I_{M}$ where $I_{M}\subset C^{\infty}(U)$ is
the ideal of functions that vanish on $M$ (recall that an ideal in a
$C^{\infty}$ ring is an ideal in the underlying $\mathbb{R}$-algebra).
Explicitly, given $n>0$, $f\in C^{\infty}(\mathbb{R}^{n})$ and
$g_{1}|_{M},\ldots,g_{n}|_{M}\in C^{\infty}(U)|_{M}$

(3.1)
$f_{C^{\infty}(U)|_{M}}(g_{1}|_{M},\ldots,g_{n}|_{M}):=(f\circ(g_{1},\ldots,g_{n}))|_{M}.$

In general $C^{\infty}(M)$ is the union of the $C^{\infty}$-rings
$C^{\infty}(U)|_{M}$:

$C^{\infty}(M)=\bigcup_{U\supset M}C^{\infty}(U)|_{M}.$

So given $n>0$, $f\in C^{\infty}(\mathbb{R}^{n})$, $h_{1},\ldots,h_{n}\in
C^{\infty}(M)$ there are open neighborhoods $U_{1},\ldots,U_{n}$ of $M$ in $N$
and smooth functions $g_{i}\in C^{\infty}(U_{i})$, $1\leq i\leq n$ so that
$h_{i}=g_{i}|_{M}$. We then define

$f_{C^{\infty}(M)}(h_{1},\ldots,h_{n}):=f\circ(h_{1},\ldots,h_{n})$

Since
$f\circ(h_{1},\ldots,h_{n})=\left(f\circ(g_{1}|_{V},\ldots,g_{n}|_{V})\right)$
where $V=\bigcap_{i=1}^{n}U_{i}$ the function $f\circ(h_{1},\ldots,h_{n})$ is
in $C^{\infty}(M)$ and so the defintion of
$f_{C^{\infty}(M)}(h_{1},\ldots,h_{n})$ does make sense. ∎

######  Definition 3.16.

Let $N$ be a $C^{\infty}$ second countable Hausdorff manifold and $M\subset N$
a subset. We define the induced smooth structure on $M$ to be the set

$C^{\infty}(M):=\\{f:M\to\mathbb{R}\mid f\textrm{ is smooth in the sense of
Definition~{}\ref{def:01}}\\}.$

######  Remark 3.17.

Note that by Lemma 3.15 the induced smooth structure $C^{\infty}(M)$ is a
$C^{\infty}$-ring. It is not difficult to check that the subspace topology on
$M$ is the smallest topology making the function in $C^{\infty}(M)$
continuous. Consequently the subset $M$ with its induced smooth structure is a
differential space.

######  Lemma 3.18.

Let $(M,C^{\infty}(M))$ be a differential space, $U\subset M$ an open set and
$x_{0}\in U$ a point. Then there is a function $\tau\in C^{\infty}(M)$ so that

  1. (3.18.i)

$0\leq\tau(x)\leq 1$ for all $x\in M$

  2. (3.18.ii)

The support of $\tau$ is contained in $U$ and

  3. (3.18.iii)

$\tau$ is identically 1 on a neighborhood of $x_{0}$.

We will refer to $\tau$ is a bump function.

###### Proof.

Let $\mathscr{T}_{M}$ denote the topology on $M$. Recall that
$\mathscr{T}_{M}$ is the smallest topology making all the functions in
$C^{\infty}(M)$ continuous. For any point $x\in M$ there is $f\in
C^{\infty}(M)$ with $f(x)\in(0,1)$. It follows that the collection

$\mathscr{B}=\\{f_{1}^{-1}((0,1))\cap\ldots\cap f_{1}^{-1}((0,1))\mid
k>0,f_{1},\ldots,f_{k}\in C^{\infty}(M)\\}$

is a basis for a topology $\mathscr{T}$ on $M$. If $\mathscr{T}^{\prime}$ is
any topology on $M$ that makes the functions in $C^{\infty}(M)$ continuous
then $\mathscr{T}\subset\mathscr{T}^{\prime}$. Therefore
$\mathscr{T}=\mathscr{T}_{M}$ and $\mathscr{B}$ is a basis for
$\mathscr{T}_{M}$.

Now let $x_{0}\in M$ be a point and $U\subseteq M$ an open set containing
$x_{0}$, as in the statement we are proving. Then since $\mathscr{B}$ is a
basis for $\mathscr{T}_{M}$ there is $k>0$ and functions
$f_{1},\ldots,f_{k}\in C^{\infty}(M)$ so that

$x_{0}\in f_{1}^{-1}((0,1))\cap\ldots\cap f_{1}^{-1}((0,1))\subset U.$

Let $F=(f_{1},\ldots,f_{k}):M\to\mathbb{R}^{k}$. Choose $\rho\in
C^{\infty}(\mathbb{R}^{k})$ so that $0\leq\rho\leq 1$,
$\operatorname{supp}\rho\subset(0,1)^{k}$ and $\rho$ is identically 1 near
$F(x_{0})$. Then $\tau=\rho\circ F\in C^{\infty}(M)$ is a desired bump
function. ∎

### Induced differential space structures on open subsets

Suppose $(M,C^{\infty}(M))$ is a differential space and $U\subset M$ an open
subset. There are several and a priori different ways to define a differential
space structure on $U$. For example we can define it to be the
$C^{\infty}$-ring of local sections $C^{\infty}_{M}(U)$ of the structure sheaf
$C^{\infty}_{M}$ (see Definition 3.9). If $M$ is a closed subset of
$\mathbb{R}^{n}$ then the following definition of $C^{\infty}(U)$ also feels
natural: choose an open set $\tilde{U}$ in $\mathbb{R}^{n}$ so that
$U=\tilde{U}\cap M$ and set $C^{\infty}(U)=C^{\infty}(\tilde{U})|_{U}$. The
lemma below shows that the two constructions/definitions give the same result.

######  Lemma 3.19.

Let $M\subset\mathbb{R}^{n}$ be a closed set, $U\subset M$ an open subset,
$\tilde{U}\subset\mathbb{R}^{n}$ an open set with $U=M\cap\tilde{U}$. Then

$C^{\infty}_{M}(U)=C^{\infty}(\tilde{U})|_{U},$

where $C^{\infty}_{M}$ the structure sheaf of the differential space
$(M,C^{\infty}(M))$ (Definition 3.9).

###### Proof.

Let $f\in C^{\infty}(\tilde{U})|_{U}$ and $p\in U$. Then there is
$\tilde{f}\in C^{\infty}(\tilde{U})$ with $\tilde{f}|_{U}=f$. Let $\rho\in
C^{\infty}(\mathbb{R}^{n})$ be a bump function with
$\operatorname{supp}\rho\subset\tilde{U}$ and $\rho$ identically 1 on a
neighborhood $\tilde{V}$ of $p$ in $\tilde{U}$. Let
$g_{p}=(\rho\tilde{f})|_{M}$. Then $g_{p}\in C^{\infty}(M)$ and
$g_{p}|_{\tilde{V}\cap U}=f|_{\tilde{V}\cap U}$. Hence $f\in
C^{\infty}_{M}(U)$.

Conversely suppose $f\in C^{\infty}_{M}(U)$. Then for any $p\in U$ there is an
open neighborhood $V_{p}$ of $p$ in $U$ and $g_{p}\in
C^{\infty}(M)=C^{\infty}(\mathbb{R}^{n})|_{M}$ with
$f|_{V_{p}}=g_{p}|_{V_{p}}$. Choose $\tilde{g}_{p}\in
C^{\infty}(\mathbb{R}^{n})$ with $\tilde{g}_{P}|_{M}=g_{p}$. Choose open sets
$\tilde{V}_{p}\subset\tilde{U}$ so that $\tilde{V}_{p}\cap U=V_{p}$. Then
$(\tilde{U}\smallsetminus U)\cup\\{\tilde{V}_{p}\\}_{p\in U}$ is an open cover
of $\tilde{U}$. Choose a partition of unity $\\{\rho_{0}\\}\cup\\{\rho_{p}\\}$
subordinate to the cover (so that
$\operatorname{supp}\rho_{p}\subset\tilde{V}_{p}$ for all $p\in U$). Let

$g=\sum_{p\in U}\rho_{p}\tilde{g}_{p}.$

Then $g\in C^{\infty}(\tilde{U})$ and $g|_{U}=f$. So $f\in
C^{\infty}(\tilde{U})|_{U}$. ∎

######  Corollary 3.20.

Let $M\subset\mathbb{R}^{n}$ be a closed set, $U\subset M$ an open subset and
$\tilde{U}_{1},\tilde{U}_{2}\subset\mathbb{R}^{n}$ two open sets with
$\tilde{U}_{i}\cap M=U$. Then
$C^{\infty}(\tilde{U}_{1})|_{U}=C^{\infty}(\tilde{U}_{2})|_{U}$. In other
words the choice of $\tilde{U}\subset\mathbb{R}^{n}$ in defining
$C^{\infty}(U)$ doesn’t matter.

######  Remark 3.21.

If $M$ is a subset of $\mathbb{R}^{n}$ then the $C^{\infty}$-ring
$C^{\infty}(M)$ need not be finitely generated. For example if
$M=\mathbb{Q}\subset\mathbb{R}$, the set of rationals, then
$C^{\infty}(\mathbb{Q})$ is not finitely generated.

###### Proof.

Suppose not. Then there is $n>0$, $f_{1},\ldots,f_{n}\in
C^{\infty}(\mathbb{Q})$ so that for any $a\in C^{\infty}(\mathbb{Q})$ there is
$h\in C^{\infty}(\mathbb{R}^{n})$ with

$a=h(f_{1},\ldots,f_{n}).$

By definition of the induced smooth structure $C^{\infty}(\mathbb{Q})$ for
each index $i$, $1\leq i\leq n$ there is an open set $U_{i}\subset\mathbb{R}$
and $\tilde{f}_{i}\in C^{\infty}(\mathbb{R})$ with
$f_{i}=\tilde{f}|_{\mathbb{Q}}$. Let $U=U_{1}\cap\ldots\cap U_{n}$. Then for
any $h\in C^{\infty}(\mathbb{R}^{n})$

$h(\tilde{f}_{1},\ldots,\tilde{f}_{n})\in C^{\infty}(U).$

Choose $x_{0}\in U\smallsetminus\mathbb{Q}$ and consider
$g(x)=\frac{1}{x-x_{0}}$. Then $g|_{\mathbb{Q}}\in C^{\infty}(\mathbb{Q})$.
but there is no $h\in C^{\infty}(\mathbb{R}^{n})$ with

$g|_{\mathbb{Q}}=h(f_{1},\ldots,f_{n})=\left(h(\tilde{f}_{1},\ldots,\tilde{f}_{n})\right)|_{\mathbb{Q}}.$

∎

### Concrete differential spaces

The world of differential spaces is at least as wild as the world of general
topological spaces. After all, any topological space $X$ is a differential
space: its $C^{\infty}$-ring of “smooth functions” is $C^{\infty}$-ring of
continuous functions $C^{0}(X)$ (cf. Example 2.4). While one can define vector
fields and their integral curves on any differential space (see Definition
3.29 below) proofs of existence of integral curves usually require more
structure. One typically assumes that the differential space is subcartesian.
Recall the definition:

######  Definition 3.22.

A differential space $(M,C^{\infty}(M))$ is subcartesian iff it is locally
isomorphic to $(N,C^{\infty}(N))$ where $N$ is a subset of some Euclidean
(i.e., Cartesian) space $\mathbb{R}^{n}$. That is, for any point $p\in M$
there is an open neighborhood $U$ of $p$, $n\geq 0$, $N\subset\mathbb{R}^{n}$
so that $(U,C^{\infty}_{M}(U))$ is isomorphic, as a differential space, to
$(N,C^{\infty}(N))$. Here $C^{\infty}_{M}(U)$ is the value of the sheaf
$C^{\infty}_{M}$ on $U$ and $C^{\infty}(N)$ is the differential space
structure on $N$ induced from its inclusion in $\mathbb{R}^{n}$.

While the assumption that the space is subcartesian makes it less wild,
subcartesian spaces can still be fairly strange. One issue is that their
“structural dimension” can be unbounded.

######  Definition 3.23.

The structural dimension of a subcartesian differential space
$(M,C^{\infty}(M))$ at a point $x\in M$ is the smallest integer $n_{x}$ so
that an open neighborhood $U$ of $x$ in $M$ is isomorphic to a subset $V$ of
$\mathbb{R}^{n_{x}}$.

The structural dimension  of $(M,C^{\infty}(M))$ is the supremum of the set
$\\{n_{x}\mid x\in M\\}$ of structural dimensions of the points of $x$ of $M$.

Note that the structural dimension of a subcartesian space can be infinite. On
the other hand if the structural dimension of a subcartesian space
$(M,C^{\infty}(M))$ is finite and the topology $M$ is nice ($M$ is Hausdorff,
second countable and locally compact) then by [Xia, Theorem 3]
$(M,C^{\infty}(M))$ is isomorphic to a subset of $\mathbb{R}^{N}$ for some $N$
(see also [Xia2]). But the $C^{\infty}$-ring of functions on a subset $X$ of
$\mathbb{R}^{N}$ can be pretty complicated. In particular it need not be
finitely generated. For instance $C^{\infty}(\mathbb{Q})$ where
$\mathbb{Q}\subset\mathbb{R}$ are the rationals is not finitely generated —
see Remark 3.21. For this reason we single out a class of subcartesian spaces.
It turns out this class of differential spaces coincides with a class of
$C^{\infty}$-schemes — see Theorem 5.21.

######  Definition 3.24.

A differential space $(M,C^{\infty}_{M})$ is concrete if it is isomorphic to a
closed subset $Z$ of some Euclidean space $\mathbb{R}^{n}$ (with $Z$ having
the induced differential structure:
$C^{\infty}_{Z}(Z)=C^{\infty}(\mathbb{R}^{n})|_{Z}$.

The category of concrete differential spaces is fairly large. In particular it
contains manifolds, closed subsets of manifolds and locally closed subsets of
manifolds. We’ll show this in a string of lemmas.

######  Lemma 3.25.

Any closed subset of a concrete differential space is concrete.

###### Proof.

The assertion holds for closed subsets of Euclidean spaces and therefore for
any concrete differential space. ∎

######  Lemma 3.26.

An open subset of a concrete differential space is a concrete differential
space.

###### Proof.

Let $M$ be a concrete differential space. It’s no loss of generality to assume
that $M$ is a closed subset of some $\mathbb{R}^{n}$. Let $U\subset M$ be an
open subset. Then there exists an open subset $W$ of $\mathbb{R}^{n}$ with
$U=M\cap W$. By Lemma 3.4 there is a smooth function
$f:\mathbb{R}^{n}\to[0,1]$ so that

$W=\\{f\not=0\\}.$

The map $F:W\to\mathbb{R}^{n}\times\mathbb{R}$, $F(x)=(x,\frac{1}{f(x)})$ is
an embedding and its image is the set

$F(W)=\\{(x,y)\in\mathbb{R}^{n}\times\mathbb{R}\mid yf(x)=1\\}.$

Hence $F$ is a closed embedding. In particular $F^{*}:C^{\infty}(F(W))\to
C^{\infty}(W)$ is an isomorphism of $C^{\infty}$-rings. Since $U$ is closed in
$W$ and $F:W\to F(W)$ is a homeomorphism, the set $F(U)$ is closed in $F(W)$.
Since $F(W)$ is closed in $\mathbb{R}^{n}\times\mathbb{R}$ so is $F(U)$.
Consequently $F(U)$ is a concrete differential space with

$C^{\infty}(F(U))=C^{\infty}(\mathbb{R}^{n})|_{F(U)}=\left(C^{\infty}(\mathbb{R}^{n})|_{F(W)}\right)|_{F(U)}=C^{\infty}(F(W))|_{F(U)}.$

The isomorphism $F^{*}$ sends the ideal of functions in $C^{\infty}(F(W))$
that vanish on $F(U)$ to the ideal of functions in $C^{\infty}(W)$ that vanish
on $U$. Hence $F|_{U}:(U,C^{\infty}(U))\to(F(U),C^{\infty}(F(U)))$ is an
isomorphism of differential spaces and $(U,C^{\infty}(U))$ is concrete. ∎

######  Lemma 3.27.

Let $N$ be a closed subset of a manifold $M$. Then the differential space
$(N,C^{\infty}(N))$ (with its induced smooth structure) is a concrete
differential space.

###### Proof.

By Whitney embedding theorem we may assume that $M$ is embedded in
$\mathbb{R}^{n}$ for some $n$. Let $U\subset\mathbb{R}^{n}$ be tubular
neighborhood of $M$. Then $M$ is closed in $U$ and therefore $N$ is closed in
$U$ as well. By Lemma 3.26 $U$ is a concrete differential space. By Lemma 3.25
$N$ is concrete. ∎

######  Corollary 3.28.

Let $N$ be a locally closed subset of a manifold $M$. Then the differential
space $(N,C^{\infty}(N))$ (with the induced smooth structure) is a concrete
differential space.

###### Proof.

If $N\subseteq M$ is locally closed then there is an open subset $U\subseteq
M$ containing $N$ so that $N$ is closed in $U$. Since an open subset of a
manifold is a manifold, Lemma 3.27 above implies that $N$ is concrete. ∎

######  Definition 3.29.

A vector field on a differential space $(M,C^{\infty}(M))$ is a $C^{\infty}$
derivation $v:C^{\infty}(M)\to C^{\infty}(M)$. That is, we require that for
any $n>0$, any $f\in C^{\infty}(\mathbb{R}^{n})$ and any
$a_{1},\ldots,a_{n}\in C^{\infty}(M)$

$v(f\circ(a_{1},\ldots,a_{n}))=\sum_{i=1}^{n}\left((\partial_{i}f)\circ(a_{1},\ldots,a_{n})\right)\,v(a_{i}).$

######  Remark 3.30.

In the differential spaces literature it is more common to view vector fields
as $\mathbb{R}$-linear maps satisfying the Leibnitz rules, i.e., as
derivations of $\mathbb{R}$-algebras. As we noted above for a differential
space $(M,C^{\infty}(M))$ the $C^{\infty}$-ring $C^{\infty}(M)$ is point
determined hence any $\mathbb{R}$-algebra derivation $v:C^{\infty}(M)\to
C^{\infty}(M)$ is automatically a $C^{\infty}$-ring derivation. See [Ya] for a
fairly general result and [KL] for differential spaces specifically.

######  Remark 3.31.

As we noted previously if the differential space $(M,C^{\infty}(M))$ is a
manifold then a vector field in the sense of Definition 3.29 is a vector field
on $M$ in the usual sense.

######  Remark 3.32.

Let $M\subset\mathbb{R}^{n}$ be a closed subset and $I_{M}:=\\{f\in
C^{\infty}(\mathbb{R}^{n})\mid f|_{M}=0\\}$, the ideal of function that vanish
on $M$. Then if a vector field $w$ on $\mathbb{R}^{n}$ preserves the ideal
$I_{M}$ (i.e., if $w(I_{M})\subset I_{M}$) the $w$ induces a derivation on the
quotient $C^{\infty}$-ring $C^{\infty}(M)=C^{\infty}(\mathbb{R}^{n})/I_{M}$.

In general a vector field on $\mathbb{R}^{n}$ does not induce a vector field
(derivation) on $M$. For example if $M$ is an embedded submanifold then a
vector field $v$ on $\mathbb{R}^{n}$ induces a vector field on $M$ if and only
if $v$ is tangent to $M$. Here is another example: let
$M=\\{(x,y)\in\mathbb{R}^{2}\mid xy=0\\}$; $M$ is the union of two coordinate
axes. One can show using Hadamard’s lemma that $I_{M}$ is the ideal generated
by the function $xy$; see Example 6.22. However $\frac{\partial}{\partial
x}(xy)=y\not\in\langle xy\rangle$. Hence $\frac{\partial}{\partial x}$ does
not induce a derivation on $C^{\infty}(M)$. This is not surprising since
$\frac{\partial}{\partial x}$ is not tangent to the $y$-axis.

However there are cases when there is a surjective restriction map from the
vector fields on $\mathbb{R}^{n}$ to the vector fields on $M$: see examples
below.

######  Example 3.33.

Let $C\subset\mathbb{R}$ denote the standard (middle-thirds) Cantor set. Then
the there is a well-defined surjective map
$\operatorname{Der}(C^{\infty}(\mathbb{R}))\to\operatorname{Der}(C^{\infty}(C))$.

###### Proof.

Let $f\in C^{\infty}(\mathbb{R})$ be a smooth function with $f|_{C}=0$. We
argue that derivative $f^{\prime}$ has to restrict to 0 on $C$ as well.

If $f^{\prime}(c)>0$ for some $c\in C$ then there is $\varepsilon>0$ so that
$f(x)>0$ for all $x\in(c,c+\varepsilon)$ and $f(x)<0$ for all
$x\in(c-\varepsilon,c)$. Since either $C\cap(c,\varepsilon)\not=\varnothing$
or $C\cap(c-\varepsilon,c)\not=\varnothing$ this contradicts $f_{C}=0$. Hence
$\frac{d}{dx}$ preserves the ideal $I_{C}$ of functions on $\mathbb{R}$ that
vanish on $C$. Consequently any other vector field on $\mathbb{R}$ preserves
the ideal $I_{C}$ and hence induces a derivation on $C^{\infty}(C)$.

Conversely given a derivation $v:C^{\infty}(C)\to C^{\infty}(C)$, consider the
value of $v$ on the restriction $x|_{C}$ of the standard coordinate function
$x:\mathbb{R}\to\mathbb{R}$. Then $v(x|_{X})=f(x)|_{C}$ for some smooth
function $f\in C^{\infty}(\mathbb{R})$. It is not hard to see that the
derivation $V:=f\frac{d}{dx}:C^{\infty}(\mathbb{R})\to C^{\infty}(\mathbb{R})$
induces the derivation $v$. This is because for any $h\in
C^{\infty}(\mathbb{R})$

$v(h|_{C})=v(h\circ
x|_{C})=(h^{\prime}\circ(x|_{C}))v(x|_{C})=h^{\prime}|_{C}f_{C}=V(h)|_{C}.$

∎

######  Definition 3.34.

An interval  is a connected subset of $\mathbb{R}$ consisting of more than 1
point.

######  Remark 3.35.

If $J\subset\mathbb{R}$ is an interval and $f\in C^{\infty}(J)$ a smooth
function (where $J$ has the induced smooth structure) then the derivative
$f^{\prime}$ is also smooth. Hence we have a well-defined $C^{\infty}$-ring
derivation $C^{\infty}(J)\to C^{\infty}(J)$, $f\mapsto f^{\prime}$ which we
denote by $\frac{d}{dx}$.

######  Definition 3.36.

Let $v:C^{\infty}(M)\to C^{\infty}(M)$ be a derivation on a differential space
$M$. An integral curve $\gamma$ of $v$ through a point $p\in M$ is either a
map $\gamma:\\{0\\}\to M$ with $\gamma(0)=p$ or a map
$\gamma:(J,C^{\infty}(J))\to(M,C^{\infty}(M))$ of differential spaces from an
interval $J\subset\mathbb{R}$ containing 0 so that $\gamma(0)=p$ and

(3.2) $\frac{d}{dx}(f\circ\gamma)=v(f)\circ\gamma$

for all functions $f\in C^{\infty}(M)$.

The following examples are meant to illustrate two points:

  * (1)

curves that only exist for time 0 should be allowed as integral curves of
derivations and

  * (2)

we should not require the domain of an integral curve to be an open interval.

######  Example 3.37.

Let $M=[0,1]$ with the differential structure inherited from $\mathbb{R}$. Let
$v=\frac{d}{dx}$. Then $\gamma:[-1/2,1/2]\to[0,1]$, $\gamma(t)=t+1/2$ is an
integral curve of $v$ through $1/2$. The curve $\gamma$ is a maximal integral
curve of $v$ and its domain is a closed interval.

######  Example 3.38.

Let $M$ be the standard closed disk in $\mathbb{R}^{2}$: $M=\\{(x,y)\mid
x^{2}+y^{2}\leq 1\\}$. Then $M$ is a manifold with boundary and a differential
subspace of $\mathbb{R}^{2}$. Consider the vector field
$v=\frac{\partial}{\partial x}$ on $M$. The integral curve of $v$ through
$(0,1)$ is $\gamma:\\{0\\}\to M$, $\gamma(0)=(0,1)$; it only exists for zero
time. Note that $v$ does have a flow. It’s a smooth map $\Phi$ from
$U=\\{((x,y),t)\in\mathbb{R}^{2}\times\mathbb{R}\mid x^{2}+y^{2}\leq
1,(x-t)^{2}+y^{2}\leq 1\\}$ to $M$. It is given by $\Phi((x,y),t)=(x+t,y)$.

Note that while $M$ is a manifold with boundary, the flow domain $U$ is not a
manifold with boundary nor a manifold with corners. The domain $U$ is a
differential space.

######  Remark 3.39.

It is natural to wonder if vector fields on (Hausdorff) differential spaces
have unique maximal integral curves and, more generally, if they have smooth
flows. Śniatycki showed that for a large interesting class of differential
spaces (the so called subCartesian spaces) vector fields have unique maximal
integral curves. See [Sn] and references therein. Another proof is given in
[KL]. If the differential spaces is diffeomorphic to a closed subset of some
$\mathbb{R}^{n}$ then vector fields have smooth flows. See [KL]. Note that any
locally closed subset of a second countable Hausdorff manifold is
diffeomorphic to a closed subset of some $\mathbb{R}^{n}$. Thus the existence
of flows is known for a large class of spaces.

##  4\. Local $C^{\infty}$-ringed spaces

We have seen that differential spaces come with a sheaf of “smooth” functions.
We now generalize.

######  Definition 4.1.

A $C^{\infty}$-ringed space is a pair $(X,\mathscr{O}_{X})$ where $X$ is a
topological space and $\mathscr{O}_{X}$ is a sheaf of $C^{\infty}$-rings on
$X$.

A local $C^{\infty}$-ringed space is a $C^{\infty}$-ringed space so that all
the stalks $\mathscr{O}_{X,x}$ of the sheaf $\mathscr{O}_{X}$ are local
$C^{\infty}$-rings in the sense of Definition 2.38.

######  Example 4.2.

Any manifold $M$ is a local $C^{\infty}$-ringed space because

  * •

smooth functions on $M$ give rise to a sheaf $C^{\infty}_{M}$,

  * •

the stalk $C^{\infty}_{M,x}$ of the sheaf $C^{\infty}_{M}$ at a point $x\in M$
is the algebra of germes of smooth functions at $x$, which is a
$C^{\infty}$-ring, and

  * •

the maximal ideal in question is the ideal $\mathfrak{m}_{x}$ of germes that
vanish at $x$. This maximal ideal is unique because any germ $f_{x}$ of a
smooth function $f$ that does not vanish at $x$ is invertible in
$C^{\infty}_{M,x}$. The uniqueness of the maximal ideal may not be entirely
obvious; see Lemma 4.3 below for a proof of that rings of germs of smooth
functions on differential spaces are local $C^{\infty}$-rings.

######  Lemma 4.3.

Let $(M,C^{\infty}(M))$ be a differential space and $C^{\infty}_{M}$ the
corresponding sheaf of smooth functions (Definition 3.9). The stalks of
$C^{\infty}_{M}$ are local $C^{\infty}$-rings, consequently
$(M,C^{\infty}_{M})$ is a local $C^{\infty}$-ringed spaces.

###### Proof.

Let $x\in M$ be a point and $C^{\infty}_{M,x}$ the stalk of $C^{\infty}_{M}$
at $x$. Recall that $C^{\infty}_{M,x}$ is the set of equivalence classes of
pairs $(U,f)$ where $U$ is an open neighborhood of $x$ and $f\in
C^{\infty}_{M}(U)$ a smooth function. We denote an element of
$C^{\infty}_{M,x}$ either by $[(U,f)]$ or by $f_{x}$, depending on whether we
want to emphasize that that the function representing the germ is defined on a
neighborhood $U$ or if we want to emphasize that germ is at the point $x$ (and
the neighborhood $U$, on which $f$ is defined, is left unmentioned).

We have a well-defined evaluation map

$\operatorname{\mathsf{ev}}_{x}:C^{\infty}_{M,x}\to\mathbb{R},\qquad\operatorname{\mathsf{ev}}_{x}(f_{x})=f(x).$

Since the evaluation map is onto, its kernel $\mathfrak{m}$ is a maximal ideal
in $C^{\infty}_{M,x}$. To prove that $C^{\infty}_{M,x}$ is local, it is enough
to show that any element of the complement of $\mathfrak{m}$ in
$C^{\infty}_{M,x}$ is a unit.

Suppose $f_{x}\in C^{\infty}_{M},x\smallsetminus\mathfrak{m}$. It is no loss
of generality that $a=\operatorname{\mathsf{ev}}_{x}(f_{x})=f(x)>0$. Choose a
smooth function $\eta$ on $\mathbb{R}$ so that $\eta>0$ and $\eta(t)=t$ for
all $t\in(a/2,3a/2)$. Since $\eta>0$, $\zeta(t)=1/\eta(t)$ is well-defined and
smooth. Consider $V=f^{-1}((a/2,3a/2))$. For any point $y\in V$,
$\eta(f(y))=f(y)$. On the other hand,

$1=(\eta\cdot\zeta)(f(y))=\eta(f(y))\cdot\zeta(f(y)).$

It follows that $[(V,\zeta\circ f)]$ is the inverse of $f_{x}$ in the stalk
$C^{\infty}_{M,x}$. Hence the gern $f_{x}$ is a unit, as we wanted to show. ∎

######  Remark 4.4.

A reader unfamiliar with ringed spaces of any kind may wonder why anyone would
single out local ringed spaces. There is a variety of reasons. One of them is
that if $(X,\mathscr{O}_{X})$ is a local $C^{\infty}$-ringed spaces, then for
any open set $U\subset X$ a section $s\in\mathscr{O}_{X}(U)$ gives rise to a
real-valued function $f_{s}$ as follows: given $x\in U$, there is a unique
$\mathbb{R}$-point $\hat{x}:\mathscr{O}_{X,x}\to\mathbb{R}$. We then define

$f_{s}(x):=\hat{x}(s_{x}),$

where, as before, $s_{x}$ is the germ of $s$ at $x$. For differential spaces,
of course, $f_{s}$ is just $s$. But in general the sheaf $\mathscr{O}_{X}$
need not be a sheaf of $C^{\infty}$-rings of functions, and locality gives us
a map from local sections to (local) functions. See the proof of Theorem 4.9
below for an example of the use of this fact.

In order to define maps (or, if you prefer, morphisms) between local
$C^{\infty}$-ringed spaces we need to recall that (pre)sheaves can be pushed
forward by continuos maps. Namely, if $\mathscr{S}$ is a presheaf on a space
$X$ and $f:X\to Y$ is a continuous map, then there is a presheaf
$f_{*}\mathscr{S}$ on $Y$ which is defined by

$f_{*}\mathscr{S}(U):=\mathscr{S}(f^{-1}(U))$

for every open set $U\subset Y$. It is not hard to check that if $\mathscr{S}$
is a sheaf then the pushforward presheaf $f_{*}\mathscr{S}$ is also a sheaf.

The following example is meant to motivate the definition of a map between two
$C^{\infty}$-ringed spaces. Consider a smooth map $f:M\to N$ between two
manifolds. It induces a map $f_{\\#}:C^{\infty}(N)\to C^{\infty}(M)$ between
$C^{\infty}$-rings; $f_{\\#}$ is given by

$f_{\\#}h:=h\circ f$

for all $h\in C^{\infty}(M)$; i.e., $f_{\\#}$ is just a pullback map. There is
also more: for every open set $U\subset M$ we have a map of $C^{\infty}$-rings

$f_{\\#,U}:C^{\infty}(U)\to C^{\infty}(f^{-1}(U)),\qquad f_{\\#}(h)=h\circ f.$

More precisely $C^{\infty}(U)=C^{\infty}_{M}(U)$ and
$C^{\infty}(f^{-1}(U))=C^{\infty}_{M}(f^{-1}(U))=(f_{*}C^{\infty}_{M})(U)$.
Moreover, the maps $\\{f_{\\#,U}\\}_{U\subset M}$ are compatible with
restrictions, and therefore pullback by $f$ defines a map of sheaves

$f_{\\#}:C^{\infty}_{M}\to f_{*}C^{\infty}_{M}.$

######  Definition 4.5.

A morphism (or a map) of local $C^{\infty}$-ringed spaces from
$(X,\mathscr{O}_{X})$ to $(Y,\mathscr{O}_{Y})$ is a pair $(f,f_{\\#})$ where
$f:X\to Y$ is continuous and $f_{\\#}:\mathscr{O}_{Y}\to f_{*}\mathscr{O}_{X}$
is a map of sheaves of $C^{\infty}$-rings on the space $Y$.

######  Remark 4.6.

Note that the map $(f,f_{\\#}):(X,\mathscr{O}_{X})\to(Y,\mathscr{O}_{Y})$ of
local $C^{\infty}$-ringed spaces induces, for every point $x\in X$, a map of
stalks $f_{x}:\mathscr{O}_{Y,f(x)}\to\mathscr{O}_{X,x}$

In algebraic geometry one defines a morphism of locally ringed spaces to be a
pair $(f,f_{\\#}):(X,\mathscr{O}_{X})\to(Y,\mathscr{O}_{Y})$ so that the
induced maps on stalks $f_{x}$ are all local: they are required to take the
unique maximal ideal to the unique maximal ideal. For local
$C^{\infty}$-ringed spaces this is unnecessary because any map of local
$C^{\infty}$-rings automatically preserves maximal ideals. This, in turn, is
because all the residue fields are just $\mathbb{R}$.

In order to define the composition of two maps of $C^{\infty}$-ringed spaces
we need to recall that pushing (pre)sheaves forward along a continuous map
$f:X\to Y$ is actually a functor from the category $\mathsf{Psh}_{X}$ of
presheaves on $X$ to the category $\mathsf{Psh}_{Y}$ of presheaves on $Y$:
given a map of presheaves $\varphi:\mathscr{O}\to\mathscr{O}^{\prime}$ in
$\mathsf{Psh}_{X}$ we get a map of presheaves
$f_{*}(\varphi):f_{*}\mathscr{O}\to f_{*}\mathscr{O}$. Namely for an open set
$U\subset Y$ the $U$-component $f_{*}(\varphi)_{U}:f_{*}\mathscr{O}(U)\to
f_{*}\mathscr{O}^{\prime}(U)$ of $f_{*}(\varphi)$ is, by construction,

$\varphi_{f^{-1}(U)}:\mathscr{O}(f^{-1}(U))\to\mathscr{O}^{\prime}(f^{-1}(U)).$

######  Definition 4.7.

Given two maps $(f,f_{\\#}):(X,\mathscr{O}_{X})\to(Y,\mathscr{O}_{Y})$ and
$(g,g_{\\#}):(Y,\mathscr{O}_{Y})\to(Z,\mathscr{O}_{Z})$ we define their
composite to be $(g\circ f,(g\circ
f)_{\\#}):(X,\mathscr{O}_{X})\to(Z,\mathscr{O}_{Z})$ where

$(g\circ
f)_{\\#}:=(g_{*}(f_{*}\mathscr{O}_{x})\xleftarrow{g_{*}(f_{\\#})}g_{*}\mathscr{O}_{Y})\circ(g_{*}\mathscr{O}_{Y}\xleftarrow{g_{\\#}}\mathscr{O}_{Z}).$

Here we tacitly used the fact that $(g\circ f)_{*}=g_{*}\circ f_{*}$.

######  Definition 4.8 (The category $\mathsf{LC^{\infty}RS}$).

Local $C^{\infty}$-ringed spaces and their morphisms form a category which we
denote by $\mathsf{LC^{\infty}RS}$.

######  Theorem 4.9.

The category $\mathsf{DiffSpace}$ of differential spaces is (isomorphic to) a
full subcategory of $\mathsf{LC^{\infty}RS}$ of local $C^{\infty}$-ringed
spaces.

###### Proof.

We construct a functor $F:\mathsf{DiffSpace}\to\mathsf{LC^{\infty}RS}$ and
check that it is fully faithful. We define $F$ on objects by sending a
differential space $(M,C^{\infty}(M))$ to the corresponding local
$C^{\infty}$-ringed space $(M,C^{\infty}_{M})$:

$F((M,C^{\infty}(M))=(M,C^{\infty}_{M}),$

see Lemma 4.3. We need to define $F$ on morphisms.

Recall that a smooth map $f:(M,C^{\infty}(M))\to(N,C^{\infty}(N))$ between two
differential spaces is a continuous map $f:M\to N$ with the property that for
all $h\in C^{\infty}(N)$ its pullback $f^{*}h$ is in $C^{\infty}(M)$. Thus in
order to define $F$ on morphisms we need to construct a map of sheaves
$f_{\\#}:C^{\infty}_{N}\to f_{*}C^{\infty}_{M}$.

Recall that $C^{\infty}_{N}$ is the sheafification of the presheaf
${\mathcal{P}}_{N}$ defined by

${\mathcal{P}}_{N}(U)=C^{\infty}_{N}(N)|_{U}$

for all open sets $U\subseteq N$. Since pullback by $f$ commutes with
restrictions we get a map of presheaves

$f_{\\#}:{\mathcal{P}}_{N}\to f_{*}({\mathcal{P}}_{M})\hookrightarrow
f_{*}C^{\infty}_{M},\qquad f_{\\#}(h)=h\circ f.$

The universal property of sheafification ${\mathcal{P}}_{N}\hookrightarrow
C^{\infty}_{N}$ uniquely extends $f_{\\#}$ to

$f_{\\#}:C^{\infty}_{N}\to f_{*}C^{\infty}_{M},$

which is given by the same formula: for any open set $U\subset N$ and for any
function $h\in C^{\infty}_{N}(U)$,

$f_{\\#}h=h\circ f\in C^{\infty}_{M}(f^{-1}(U)).$

We now define $F$ on morphisms by

$F((M,C^{\infty}(M))\xrightarrow{f}(N,C^{\infty}(N)))=(f,f_{\\#}):(M,C^{\infty}_{M})\to(N,C^{\infty}_{N}),$

where as above $f_{\\#}h=h\circ f$.

It is not hard to check that $F$ preserves identity maps and composition.
Therefore $F$ is in fact a functor.

If $F(f)=(f,f_{\\#})=(g,g_{\\#})=F(g)$ for two maps
$f,g:(M,C^{\infty}(M))\to(N,C^{\infty}(N))$ then clearly $f=g$. Therefore $F$
is injective on $\operatorname{Hom}$ sets, i.e, the functor $F$ is faithful.

It remains to prove that $F$ is surjective on $\operatorname{Hom}$ sets. This
is where locality has to come in. Suppose we have a continuous map $f:M\to N$
and a map of sheaves $\psi:C^{\infty}_{N}\to f_{*}C^{\infty}_{M}$. We want to
show that for any open set $U\subset N$ and any $h\in C^{\infty}_{N}(U)$

$\psi(h)=h\circ f,$

for then $(f,\psi)=F(f)$.

Consider a point $p\in U$ and a point $q\in f^{-1}(p)$. The map $\psi$ induces
a map on stalks

$\psi_{p}:C^{\infty}_{N,p}\to(f_{*}C^{\infty}_{M})_{p}.$

Since $q\in f^{-1}(p)$, there is a canonical map of stalks
$(f_{*}C^{\infty}_{M})_{p}\to C^{\infty}_{M,q}$. Precomposing this canonical
map with $\psi_{p}$ we get a map

$\varphi:C^{\infty}_{N,p}\to C^{\infty}_{M,q}.$

We can give this map $\varphi$ a concrete description: $U$ is a neighborhood
of $p$ and $h\in C^{\infty}_{N}(U)$ then

$\varphi([(U,h)]=[(f^{-1}(U),\psi(h))]\in C^{\infty}_{M,q}.$

The stalk $C^{\infty}_{M,q}$ is a local $C^{\infty}$-ring, hence has the
unique $\mathbb{R}$-point $\hat{q}:C^{\infty}_{M,q}\to\mathbb{R}$:
$\hat{q}(g_{q})=g(q)$ for all germs $g_{q}$; it is the evaluation at $q$. The
composite $\hat{q}\circ\varphi:C^{\infty}_{N,p}\to\mathbb{R}$ is also an
$\mathbb{R}$-point. Since the $C^{\infty}$-ring $C^{\infty}_{N,p}$ is local,
it has only one $\mathbb{R}$-point, namely $\hat{p}$, the evaluation at $p$.
Therefore

$\hat{q}\circ\varphi=\hat{p}.$

We are almost done. Recall that any nonzero $C^{\infty}$-ring $\mathscr{A}$
contains a copy of $\mathbb{R}$. This copy correspond to operations of arity
$0$, i.e., to constants, and any map of $C^{\infty}$-rings preserves these
constants. Now for any germ $h_{p}\in C^{\infty}_{M,p}$

$0=\hat{p}(h_{p}-h(p))=\hat{q}(\varphi(h_{p}-h(p)))=\hat{q}(\varphi(h_{p}))-h(p)$

since $h(p)$ is a constant. Since $h(p)=h(f(q))$ and since
$\hat{q}(\varphi(h_{p}))=\psi(h)(q)$ we have

$\psi(h)(q)=h(f(q))$

for all $q\in f^{-1}(U)$. ∎

##  5\. $C^{\infty}$-schemes

There is a (contravariant) functor $\Gamma$, the global sections functor, from
the category of local $C^{\infty}$-ringed spaces $\mathsf{LC^{\infty}RS}$ (see
Definition 4.8 to the (opposite) category of $C^{\infty}$-rings
$C^{\infty}\mathsf{Ring}$. Namely, given a $C^{\infty}$-ringed space
$(M,\mathscr{A})$, $\Gamma$ sends it to the ring $\mathscr{A}(M)$ of global
sections. Given a map $(f,f_{\\#}):(N,\mathscr{B})\to(M,\mathscr{A})$ of local
$C^{\infty}$-ringed spaces we have a map of $C^{\infty}$-rings
$f_{\\#}:\mathscr{A}(M)=f_{*}(\mathscr{B})(M)=\mathscr{B}(f^{-1}(M))=\mathscr{B}(N)$.
In other words

$\Gamma:\mathsf{LC^{\infty}RS}\to{C^{\infty}\mathsf{Ring}}^{\mbox{\sf{\tiny{op}}}}$

is given by

$\Gamma((N,\mathscr{B}))\xrightarrow{(f,f_{\\#})}(M,\mathscr{A})):=(\mathscr{A}(M)\xrightarrow{f_{\\#}}\mathscr{B}(N)).$

It is easy to see that $\Gamma$ is in fact a functor. Thanks to a theorem of
Dubuc [D], the global section functor $\Gamma$ has a right adjoint
$\operatorname{Spec}:{C^{\infty}\mathsf{Ring}}^{\mbox{\sf{\tiny{op}}}}\to\mathsf{LC^{\infty}RS}$.
An affine $C^{\infty}$-scheme is, by definition, a local $C^{\infty}$-ringed
space isomorphic to $\operatorname{Spec}(\mathscr{A})$ for some
$C^{\infty}$-ring $\mathscr{A}$. That is, the category of affine schemes
$\mathsf{C^{\infty}Aff}$ is the essential image of the functor
$\operatorname{Spec}$. Unlike the spectrum functor in algebraic geometry
$\operatorname{Spec}:{C^{\infty}\mathsf{Ring}}^{\mbox{\sf{\tiny{op}}}}\to\mathsf{C^{\infty}Aff}$
is not an equivalence of categories.

Dubuc’s proof of the existence of $\operatorname{Spec}$ is terse and uses
topos theory. Joyce provides another proof of Dubuc’s theorem in [J]. We
present (a version of) Joyce’s proof below. We shorten the proof in [J] by
appealing to a category-theoretic fact. Recall:

######  Lemma 5.1.

Let $L:\mathsf{A}\to\mathsf{B}$ be a functor. Suppose for every object $b$ of
$\mathsf{B}$ there is an object $Rb$ of $\mathsf{A}$ and a morphism
$\varepsilon_{b}:LRb\to b$ with the following universal property: for any
object $a$ of $\mathsf{A}$ and any morphism $\varphi:La\to b$ there is a
unique morphism $f:a\to Rb$ making the diagram

$\textstyle{b}$$\textstyle{LRb}$$\textstyle{La}$$\scriptstyle{\varepsilon_{b}}$$\scriptstyle{\varphi}$$\scriptstyle{Lf}$

commute. Then

  * (i)

the function $b\mapsto Rb$ from the objects of $\mathsf{B}$ to the objects of
$\mathsf{A}$ extends to a functor $R:\mathsf{B}\to\mathsf{A}$;

  * (ii)

the functor $R$ is right adjoint to $L$ and

  * (iii)

the collection $\\{\varepsilon_{b}:LRb\to b\\}_{b\in\mathsf{B}}$ assemble into
a natural transformation
$\varepsilon:LR\Rightarrow\operatorname{id}_{\mathsf{B}}$, which is the counit
of the adjunction $L\dashv R$.

###### Proof.

Omitted. ∎

It follows from Lemma 5.1 that in order to construct a right adjoint
$\operatorname{Spec}:{C^{\infty}\mathsf{Ring}}^{\mbox{\sf{\tiny{op}}}}\to\mathsf{LC^{\infty}RS}$
to the global section functor
$\Gamma:\mathsf{LC^{\infty}RS}\to{C^{\infty}\mathsf{Ring}}^{\mbox{\sf{\tiny{op}}}}$
it is enough to prove the following proposition.

######  Proposition 5.2.

For every $C^{\infty}$-ring $\mathscr{A}$ there is a local $C^{\infty}$-ringed
space $\operatorname{Spec}(\mathscr{A})$ and a map of $C^{\infty}$-rings
$\varepsilon_{\mathscr{A}}:\mathscr{A}\to\Gamma(\operatorname{Spec}A)$ with
the following universal property:
for any local $C^{\infty}$-ringed space $(Y,{\mathcal{O}}_{Y})$ and any map
$\varphi:\mathscr{A}\to\Gamma(Y,{\mathcal{O}}_{Y})$ of $C^{\infty}$-rings,
there is a unique map
$(f,f_{\\#}):(Y,{\mathcal{O}}_{Y})\to\operatorname{Spec}(\mathscr{A}))$ of
$C^{\infty}$-ringed spaces with the property that

$\Gamma(f,f_{\\#})\circ\varepsilon_{\mathscr{A}}=\varphi.$

Our proof of Proposition 5.2 will take several pages. We mostly follow Joyce,
but not everywhere. We first put a topology on the set
${\mathcal{X}}_{\mathscr{A}}$ of $\mathbb{R}$-points of a $C^{\infty}$-ring
$\mathscr{A}$. We then construct a presheaf ${\mathcal{P}}_{\mathscr{A}}$ of
$C^{\infty}$-rings on ${\mathcal{X}}_{\mathscr{A}}$ and sheafify it to obtain
a sheaf ${\mathcal{O}}_{\mathscr{A}}$. The presheaf
${\mathcal{P}}_{\mathscr{A}}$ will come with a map
$\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}:\mathscr{A}\to{\mathcal{P}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
from the ring $\mathscr{A}$ to its global sections. We show that the stalks of
${\mathcal{O}}_{\mathscr{A}}$ are local. We define
$\operatorname{Spec}(\mathscr{A})$ to be
$({\mathcal{X}}_{\mathscr{A}},{\mathcal{O}}_{\mathscr{A}})$. The presheaf
${\mathcal{P}}_{\mathscr{A}}$ will come with a map
$\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}:\mathscr{A}\to{\mathcal{P}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
from the ring $\mathscr{A}$ to the ring of global sections of
${\mathcal{P}}_{\mathscr{A}}$. Since by definition
${\mathcal{O}}_{\mathscr{A}}$ is the sheafification of
${\mathcal{P}}_{\mathscr{A}}$ there is a canonical map
$\eta_{\mathscr{A}}:{\mathcal{P}}_{\mathscr{A}}\to\mathsf{sh}({\mathcal{P}}_{\mathscr{A}})={\mathcal{O}}_{\mathscr{A}}$
of presheaves. Composing the global sections part
$\eta_{\mathscr{A},{\mathcal{X}}_{\mathscr{A}}}:{\mathcal{P}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
of $\eta_{\mathscr{A}}$ with
$\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}$ we get
$\varepsilon_{\mathscr{A}}=\eta_{\mathscr{A},{\mathcal{X}}_{\mathscr{A}}}\circ\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}:\mathscr{A}\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$.
Finally we prove that the
$\varepsilon_{\mathscr{A}}:\mathscr{A}\to\Gamma(\operatorname{Spec}A)={\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
of $C^{\infty}$-rings has the desired universal property.

######  Notation 5.3.

Given a $C^{\infty}$-ring $\mathscr{A}$ denote the set of $\mathbb{R}$-points
of $\mathscr{A}$ by ${\mathcal{X}}_{\mathscr{A}}$. Thus

${\mathcal{X}}_{\mathscr{A}}=\\{p:\mathscr{A}\to\mathbb{R}\mid p\textrm{ is a
map of $C^{\infty}$-rings}\\}.$

######  Example 5.4.

Let $M$ be a (second countable Hausdorff) manifold and $C^{\infty}(M)$ the
$C^{\infty}$-ring of smooth functions on $M$. Then by a theorem of Pursell the
set ${\mathcal{X}}_{C^{\infty}(M)}$ of $\mathbb{R}$-points of $M$ is the set
of points of $M$ (cf. Remark 2.21).

######  Example 5.5.

If $\mathscr{A}$ is the zero ring then
${\mathcal{X}}_{\mathscr{A}}=\varnothing$.

######  Lemma 5.6.

Let $I$ be an ideal in the $\mathbb{R}$-algebra $C^{\infty}(\mathbb{R}^{n})$
and $\mathscr{A}=C^{\infty}(\mathbb{R}^{n})/I$ the corresponding quotient
$C^{\infty}$-ring. The $\mathbb{R}$-points of $\mathscr{A}$ are in bijection
with the set

$Z(I)=\\{x\in\mathbb{R}^{n}\mid f(x)=0\textrm{ for all }f\in I\\}.$

###### Proof.

Given a point $p\in Z(I)$ the evaluation homomorphism
$\operatorname{\mathsf{ev}}_{p}:C^{\infty}(\mathbb{R}^{n})\to\mathbb{R}$,
$\operatorname{\mathsf{ev}}_{p}(f):=f(p)$ contains the ideal $I$ in its
kernel. Hence it induces an $\mathbb{R}$-point

$\overline{\operatorname{\mathsf{ev}}}_{p}:\mathscr{A}=C^{\infty}(\mathbb{R}^{n})/I\to\mathbb{R}.$

If
$\overline{\operatorname{\mathsf{ev}}}_{p}=\overline{\operatorname{\mathsf{ev}}}_{q}$
for some $q\in\mathbb{R}^{n}$ then

$ev_{p}=\overline{\operatorname{\mathsf{ev}}}_{p}\circ\pi=\overline{\operatorname{\mathsf{ev}}}_{q}\circ\pi=\operatorname{\mathsf{ev}}_{q},$

where $\pi:C^{\infty}(\mathbb{R}^{n})\to C^{\infty}(\mathbb{R}^{n})/I$ is the
canonical projection. Since the functions in $C^{\infty}(\mathbb{R}^{n})$
separate points, $p=q$ and consequently the map

$F:Z(I)\to\operatorname{Hom}(\mathscr{A},\mathbb{R}),\quad
F(p)=\overline{\operatorname{\mathsf{ev}}}_{p}$

is injective.

Given an $\mathbb{R}$-point
$\varphi\in\operatorname{Hom}(\mathscr{A},\mathbb{R})$ the composite
$\varphi\circ\pi:C^{\infty}(\mathbb{R}^{n})\to\mathbb{R}$ is an
$\mathbb{R}$-point of $C^{\infty}(\mathbb{R}^{n})$. By Lemma 2.22 there is
$p\in\mathbb{R}^{n}$ so that $\varphi\circ\pi=\operatorname{\mathsf{ev}}_{p}$.
Consequently for any $f\in I$

$0=f(p)=\operatorname{\mathsf{ev}}_{p}(f)=\varphi(\pi(f))=\varphi(0)=0.$

Hence $p\in Z(I)$ and the function
$F:Z(I)\to\operatorname{Hom}(\mathscr{A},\mathbb{R})$ is also surjective. ∎

######  Example 5.7.

Let $I\subset C^{\infty}(\mathbb{R}^{n})$ be the set of compactly supported
functions. It’s easy to see that $I$ is an ideal of
$C^{\infty}(\mathbb{R}^{n})$ hence $C^{\infty}(\mathbb{R}^{n})/I$ is a
$C^{\infty}$-ring. By Lemma 5.6 there is a bijection

${\mathcal{X}}_{\mathscr{A}}\to Z(I):=\\{x\in\mathbb{R}^{n}\mid f(x)=0\textrm{
for all }f\in I\\}.$

In our case $Z(I)=\varnothing$ since for any point $x\in\mathbb{R}^{n}$ there
is a compactly supported function $f$ with $f(x)\not=0$. Hence in this case as
well ${\mathcal{X}}_{\mathscr{A}}=\varnothing$.

It follows that the map
${\mathcal{X}}:\mathscr{A}\mapsto{\mathcal{X}}_{\mathscr{A}}$ from
$C^{\infty}$-rings to sets is not injective.

######  Remark 5.8.

Example I of [CS] shows that the set $M$ of points of a differentials space
$(M,C^{\infty}(M))$ maybe strictly less than the set
${\mathcal{X}}_{C^{\infty}(M)}$ of $\mathbb{R}$-points of the
$C^{\infty}$-ring $C^{\infty}(M)$. This implies that there are differential
spaces which are not affine schemes.

Conversely there are affine schemes $\operatorname{Spec}(\mathscr{A})$ whose
structure sheaves are not $C^{\infty}$-rings of functions on the underlying
space ${\mathcal{X}}_{\mathscr{A}}$. For example we can view any local
$C^{\infty}$-ring $\mathscr{A}$ as a sheaf over a point $p$. If
$\mathscr{A}\not=\mathbb{R}$ then it is not a sheaf of functions on a one-
point space. Consequently there are affine $C^{\infty}$-schemes that are not
differential spaces.

On the other hand if $\mathscr{A}$ is finitely generated and point determined
then $\operatorname{Spec}(\mathscr{A})$ is a differential space isomorphic to
a closed subset of some $\mathbb{R}^{n}$ with the induced differential
structure (see Definition 3.16 and Theorem 5.21).

######  Lemma 5.9.

Let $\mathscr{A}$ be a (nonzero) $C^{\infty}$-ring with the set of points
${\mathcal{X}}_{\mathscr{A}}$. For each $a\in\mathscr{A}$ define

$U_{a}:=\\{p\in{\mathcal{X}}_{\mathscr{A}}\mid p(a)\not=0\\}.$

The set $\\{U_{a}\\}_{a\in\mathscr{A}}$ is a basis for a topology
${\mathcal{T}}_{\mathscr{A}}$ on ${\mathcal{X}}_{\mathscr{A}}$.

###### Proof.

The $C^{\infty}$-ring $\mathscr{A}$ contains 0 and 1. $U_{0}=\varnothing$,
$U_{1}={\mathcal{X}}$. For any $a,b\in\mathscr{A}$ and any
$p\in{\mathcal{X}}_{\mathscr{A}}$, $p(ab)=p(a)p(b)$. It follows that
$U_{a}\cap U_{b}=U_{ab}$. Therefore $\\{U_{a}\\}_{a\in\mathscr{A}}$ is a basis
for a topology. ∎

######  Definition 5.10.

We will refer to the topology ${\mathcal{T}}_{\mathscr{A}}$ of Lemma 5.9 as
the Zariski topology on ${\mathcal{X}}_{\mathscr{A}}$.

######  Notation 5.11.

For any $C^{\infty}$-ring $\mathscr{A}$ an element $a\in\mathscr{A}$ defines a
function $a_{*}:{\mathcal{X}}_{\mathscr{A}}\to\mathbb{R}$:

$a_{*}(p):=p(a)$

for all $p\in{\mathcal{X}}_{\mathscr{A}}$.

######  Lemma 5.12.

Let $\mathscr{A}$ be a $C^{\infty}$-ring and ${\mathcal{X}}_{\mathscr{A}}$ be
its set of points. The Zariski topology ${\mathcal{T}}_{\mathscr{A}}$ is the
smallest topology on the set ${\mathcal{X}}_{\mathscr{A}}$ so that the
functions $a_{*}:{\mathcal{X}}_{\mathscr{A}}\to\mathbb{R}$, $a_{*}(p):=p(a)$
are continuous for all $a\in\mathscr{A}$.

###### Proof.

We write $\mathbb{R}^{\times}$ for the set of nonzero real numbers. For any
function $h\in C^{\infty}(\mathbb{R})$, for any element $a\in\mathscr{A}$ and
any point $x\in{\mathcal{X}}_{\mathscr{A}}$

$(h_{\mathscr{A}}(a))_{*}(x)=x(h_{\mathscr{A}}(a))=h_{\mathbb{R}}(x(a))=h(a_{*}(x))=(h\circ
a_{*})(x).$

Given an open set $V\subseteq\mathbb{R}$ there is a function $f\in
C^{\infty}(\mathbb{R})$ so that $V=\\{f\not=0\\}\equiv
f^{-1}(\mathbb{R}^{\times})$ (this again follows from Whitney’s theorem: any
closed set in $\mathbb{R}$ is the zero set of some smooth function). And then

$(a_{*})^{-1}(V)=(a_{*})^{-1}(f^{-1}(\mathbb{R}^{\times}))=(f\circ
a_{*})^{-1}(\mathbb{R}^{\times})=((f_{\mathscr{A}}(a))_{*})^{-1}(\mathbb{R}^{\times})=\\{x\in{\mathcal{X}}_{\mathscr{A}}\mid
x(f_{\mathscr{A}}(a))\not=0\\}=U_{a}.$

Since the collection $\\{U_{b}\\}_{b\in\mathscr{A}}$ of sets forms a basis of
the topology ${\mathcal{T}}_{\mathscr{A}}$, the function $a_{*}$ is
continuous.

Conversely if ${\mathcal{T}}$ is any other topology so that all the functions
$a_{*}$ are continuous, then for any $a\in\mathscr{A}$

$U_{a}=(a_{*})^{-1}(\mathbb{R}^{\times})\in{\mathcal{T}},$

and consequently ${\mathcal{T}}_{\mathscr{A}}\subseteq{\mathcal{T}}$. ∎

######  Lemma 5.13.

For any $C^{\infty}$-ring $\mathscr{A}$ the topological space
${\mathcal{X}}_{\mathscr{A}}$ is Hausdorff and completely regular (points and
closed sets can be separated by continuous functions).

###### Proof.

We first check that ${\mathcal{X}}$ is Hausdorff. Suppose
$x,y\in{\mathcal{X}}_{\mathscr{A}}$ and $x\not=y$. Then there is
$a\in\mathscr{A}$ so that $x(a)\not=y(a)$. Choose disjoint open neighborhoods
$U$ of $x(a)$, $V$ of $y(a)$ in $\mathbb{R}$. Then, since
$a_{*}:{\mathcal{X}}_{\mathscr{A}}\to\mathbb{R}$ is continuous,
$(a_{*})^{-1}(U)$, $(a_{*})^{-1}(V)$ are disjoint open neighborhoods of $x$
and $y$, respectively, in ${\mathcal{X}}_{\mathscr{A}}$.

Now suppose $S\subset{\mathcal{X}}_{\mathscr{A}}$ is closed,
$x\in{\mathcal{X}}_{\mathscr{A}}$ and $x\not\in S$. Since $S$ is closed and
$\\{U_{a}\\}_{a\in\mathscr{A}}$ is a basis for the topology on
${\mathcal{X}}_{\mathscr{A}}$ there is $a\in\mathscr{A}$ so that $x\in U_{a}$
and $U_{a}\cap S=\varnothing$. Then $a_{*}(x)=x(a)\not=0$ and $a_{*}(s)=0$ for
all $s\in S$. ∎

We now start constructing a sheaf ${\mathcal{O}}_{\mathscr{A}}$ of local
$C^{\infty}$-rings on the space ${\mathcal{X}}_{\mathscr{A}}$ of points of a
ring $\mathscr{A}$. Again, we follow Joyce [J]. It will turn out that the
local ring $\mathscr{A}_{x}$ constructed in the next lemma is the stalk of the
structure sheaf ${\mathcal{O}}_{\mathscr{A}}$ of the $C^{\infty}$-affine
scheme $\operatorname{Spec}(\mathscr{A})$.

######  Lemma 5.14.

Let $\mathscr{A}$ be a $C^{\infty}$-ring and $x:\mathscr{A}\to\mathbb{R}$ a
point. Define

(5.1) $I_{x}:=\\{a\in\mathscr{A}\mid\textrm{ there is }d\in\mathscr{A}\textrm{
so that }x(d)\not=0\textrm{ and }ad=0\\}.$

Then

  1. (5.14.i)

$I_{x}$ is an ideal in $\mathscr{A}$.

  2. (5.14.ii)

$\mathscr{A}_{x}:=\mathscr{A}/I_{x}$ has exactly one $\mathbb{R}$-point
$\bar{x}$ which is given by $\bar{x}(a+I_{x})=x(a)$. Moreover
$\bar{x}(c)\not=0$ implies that $c$ is invertible in $\mathscr{A}_{x}$.

  3. (5.14.iii)

The quotient map $\pi_{x}:\mathscr{A}\to\mathscr{A}_{x}$ is the localization
of the $C^{\infty}$-ring $\mathscr{A}$ at the set $S:=\\{s\in\mathscr{A}\mid
x(s)\not=0\\}$.

###### Proof.

If $a_{1},a_{2}\in I_{x}$ then there are $d_{1},d_{2}\in\mathscr{A}$ so that
$x(d_{i})\not=0$ and $a_{i}d_{i}=0$, $i=1,2$. Then

$(a_{1}-a_{2})d_{1}d_{2}=(a_{1}d_{1})d_{2}-d_{1}(a_{2}d_{2})=0,$

while $x(d_{1}d_{2})=x(d_{1})x(d_{2})\not=0$. Also for any $b\in\mathscr{A}$

$(ba_{1})d_{1}=b(a_{1}d_{1})=b\cdot 0=0.$

We conclude that $I_{x}$ is, in fact, an ideal.

For any $c\in I_{x}$ there is $d\in\mathscr{A}$ so that $cd=0$ and
$x(d)\not=0$. Then $0=x(cd)=x(c)x(d)$ which forces $x(c)=0$ (since
$x(c),x(d)\in\mathbb{R}$). Thus $x|_{I_{x}}=0$ hence induces a unique map
$\bar{x}:\mathscr{A}/I_{x}\to\mathbb{R}$ of $C^{\infty}$-rings with
$\bar{x}(a+I_{x})=x(a)$ for all $a+I_{x}\in\mathscr{A}/I_{x}$.

We now argue that if $\bar{x}(a+I_{x})\not=0$ then $a+I_{x}\in\mathscr{A}_{x}$
is a unit. This easily implies that $\bar{x}$ is a unique $\mathbb{R}$-point
of $\mathscr{A}_{x}$, i.e., that $\mathscr{A}_{x}$ is local.

Since $0\not=\bar{x}(a+I_{x})=x(a)$ we may assume that $x(a)>0$ (otherwise
replace $a$ by $-a$). Choose $\eta\in C^{\infty}(\mathbb{R})$ so that $\eta>0$
and that $\eta(t)=t$ for all $t\in I:=(x(a)/2,3x(a)/2)$. That is, we require
that $\eta|_{I}=\operatorname{id}_{\mathbb{R}}|_{I}$. Choose $\zeta\in
C^{\infty}(\mathbb{R})$ so that $\zeta(x(a))=1$ and
$\operatorname{supp}\zeta\subseteq I$. Then

$(\eta-\operatorname{id}_{\mathbb{R}})\cdot\zeta=0.$

Since $\eta>0$ $\eta$ is invertible in $C^{\infty}(\mathbb{R})$. Denote its
inverse by $\eta^{-1}$. Let $c_{1}=\eta_{\mathscr{A}}(a)$,
$c_{2}=(\eta^{-1})_{\mathscr{A}}(a)$, $d=\zeta_{\mathscr{A}}(a)$. Then

$\displaystyle c_{1}c_{2}$ $\displaystyle=$
$\displaystyle\eta_{\mathscr{A}}(a)(\eta^{-1})_{\mathscr{A}}(a)=1_{\mathscr{A}}(a)=1\qquad$
$\displaystyle x(d)$ $\displaystyle=$ $\displaystyle
x(\zeta_{\mathscr{A}}(a))=\zeta(x(a))=1\qquad$ $\displaystyle(c_{1}-a)d$
$\displaystyle=$
$\displaystyle(\eta_{\mathscr{A}}(a)-(\operatorname{id}_{\mathbb{R}})_{\mathscr{A}}(a))\zeta_{\mathscr{A}}(a)=((\eta-\operatorname{id}_{R})\zeta)_{\mathscr{A}}(a)=0_{\mathscr{A}}(a)=0.$

Since $x(d)\not=0$ we have $c_{1}-a\in I_{x}$. Hence

$c_{1}+I_{x}=a+I_{x}.$

On the other hand, since $c_{1}c_{2}=1$, $(c_{1}+I_{x})(c_{2}+I_{x})=1+I_{x}$.
Therefore $a+I_{x}$ is invertible.

It remains to show that the quotient map
$\pi_{x}:\mathscr{A}\to\mathscr{A}_{x}=\mathscr{A}/I_{x}$,
$\pi_{x}(a)=a+I_{x}$ is the localization of $\mathscr{A}$ at the set
$S=\\{x\not=0\\}$. We have already checked that $\pi_{x}(a)$ is a unit in
$\mathscr{A}_{x}$ for all $a\in S$. So suppose $f:\mathscr{A}\to\mathscr{B}$
is a map of $C^{\infty}$-rings so that $f(a)$ is invertible for all $a\in S$.
We would like to show that $f$ descends to
$\bar{f}:\mathscr{A}_{x}\to\mathscr{B}$ with

$\bar{f}\circ\pi_{x}=f.$

Note that since $\pi_{x}$ is surjective such map $\bar{f}$ is automatically
unique. To check the existence of $\bar{f}$ it is enough to show that $\ker
f\supseteq I_{x}=\ker\pi_{x}$.

If $c\in I_{x}$ then there is $d\in\mathscr{A}$ so that $x(d)\not=0$ (i.e.,
$d\in S$) and $cd=0$. Then $0=f(cd)=f(c)f(d)$. By assumption on $f$, $f(d)$ is
a unit in $\mathscr{B}$. Hence $f(c)=0$ and we are done. ∎

######  Notation 5.15.

For any element $a$ of a $C^{\infty}$-ring $\mathscr{A}$ and any point
$x:\mathscr{A}\to\mathbb{R}$ we write $a_{x}$ for the image of $a$ in the
quotient $\mathscr{A}/I_{x}$, where as before $I_{x}$ is defined by (5.1). We
think of $a_{x}$ as an (abstract) germ of $a$ at $x$.

######  Construction 5.16 (Construction of
$({\mathcal{X}}_{\mathscr{A}},{\mathcal{O}}_{\mathscr{A}})=\operatorname{Spec}(\mathscr{A})$
and of
$\varepsilon_{\mathscr{A}}:\mathscr{A}\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$).

Let $\mathscr{A}$ be a $C^{\infty}$-ring, ${\mathcal{X}}_{\mathscr{A}}$ its
space of points with the Zariski topology. For an open set
$U\subset{\mathcal{X}}_{\mathscr{A}}$ consider the ideal $I_{U}:=\bigcap_{x\in
U}I_{x}$, where $I_{x}$ are the ideals defined by (5.1). Let
${\mathcal{P}}_{\mathscr{A}}(U)=\mathscr{A}/I_{U}$ if $U\not=\varnothing$ and
0 otherwise. Let
$\operatorname{\mathsf{pr}}_{U}:\mathscr{A}\to{\mathcal{P}}_{\mathscr{A}}(U)$
denote the canonical surjective map. For any open set $V\subset U$ there is an
evident “restriction” map $\mathscr{A}/I_{U}\to\mathscr{A}/I_{V}$. Hence
$U\mapsto{\mathcal{P}}_{\mathscr{A}}(U)$ is a presheaf of $C^{\infty}$-rings
on the space ${\mathcal{X}}_{\mathscr{A}}$. Note the the presheaf
${\mathcal{P}}_{\mathscr{A}}$ is separated: for any open set $U$, any open
cover $\\{U_{\alpha}\\}_{\alpha\in A}$ and any section
$s\in{\mathcal{P}}_{\mathscr{A}}(U)$ with $s|_{U_{\alpha}}=0$ we must have
$s=0$.

We define the structure sheaf ${\mathcal{O}}_{\mathscr{A}}$ to be the
sheafification of the presheaf ${\mathcal{P}}_{\mathscr{A}}$:

${\mathcal{O}}_{\mathscr{A}}:=\mathsf{sh}({\mathcal{P}}_{\mathscr{A}}).$

We thus obtain a $C^{\infty}$-ringed space
$({\mathcal{X}}_{\mathscr{A}},{\mathcal{O}}_{\mathscr{A}})$.

The presheaf ${\mathcal{P}}_{\mathscr{A}}$ has the following geometric
interpretation. For any open set $U\subset{\mathcal{X}}_{\mathscr{A}}$ we have
a product $\prod_{x\in U}A_{x}$ of local $C^{\infty}$-rings.111 If we think of
$C^{\infty}$-rings as sets with operations, then the product in question is
the product of underlying sets with the operations defined “coordinate-wise”.
Equivalently the category $C^{\infty}\mathsf{Ring}$ of $C^{\infty}$-rings has
products and $\prod_{x\in U}A_{x}$ is a product in $C^{\infty}\mathsf{Ring}$.
The first description is then a construction of this product. We can think of
elements of the product $\prod_{x\in U}A_{x}$ as functions
$s:U\to\bigcup_{x\in U}A_{x}$ with the property that $s(x)\in A_{x}$ for all
$x$. For every element $a\in\mathscr{A}$ we have an element
$\underline{a}\in\prod_{x\in U}A_{x}$, which is defined by

$\underline{a}(x):=a_{x},$

the image of $a$ in the ring $\mathscr{A}_{x}$ or, equivalently, the germ of
$a$ at $x$. The kernel of the map $a\mapsto\underline{a}$ is $I_{U}$. Hence
${\mathcal{P}}(U)=\mathscr{A}/I_{U}$ is (isomorphic to) a $C^{\infty}$-subring
of $\prod_{x\in U}A_{x}$.

The elements of ${\mathcal{O}}_{\mathscr{A}}(U)$ have an analogous
description: they are elements $s\in\prod_{x\in U}A_{x}$ so that for any $y\in
U$ there is an open neighborhood $V$ of $y$ in $U$ and $c\in\mathscr{A}$ so
that

$s(x)=c_{x}$

for all $x\in V$. Note that the canonical map
$\eta_{\mathscr{A}}:{\mathcal{P}}_{\mathscr{A}}\to\mathsf{sh}({\mathcal{P}}_{\mathscr{A}})={\mathcal{O}}_{\mathscr{A}}$
has the property that the maps
$\eta_{\mathscr{A},U}:{\mathcal{P}}_{\mathscr{A}}(U)\to{\mathcal{O}}_{\mathscr{A}}(U)$
are all injective. This is not surprising since ${\mathcal{P}}_{\mathscr{A}}$
is separated.

Finally, as promised, we define
$\varepsilon_{\mathscr{A}}:\mathscr{A}\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
to be the composite

$\varepsilon_{\mathscr{A}}:=\eta_{\mathscr{A},{\mathcal{X}}_{\mathscr{A}}}\circ\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}:\mathscr{A}\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}}).$

Construction 5.16 produces for every $C^{\infty}$-ring $\mathscr{A}$ a sheaf
${\mathcal{O}}_{\mathscr{A}}$ on the space of points
${\mathcal{X}}_{\mathscr{A}}$ of $\mathscr{A}$. We still need to show that the
stalks of the sheaf $\mathscr{O}_{\mathscr{A}}$ are local in order to
guarantee that $({\mathcal{X}},{\mathcal{O}}_{\mathscr{A}})$ is a local
$C^{\infty}$-ringed space. This follows from the next lemma.

######  Lemma 5.17.

Let $\mathscr{A}$ be a $C^{\infty}$-ring and let ${\mathcal{O}}_{\mathscr{A}}$
be a sheaf on the space ${\mathcal{X}}_{\mathscr{A}}$ produced by Construction
5.16. Then for any point $x\in{\mathcal{X}}_{\mathscr{A}}$ the stalk of
${\mathcal{O}}_{\mathscr{A}}$ at $x$ is the ring $\mathscr{A}_{x}$ of Lemma
5.14 hence is a local $C^{\infty}$-ring.

###### Proof.

Since ${\mathcal{O}}_{\mathscr{A}}$ is the sheafification of a presheaf of
${\mathcal{P}}$, it is enough to show that the stalk ${\mathcal{P}}_{x}$ of
${\mathcal{P}}$ at $x$ is $\mathscr{A}_{x}$. The stalk, by definition is the
colimit $\operatorname{colim}_{x\in U}{\mathcal{P}}(U)$ taken over all open
sets containing $x$. Since ${\mathcal{P}}(U)=\mathscr{A}/\bigcap_{y\in
U}I_{y}$ and since $x\in U$, there is an evident map
$\imath_{U}:\mathscr{A}/\bigcap_{y\in
U}I_{y}\to\mathscr{A}/I_{x}=\mathscr{A}_{x}$. It is easy to see that for any
pair of open sets $U$, $V$ with $V\subset U$ the diagram

$\textstyle{\mathscr{A}/I_{U}}$$\textstyle{\mathscr{A}/I_{V}}$$\textstyle{\mathscr{A}_{x}}$$\scriptstyle{\imath_{U}}$$\scriptstyle{\imath_{V}}$

commutes. We need to check that it is a colimit diagram, i.e., we need to
check its universal property. Note that the canonical projection
$\varphi_{U}:\mathscr{A}\to\mathscr{A}/I_{U}$ makes the diagram

$\textstyle{\mathscr{A}/I_{U}}$$\textstyle{\mathscr{A}}$$\textstyle{\mathscr{A}_{x}}$$\scriptstyle{\varphi_{U}}$$\scriptstyle{\imath_{U}}$$\scriptstyle{\pi_{x}}$

commute for all open neighborhoods $U$ of $x$ and that
$\textstyle{\mathscr{A}}$$\textstyle{\mathscr{A}/I_{U}}$$\textstyle{\mathscr{A}/I_{V}}$$\scriptstyle{\varphi_{U}}$$\scriptstyle{\varphi_{V}}$
commutes as well.

Suppose $\mathscr{B}$ is another $C^{\infty}$-ring and
$\\{f_{U}:\mathscr{A}/I_{U}\to\mathscr{B}\\}$ is another cocone. Then

$f_{U}\circ\varphi_{U}=f_{V}\circ\varphi_{V}$

for all $V,U$ with $V\subset U$. It follows that there is a well-defined map
$\tilde{f}:\mathscr{A}\to\mathscr{B}$ given by
$\tilde{f}=f_{U}\circ\varphi_{U}$ for any neighborhood $U$. It remains to show
that $\ker\tilde{f}\supseteq I_{x}$.

We know that $\ker\tilde{f}\subseteq\ker\varphi_{U}=I_{U}=\bigcap_{y\in
U}I_{y}$ for any neighborhood $U$. Suppose $a\in I_{x}$. Then there is
$d\in\mathscr{A}$ so that $da=0$ and $x(d)\not=0$. Consider
$U_{d}=\\{y\in{\mathcal{X}}_{\mathscr{A}}\mid y(d)\not=0\\}$; it is a
neighborhood of $x$. Then $a\in I_{y}$ for any $y\in U_{d}$ by definition of
$I_{y}$ and $U_{d}$. Hence $a\in\bigcap_{y\in
U_{d}}I_{y}=I_{U_{d}}\subset\ker\tilde{f}$ and we are done. ∎

To finish our proof of Proposition 5.2 it is enough to show that the map
$\varepsilon_{\mathscr{A}}:\mathscr{A}\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
has the desired universal property. This would finish our construction of the
functor $\operatorname{Spec}$, prove the the global section functor $\Gamma$
has a right adjoint and that $\varepsilon_{\mathscr{A}}$ is the
$\mathscr{A}$-component of the counit of adjunction.

######  Lemma 5.18.

Let $\mathscr{A}$ be a $C^{\infty}$-ring,
$\operatorname{Spec}(\mathscr{A})=({\mathcal{X}}_{\mathscr{A}},{\mathcal{O}}_{\mathscr{A}})$
the local $C^{\infty}$-ringed space and
$\varepsilon_{\mathscr{A}}:\mathscr{A}\to{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})$
the map of Construction 5.16. Then for any local $C^{\infty}$-ringed space
$(Y,{\mathcal{O}}_{Y})$ and any map
$\varphi:\mathscr{A}\to\Gamma(Y,{\mathcal{O}}_{Y})={\mathcal{O}}_{Y}(Y)$ of
$C^{\infty}$-rings, there is a unique map
$(f,f_{\\#}):(Y,{\mathcal{O}}_{Y})\to\operatorname{Spec}(\mathscr{A}))$ of
$C^{\infty}$-ringed spaces so that the diagram

(5.2)
$\textstyle{\mathscr{A}}$$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\scriptstyle{\varepsilon_{\mathscr{A}}\quad}$$\scriptstyle{\varphi}$$\scriptstyle{\Gamma(f,f_{\\#})}$

commutes.

Before we prove Lemma 5.18 we prove two elementary results.

######  Lemma 5.19.

Let $(Q,\mathscr{S})$ be a local $C^{\infty}$-ringed space and
$s\in\mathscr{S}(Q)$ a global section. Then the set

$U:=\\{q\in Q\mid\bar{q}(s_{q})\not=0\\}$

is open in $Q$. Here as before $s_{q}=\operatorname{\mathsf{germ}}_{q}(s)$ is
the germ of $s$ at $q$ and $\bar{q}:\mathscr{S}_{q}\to\mathbb{R}$ is the
unique $\mathbb{R}$-point.

###### Proof.

For any point $q\in Q$, $\ker\bar{q}\subset\mathscr{S}_{q}$ is the unique
maximal ideal. Hence $z\in\mathscr{S}_{q}$ is a unit if and only if
$\bar{q}(z)\not=0$. Now suppose $q\in U$. Then $s_{q}\in\mathscr{S}_{q}$ is a
unit. Hence there is an open neighborhood $W$ of $q$, and $t\in\mathscr{S}(W)$
so that $(s|_{W})t=1$. But then $s_{x}t_{x}=1$ for all $x\in W$. Consequently
$W\subset U$. ∎

######  Lemma 5.20.

Let $(Q,\mathscr{S})$ be a local $C^{\infty}$-ringed space, $U\subset Q$ an
open subset and $s\in\mathscr{S}(U)$. Suppose $\bar{q}(s_{q})=0$ for all $q\in
U$ (as before $s_{q}=\operatorname{\mathsf{germ}}_{q}(s)$ is the germ of $s$
at $q$ and $\bar{q}:\mathscr{S}_{q}\to\mathbb{R}$ is the unique
$\mathbb{R}$-point). Then $s=0$.

###### Proof.

If $\bar{q}(s_{q})=0$ then there is an open neighborhood $V_{q}$ of $q$ so
that $s|_{V_{q}}=0$. Since $\bar{q}(s_{q})=0$ for all $q\in U$ there is an
open cover $\\{V_{q}\\}_{q\in U}$ of $U$ so that $s|_{V_{q}}=0$. Since
$\mathscr{S}$ is a sheaf, $s=0$. ∎

###### Proof of Lemma 5.18.

We first assume that the map $(f,f_{\\#})$ with the desired property exists
and prove that it has to be unique. We start by proving the uniqueness of the
map of sheaves $f_{\\#}:{\mathcal{O}}_{\mathscr{A}}\to
f_{*}{\mathcal{O}}_{Y}$. We write
$f_{\\#,{\mathcal{X}}_{\mathscr{A}}}:{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})\to
f_{*}{\mathcal{O}}_{Y}({\mathcal{X}}_{\mathscr{A}})={\mathcal{O}}_{Y}(U)$ for
$\Gamma(f,f_{\\#})$ since it is shorter. By the universal property of
$\eta:{\mathcal{P}}_{\mathscr{A}}\to{\mathcal{O}}_{\mathscr{A}}$, the pullback
by $\eta$

$\eta^{*}:\operatorname{Hom}({\mathcal{O}}_{\mathscr{A}},f_{*}{\mathcal{O}}_{Y})\to\operatorname{Hom}({\mathcal{P}}_{\mathscr{A}},f_{*}{\mathcal{O}}_{Y}),\qquad\tau\mapsto\bar{\tau}:=\tau\circ\eta$

is a bijection. Since (5.2) commutes,
$f_{\\#,{\mathcal{X}}_{\mathscr{A}}}=\Gamma(f,f_{\\#})$,
$\bar{f}_{\\#}=f_{\\#}\circ\eta_{\mathscr{A}}$,
$f_{*}{\mathcal{O}}_{Y}({\mathcal{X}}_{\mathscr{A}})={\mathcal{O}}_{Y}(Y)$,
and
$\varepsilon_{\mathscr{A}}=\eta_{\mathscr{A},{\mathcal{X}}_{\mathscr{A}}}\circ\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}$
the left triangle in the diagram

$\textstyle{\mathscr{A}}$$\textstyle{{\mathcal{P}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\scriptstyle{\eta_{\mathscr{A},{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{\bar{f}_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{\quad
f_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{\varphi}$$\scriptstyle{\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}}$

commutes. Since $\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}$ is
surjective the map $\bar{f}_{\\#,{\mathcal{X}}_{\mathscr{A}}}$ is unique. More
generally, for any open subset $U\subset{\mathcal{X}}_{\mathscr{A}}$ the
diagram

$\textstyle{\mathscr{A}}$$\textstyle{{\mathcal{P}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\textstyle{{\mathcal{O}}_{Y}(f^{-1}(U))}$$\scriptstyle{|_{U}}$$\scriptstyle{|_{U}}$$\scriptstyle{\bar{f}_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{\bar{f}_{\\#,U}}$$\scriptstyle{\varphi}$$\scriptstyle{\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}}$

commutes. Now
$|_{U}\circ\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}=\operatorname{\mathsf{pr}}_{U}$,
which is surjective.
Since$\bar{f}_{\\#,U}\circ\operatorname{\mathsf{pr}}_{U}=|_{U}\circ\varphi$,
$\bar{f}_{\\#,U}$ is unique. We conclude that
$f_{\\#}:{\mathcal{O}}_{\mathscr{A}}\to f_{*}{\mathcal{O}}_{Y}$ is unique.

We next argue that $f:Y\to{\mathcal{X}}_{\mathscr{A}}$ is unique. We will need
more notation. For any (pre)sheaf $\mathscr{S}$ on a space $Q$ and any point
$q\in Q$ we have a canonical map from the space $\mathscr{S}(Q)$ of global
sections to the stalk $\mathscr{S}_{q}$ at $q$. We denote it by
$\operatorname{\mathsf{germ}}_{q}$. Then for any map
$g:\mathscr{S}\to\mathscr{S}^{\prime}$ of presheaves on $Q$ then the diagram

(5.3)
$\textstyle{\mathscr{S}(Q)}$$\textstyle{\mathscr{S}^{\prime}(Q)}$$\textstyle{\mathscr{S}_{q}}$$\textstyle{\mathscr{S}^{\prime}_{q}}$$\scriptstyle{g}$$\scriptstyle{\operatorname{\mathsf{germ}}_{q}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{q}}$$\scriptstyle{g_{q}}$

commutes, where $g_{q}$ is the induced map on stalks. If $h:Q\to Q^{\prime}$
is a continuous map and $\mathscr{S}$ is a sheaf on $Q$ then for every $q\in
Q$ there is a canonical map
$\varpi_{q}:(h_{*}\mathscr{S})_{q}\to\mathscr{S}_{q}$ and the diagram

(5.4)
$\textstyle{(h_{*}\mathscr{S})(Q^{\prime})}$$\textstyle{(h_{*}\mathscr{S})_{h(q)}}$$\textstyle{\mathscr{S}(Q)}$$\textstyle{\mathscr{S}_{q}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{h(q)}}$$\scriptstyle{=}$$\scriptstyle{\varpi_{q}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{q}}$

commutes. By Lemma 5.17 the germ ${\mathcal{O}}_{\mathscr{A},x}$ at
$x\in{\mathcal{X}}_{\mathscr{A}}$ is $A_{x}=\mathscr{A}/I_{x}$. It also
follows from the proof of Lemma 5.17 that the composite
$\mathscr{A}\xrightarrow{\varepsilon_{\mathscr{A}}}{\mathcal{O}}_{\mathscr{A}}\xrightarrow{\operatorname{\mathsf{germ}}_{x}}{\mathcal{O}}_{\mathscr{A},x}$
is the canonical projection $\pi_{x}:\mathscr{A}\to\mathscr{A}_{x}$. The
commutativity of (5.3) now tells us that

(5.5)
$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{\mathscr{A},x}}$$\textstyle{f_{*}{\mathcal{O}}_{Y}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{(f_{*}{\mathcal{O}}_{Y})_{x}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{x}}$$\scriptstyle{f_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{f_{\\#,x}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{x}}$

commutes. Commutativity of (5.4) and (5.5) tells us that

(5.6)
$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{\mathscr{A},f(y)}}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\textstyle{{\mathcal{O}}_{Y,y}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{f(y)}}$$\scriptstyle{f_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{\varpi_{\circ}(f_{\\#,f(y)})}$$\scriptstyle{\operatorname{\mathsf{germ}}_{y}}$

commutes. Since
$\textstyle{\mathscr{A}}$$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\scriptstyle{\varepsilon_{\mathscr{A}}\quad}$$\scriptstyle{\varphi}$$\scriptstyle{f_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$
commutes, the combined diagram

$\textstyle{\mathscr{A}}$$\textstyle{{\mathcal{O}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})}$$\textstyle{{\mathcal{O}}_{\mathscr{A},f(y)}}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\textstyle{{\mathcal{O}}_{Y,y}}$$\scriptstyle{\operatorname{\mathsf{germ}}_{f(y)}}$$\scriptstyle{f_{\\#,{\mathcal{X}}_{\mathscr{A}}}}$$\scriptstyle{\varpi_{\circ}(f_{\\#,f(y)})}$$\scriptstyle{\operatorname{\mathsf{germ}}_{y}}$$\scriptstyle{\varepsilon_{\mathscr{A}}\quad}$$\scriptstyle{\varphi}$

commutes. Since
$\operatorname{\mathsf{germ}}_{f(y)}\circ\varepsilon_{\mathscr{A}}=\pi_{f(y)}:\mathscr{A}\to\mathscr{A}_{f(y)}$
it follows that

$\textstyle{\mathscr{A}}$$\textstyle{\mathscr{A}_{f(y)}}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\textstyle{{\mathcal{O}}_{Y,y}}$$\scriptstyle{\pi_{f(y)}}$$\scriptstyle{\varphi}$$\scriptstyle{k}$$\scriptstyle{\operatorname{\mathsf{germ}}_{y}}$

commutes, where $k:=\varpi_{y}\circ(f_{\\#,f(y)})$. That is,

(5.7) $k\circ\pi_{f(y)}=\operatorname{\mathsf{germ}}_{y}\circ\varphi.$

We now argue that

(5.8) $f(y)=\bar{y}\circ\operatorname{\mathsf{germ}}_{y}\circ\varphi,$

where $\bar{y}:{\mathcal{O}}_{Y,y}\to\mathbb{R}$ is the unique
$\mathbb{R}$-point of the local ring ${\mathcal{O}}_{Y,y}$. Since $y\in Y$ is
arbitrary this would prove that $f:Y\to{\mathcal{X}}_{\mathscr{A}}$ is
uniquely determined by $\varphi$. The key here is that since
$\mathscr{A}_{f(y)}$ is a local ring, the $\mathbb{R}$-point $\bar{y}\circ k$
has to be the point $\bar{f}(y)$.

Recall that for any $\mathbb{R}$ point $x:\mathscr{A}\to\mathbb{R}$,
$x=\bar{x}\circ\pi_{x}$ where $\bar{x}:\mathscr{A}_{x}\to\mathbb{R}$ is the
unique $\mathbb{R}$ point of $\mathscr{A}_{x}$. Hence

$f(y)=\bar{f}(y)\circ\pi_{f(y)}=(\bar{y}\circ
k)\circ\pi_{f(y)}\xrightarrow{\eqref{eq:5.37}}{=}\bar{y}\circ\operatorname{\mathsf{germ}}_{y}\circ\varphi.$

This finishes our proof of uniqueness of
$(f,f_{\\#}):(Y,{\mathcal{O}}_{Y})\to({\mathcal{X}}_{\mathscr{A}},{\mathcal{O}}_{\mathscr{A}})$.

It remains to prove existence of $(f,f_{\\#})$. We define a function
$f:Y\to{\mathcal{X}}_{\mathscr{A}}$ by (5.8). We need to check that $f$ is
continuous. Since the Zariski topology on ${\mathcal{X}}_{\mathscr{A}}$ is
generated by the sets $U_{a}:=\\{x\in{\mathcal{X}}_{\mathscr{A}}\mid
x(a)\not=0\\}$, $a\in\mathscr{A}$, it is enough to prove that $f^{-1}(U_{a})$
is open for any $a\in\mathscr{A}$. By definition of $f$

$f^{-1}(U_{a})=\\{y\in Y\mid
0\not=(\bar{y}\circ\operatorname{\mathsf{germ}}_{y}\circ\varphi)(a)\\}=\\{y\in
Y\mid\bar{y}(\varphi(a)_{y})\not=0\\}.$

The last set is open by Lemma 5.19.

Finally we construct $f_{\\#}:{\mathcal{O}}_{\mathscr{A}}\to
f_{*}{\mathcal{O}}_{Y}$ with the property that
$f_{\\#,{\mathcal{X}}_{\mathscr{A}}}\circ\varepsilon_{A}=\varphi$. Since
${\mathcal{O}}_{\mathscr{A}}$ is a sheafification of
${\mathcal{P}}_{\mathscr{A}}$ it is enough to construct a map of presheaves
$\bar{f}_{\\#}:{\mathcal{P}}_{\mathscr{A}}\to\to f_{*}{\mathcal{O}}_{Y}$ so
that
$\bar{f}_{\\#,{\mathcal{X}}_{\mathscr{A}}}\circ\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}=\varphi$,
where as before
$\operatorname{\mathsf{pr}}_{{\mathcal{X}}_{\mathscr{A}}}:\mathscr{A}\to{\mathcal{P}}_{\mathscr{A}}({\mathcal{X}}_{\mathscr{A}})=\mathscr{A}/\bigcap_{x\in{\mathcal{X}}_{\mathscr{A}}}I_{x}$
is the surjective map. Thus we need to construct a collection of compatible
maps
$\bar{f}_{\\#,U}:{\mathcal{P}}_{\mathscr{A}}(U)\to{\mathcal{O}}_{Y}(f^{-1}(U)$,
one for each open set $U\subset{\mathcal{X}}_{\mathscr{A}}$. As we have seen
in the proof of the uniqueness of $\bar{f}_{\\#}$ these maps have to make the
diagrams

$\textstyle{\mathscr{A}}$$\textstyle{{\mathcal{P}}_{\mathscr{A}}(U)}$$\textstyle{{\mathcal{O}}_{Y}(Y)}$$\textstyle{{\mathcal{O}}(f^{-1}(U))}$$\scriptstyle{\operatorname{\mathsf{pr}}_{U}}$$\scriptstyle{\varphi}$$\scriptstyle{\bar{f}_{\\#,U}}$$\scriptstyle{|_{U}}$

commute. Consequently to construct $\bar{f}_{\\#,U}$ it is enough to show
$\ker(|_{U}\circ\varphi)\subseteq\ker\operatorname{\mathsf{pr}}_{U}=\bigcap_{x\in
U}I_{x}$. Suppose $a\in\bigcap_{x\in U}I_{x}$. For any $y\in f^{-1}(U)$,
$f(y)\in U$, hence $a\in I_{f(y)}$. Therefore there is $d\in\mathscr{A}$ so
that $f(y)(d)\not=0$ and $da=0$. Consequently

$0=(\bar{y}\circ\operatorname{\mathsf{germ}}_{y}\varphi)(ad)=\bar{y}(\varphi(a)_{y})f(d).$

Since $f(d)\not=0$, $\bar{y}(\varphi(a)_{y})=0$. Since this is true for all
$y\in f^{-1}(U)$, Lemma 5.20 now implies that $\varphi(a)|_{U}=0$, i.e.,
$a\in\ker(|_{U}\circ\varphi)$. It remains to check that the collection of maps
$\\{\bar{f}_{\\#,U}\\}_{U\subseteq X}$ are compatible with restrictions. The
latter follows from the fact that the maps
$\operatorname{\mathsf{pr}}_{U}:\mathscr{A}\to{\mathcal{P}}_{\mathscr{A}}(U)$
are surjective and that for any $a\in\mathscr{A}$, and $V\subset U$,
$\operatorname{\mathsf{pr}}_{U}(a)|_{V}=\operatorname{\mathsf{pr}}_{V}(a)$.
This finishes the construction of the map $\bar{f}_{\\#}$ and thereby the map
$(f,f_{\\#})$ of locally ringed spaces with the desired property. ∎

We end our discussion of the functor $\operatorname{Spec}$ and affine
$C^{\infty}$-schemes by singling out a class of local $C^{\infty}$-ringed
spaces that are both differential spaces and affine schemes. Recall that in
general the two classes of spaces are distinct — see Remark 5.8. However
concrete differential spaces (Definition 3.24) are exactly the same as affine
schemes defined by finitely generated and point determined $C^{\infty}$-rings.

######  Theorem 5.21.

Let $\mathscr{A}$ be a finitely generated point determined $C^{\infty}$-ring.
Then its spectrum
$\operatorname{Spec}(\mathscr{A})=({\mathcal{X}}_{\mathscr{A}},{\mathcal{O}}_{\mathscr{A}})$
is isomorphic, as a local $C^{\infty}$-ringed spaces, to a closed subset $Z$
of some $\mathbb{R}^{n}$ with the induced differential space structure
(Definition 3.16).

###### Proof.

Since the $C^{\infty}$-ring $\mathscr{A}$ is finitely generated we may assume
that $\mathscr{A}=C^{\infty}(\mathbb{R}^{n})/J$ for some ideal $J\subseteq
C^{\infty}(\mathbb{R}^{n})$. By Lemma 5.6 the set
${\mathcal{X}}_{\mathscr{A}}$ of $\mathbb{R}$-points of $\mathscr{A}$ is in
bijection with the set

$Z(J)=\\{x\in\mathbb{R}^{n}\mid f(x)=0\textrm{ for all }f\in J\\};$

the bijection $Z\to{\mathcal{X}}_{\mathscr{A}}$ is given by
$x\mapsto\operatorname{\mathsf{ev}}_{x}$. We will now suppress the
identification and write ${\mathcal{X}}_{\mathscr{A}}=Z$.

Let $K=\\{f\in C^{\infty}(\mathbb{R}^{n})\mid f|_{Z}=0$. Then
$C^{\infty}_{Z}(Z)=C^{\infty}(\mathbb{R}^{n})/K$. We next argue that $J=K$
hence $\mathscr{A}=C^{\infty}_{Z}(Z)$. Since for any $x\in Z$ and any $h\in J$
we have $h(x)=0$, $J\subset K$. If $J\not=K$ there is $g\in K\smallsetminus
J$. Then $0\not=\pi(g)\in\mathscr{A}$. Since $\mathscr{A}$ is point determined
there is an $\mathbb{R}$-point $p:\mathscr{A}\to\mathbb{R}$ with $p(g)\not=0$.
Under the identification ${\mathcal{X}}_{\mathscr{A}}=Z$ the last condition
translates into: there is $x\in Z$ so that $g(x)\not=0$, which contradicts the
definition of $Z$. Hence $J=K$ and $\mathscr{A}=C^{\infty}_{Z}(Z)$.

We need to show that the Zariski topology ${\mathcal{T}}_{\mathscr{A}}$ on
${\mathcal{X}}_{\mathscr{A}}=Z$ (cf. Lemma 5.9) agrees with the subspace
topology on $Z\subseteq\mathbb{R}^{n}$. Let $V\subset Z$ be a set open in the
subspace topology. Then there is an open set $\tilde{V}\subset\mathbb{R}^{n}$
with $\tilde{V}\cap Z=V$. Since $\mathbb{R}^{n}\smallsetminus\tilde{V}$ is
closed, there is, by a theorem of Whitney, a function $f\in
C^{\infty}(\mathbb{R}^{n})$ so that
$f^{-1}(0)=\mathbb{R}^{n}\smallsetminus\tilde{V}$. Therefore

$V=\\{x\in
Z\mid(f|_{Z})(x)\not=0\\}=\\{\operatorname{\mathsf{ev}}_{x}:\mathscr{A}\to\mathbb{R}\mid\operatorname{\mathsf{ev}}_{x}(f|_{Z})\not=0\\}=U_{f|_{Z}},$

where $U_{f|_{Z}}$ is a basic open set in ${\mathcal{T}}_{\mathscr{A}}$
defined by $f|_{Z}\in\mathscr{A}$. It follows that the Zariski and subspace
topologies on $Z$ agree.

It remains to check that the structure sheaf ${\mathcal{O}}_{\mathscr{A}}$ of
$\operatorname{Spec}(\mathscr{A})$ is (isomorphic to) the structure sheaf
$C^{\infty}_{Z}$ of the differential space $Z\subseteq\mathbb{R}^{n}$. Recall
that ${\mathcal{O}}_{\mathscr{A}}$ is a sheafification of the presheaf
${\mathcal{P}}={\mathcal{P}}_{\mathscr{A}}$ defined by

${\mathcal{P}}_{\mathscr{A}}(U)=\mathscr{A}/\bigcap_{x\in U}I_{x},$

for all open $U\subset{\mathcal{X}}_{\mathscr{A}}$. Here as before the ideals
$I_{x}$ are defined by (5.1):

$I_{x}=\\{a\in\mathscr{A}\mid\textrm{ there is }d\in\mathscr{A}\textrm{ so
that }x(d)\not=0\textrm{ and }ad=0\\}.$

The sheaf $C^{\infty}_{Z}$ is a sheafification of the presheaf
${\mathcal{P}}_{Z}$ which is given by

${\mathcal{P}}_{Z}(U)=C^{\infty}_{Z}(Z)|_{U}=\left(C^{\infty}(\mathbb{R}^{n})|_{Z}\right)|_{U}=C^{\infty}(\mathbb{R}^{n})|_{Z}/K_{U},$

where $K_{U}=\\{f\in C^{\infty}(\mathbb{R}^{n})|_{Z}\mid f|_{U}=0\\}$. Thus in
order to prove that the two sheaves ${\mathcal{O}}_{\mathscr{A}}$ and
$C^{\infty}_{Z}$ are isomorphic, it is enough to prove that the presheaves
${\mathcal{P}}_{\mathscr{A}}$ and ${\mathcal{P}}_{Z}$ are isomorphic. For the
latter it is enough to show that

$K_{U}=\bigcap_{x\in U}I_{x}$

for all open sets $U\subseteq Z$.

Suppose $k\in\bigcap_{x\in U}I_{x}$. Then for every point $x\in U$ there is
$g_{x}\in C^{\infty}(\mathbb{R}^{n})|_{Z}=\mathscr{A}$ so that $kg=0$ and
$\operatorname{\mathsf{ev}}_{x}(g_{x})\not=0$. And then
$0=\operatorname{\mathsf{ev}}_{x}(kg)=\operatorname{\mathsf{ev}}_{x}(k)\operatorname{\mathsf{ev}}_{x}(g)$.
It follows that $\operatorname{\mathsf{ev}}_{x}(k)=0$ for all $x\in U$, i.e.,
$k\in K_{U}$.

Conversely suppose $k\in K_{U}$. Then for any $x\in U$ there is a function
$\tau_{x}\in C^{\infty}_{Z}(Z)$ so that $\tau_{x}(x)=1$ and
$\operatorname{supp}\tau_{x}\subseteq U$ (for example one can choose an open
set $\tilde{U}\subset\mathbb{R}^{n}$ with $\tilde{U}\cap Z=U$, choose
$\tilde{\tau}_{x}\in C^{\infty}(\mathbb{R}^{n})$ with $\tilde{\tau}_{x}(x)=1$
and $\operatorname{supp}\tilde{\tau}\subseteq\tilde{U}$ and set
$\tau_{x}:=\tilde{\tau}_{x}|_{Z}$). Then $k\tau_{x}=0$ and
$\operatorname{\mathsf{ev}}_{x}(\tau_{x})=1$. Hence $k\in I_{x}$. Consequently
$K_{U}\subseteq\bigcap_{x\in U}I_{x}$. ∎

##  6\. Kähler differentials of $C^{\infty}$-rings

In this section we recall the construction of the $C^{\infty}$-ring analogue
of Kähler differentials of commutative algebra. We start by reviewing the
notion of a module of a $C^{\infty}$-ring.

######  Definition 6.1.

A module over a $C^{\infty}$-ring $\mathscr{A}$ is a module over the
$\mathbb{R}$-algebra underlying $\mathscr{A}$.

######  Example 6.2.

Let $E\to M$ be a (smooth) vector bundle over a manifold $M$. Then the set of
sections $\Gamma(E)$ is a module over the $C^{\infty}$-ring $C^{\infty}(M)$.

######  Example 6.3.

Let $(M,C^{\infty}(M))$ be a differential space and $p\in M$ a point. The
evaluation at $p$

$ev_{p}:C^{\infty}(M)\to\mathbb{R},\qquad ev_{p}(f)=f(p)$

makes $\mathbb{R}$ into a $C^{\infty}(M)$ module. Note that the
$C^{\infty}(M)$-module structure on $\mathbb{R}$ depends on the choice of a
point $p$.

######  Definition 6.4.

Let $\mathscr{A}$ be a $C^{\infty}$-ring and $\mathscr{M}$ an
$\mathscr{A}$-module. A $C^{\infty}$ derivation of $\mathscr{A}$ with values
in the module $\mathscr{M}$ is a map $X:\mathscr{A}\to\mathscr{M}$ so that for
any $n>0$, any $f\in C^{\infty}(\mathbb{R}^{n})$ and any
$a_{1},\ldots,a_{n}\in\mathscr{A}$

(6.1)
$X(f_{\mathscr{A}}(a_{1},\ldots,a_{n}))=\sum_{i=1}^{n}(\partial_{i}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\cdot
X(a_{i}).$

######  Notation 6.5.

Given a $C^{\infty}$-ring $\mathscr{A}$ and an $\mathscr{A}$-module
$\mathscr{M}$ we denote the set of all $C^{\infty}$-derivations with values in
$\mathscr{M}$ by $\operatorname{Der}(\mathscr{A},\mathscr{M})$.

The set of derivations $\operatorname{Der}(\mathscr{A},\mathscr{M})$ is an
$\mathscr{A}$-module.

######  Example 6.6.

Let $M$ be a smooth manifold. A $C^{\infty}$ derivations $X:C^{\infty}(M)\to
C^{\infty}(M)$ of the $C^{\infty}$-ring of smooth functions $C^{\infty}(M)$
with values in $C^{\infty}(M)$ is an ordinary vector field.

The exterior derivative $C^{\infty}(M)\to\bm{\Upomega}^{1}(M)$ is a
$C^{\infty}$ derivation of $C^{\infty}(M)$ with values in the module
$\bm{\Upomega}^{1}(M)$ of ordinary 1-forms.

######  Example 6.7.

Let $(M,C^{\infty}(M))$ be a differential space and $p\in M$ a point,
$ev_{p}:C^{\infty}(M)\to\mathbb{R}$ the evaluation map that makes $\mathbb{R}$
into a $C^{\infty}(M)$-module. Then a derivation
$v:C^{\infty}(M)\to\mathbb{R}$ of $C^{\infty}(M)$ with values in the module
$\mathbb{R}$ is by definition a Zariski tangent vector at the point $p$.

There are several reasons why Definition 6.1 is reasonable. We list three:
(i) If $M$ is a manifold, $E\to M$ a vector bundle over $M$ then the vector
space $\Gamma(E)$ of sections of the bundle is a module over the
$\mathbb{R}$-algebra $C^{\infty}(M)$. It is reasonable to consider it as a
module over the $C^{\infty}$-ring $C^{\infty}(M)$. In particular if $E$ is the
tangent bundle $TM$ then the space of sections $\Gamma(TM)$, which is
isomorphic to the space of derivations
$\operatorname{Der}(C^{\infty}(M),C^{\infty}(M))$, is a module over
$C^{\infty}(M)$. And if $E$ is the cotangent bundle then the space of sections
$\Gamma(E)$ is the space of 1-forms $\Omega^{1}(M)$ on $M$, which is a module
over $C^{\infty}(M)$.

(ii) Given a $C^{\infty}$-ring $\mathscr{A}$ and an $\mathscr{A}$-module
$\mathscr{M}$ the product $\mathscr{A}\times\mathscr{M}$ has the structure of
a $C^{\infty}$-ring. Namely, for any $n>0$, any $f\in
C^{\infty}(\mathbb{R}^{n})$ and any
$(a_{1},m_{1}),\ldots(a_{n},m_{n})\in\mathscr{A}\times\mathscr{M}$

$f_{\mathscr{A}\times\mathscr{M}}((a_{1},m_{1}),\ldots,(a_{n},m_{n})):=(f_{\mathscr{A}}(a_{1},\ldots,a_{n}),\sum_{i=1}^{n}(\partial
f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\cdot m_{i}).$

Note that the multiplication in the $\mathbb{R}$-algebra underlying
$\mathscr{A}\times\mathscr{M}$ comes from the function $f(x,y)=xy\in
C^{\infty}(\mathbb{R}^{2})$. Consequently for any
$(a_{1},m_{1}),(a_{2},m_{2})\in\mathscr{A}\times\mathscr{M}$

$(a_{1},m_{1})\cdot(a_{2},m_{2})=(a_{1}a_{2},a_{2}m_{1}+a_{1}m_{2}).$

In particular

$(0,m_{1})\cdot(0,m_{2})=(0\cdot 0,0m_{1}+0m_{2})=(0,0)$

Therefore, just as in the case of commutative rings and commutative
$\mathbb{R}$-algebras, modules over $C^{\infty}$-rings correspond to square
zero extensions.

(iii) A module $\mathscr{M}$ over a $C^{\infty}$-ring $\mathscr{A}$ is a Beck
module [Be, Ba]: just as in the case of commutative rings given a
$C^{\infty}$-ring $\mathscr{A}$ and an $\mathscr{A}$-module $\mathscr{M}$ the
projection $p:\mathscr{A}\times\mathscr{M}\to\mathscr{A}$ on the first factor
is an abelian group object in the slice category
$C^{\infty}\mathsf{Ring}/\mathscr{A}$ of $C^{\infty}$-rings over
$\mathscr{A}$.

######  Remark 6.8.

Let $\mathscr{A}$ be a $C^{\infty}$-ring. The collection of all
$\mathscr{A}$-modules forms a small category. We denote the set of all
morphism from an $\mathscr{A}$-module $\mathscr{N}$ to an $\mathscr{A}$-module
$\mathscr{M}$ by
$\operatorname{Hom}_{\mathscr{A}-mod}(\mathscr{N},\mathscr{M})$.

######  Definition 6.9.

A module of Kähler differentials over a $C^{\infty}$-ring $\mathscr{A}$ is an
$\mathscr{A}$-module $\Omega^{1}_{\mathscr{A}}$ together with a
$C^{\infty}$-ring derivation
$d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ so that for any
$\mathscr{A}$-module $\mathscr{N}$ and any derivation
$X:\mathscr{A}\to\mathscr{N}$ there exists a unique map of modules
$\varphi_{X}:\Omega^{1}_{A}\to\mathscr{N}$ with

$\varphi_{X}\circ d=X.$

######  Remark 6.10.

The universal property of the derivation
$d:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ says that for any
$\mathscr{A}$-module $\mathscr{M}$, the map of modules

$d^{*}:\operatorname{Hom}_{\mathscr{A}{-}mod}(\Omega^{1}_{\mathscr{A}},\mathscr{M})\to\operatorname{Der}(\mathscr{A},\mathscr{M}),\quad
d^{*}(\varphi):=\varphi\circ d$

is an isomorphism (of $\mathscr{A}$-modules).

We’ll prove existence of Kähler differentials in Theorem 6.14 below. The
result is known — it was proved by Dubuc and Kock [DK] over 30 years ago. The
proof we will present is essentially the one in [J]. Our excuses for providing
the proof are that we (1) are trying to keep the paper self-contained and (2)
we will need the the details of the proof for the construction of the
$C^{\infty}$-algebraic de Rham complex of a $C^{\infty}$-ring. The uniqueness
of Kähler differentials follows easily from their universal properties — see
Remark 6.13. Before proving Theorem 6.14 we give two examples of modules of
Kähler differentials.

######  Example 6.11.

The module $\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})$ of ordinary differential
1-forms together with the exterior derivative
$d:C^{\infty}(\mathbb{R}^{n})\to\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})$ is a
module of Kähler differentials over the $C^{\infty}$-ring
$C^{\infty}(\mathbb{R}^{n})$:
$\Omega^{1}_{C^{\infty}(\mathbb{R}^{n})}=\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})$.

###### Proof.

We need to show that for any $C^{\infty}(\mathbb{R}^{n})$-module $\mathscr{N}$
and for any derivation $X:C^{\infty}(\mathbb{R}^{n})\to\mathscr{N}$ there is a
unique map of modules
$\varphi_{X}:\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})\to\mathscr{N}$ so that

$\varphi_{X}(dg)=X(g)$

for all $g\in C^{\infty}(\mathbb{R}^{n})$. Since
$\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})$ is freely generated, as a module, by
the differentials $dx_{1},\ldots,dx_{n}$ (where
$x_{1},\ldots,x_{n}:\mathbb{R}^{n}\to\mathbb{R}$ are the coordinate
functions), the map $\varphi_{X}$ given by

$\varphi_{X}(\sum a_{i}dx_{i}):=\sum a_{i}X(x_{i})$

is a well-defined map of $C^{\infty}(\mathbb{R}^{n})$ modules. And it is the
unique map from $\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})$ to $\mathscr{N}$
satisfying $\varphi_{X}(dx_{i})=X(x_{i})$.

It is not hard to check that the exterior derivative
$d:C^{\infty}(\mathbb{R}^{n})\to\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})$ is a
$C^{\infty}$-ring derivation: any $k>0$, any $h\in C^{\infty}(\mathbb{R}^{k})$
and any $f_{1},\ldots,f_{k}\in C^{\infty}(\mathbb{R}^{n})$

$\displaystyle
d\left(h_{C^{\infty}(\mathbb{R}^{n})}(f_{1},\ldots,f_{k})\right)=\sum\partial_{i}(h(f_{1},\ldots,f_{k}))\,dx_{i}=\sum_{i}\left(\sum_{j}(\partial_{j}h)(f_{1},\ldots,f_{k})\cdot\partial_{i}f_{j}\right)\,dx_{i}$
$\displaystyle=\sum_{j}\left((\partial_{j}h)(f_{1},\ldots,f_{k})\cdot\sum_{i}\partial_{i}f_{j}\,dx_{i}\right)=\sum_{j}(\partial_{j}h)_{C^{\infty}(\mathbb{R}^{n})}(f_{1},\ldots,f_{k})\cdot
df_{j}.$

It remains to check that $\varphi_{X}(dg)=X(g)$ for all $g\in
C^{\infty}(\mathbb{R}^{n})$. The $C^{\infty}$-ring
$C^{\infty}(\mathbb{R}^{n})$ is a free ring on the generators
$x_{1},\ldots,x_{n}$ (see Definition 2.28 and Lemma 2.30). In particular
$g_{C^{\infty}(\mathbb{R}^{n})}(x_{1},\ldots,x_{n})=g$ for all $g\in
C^{\infty}(\mathbb{R}^{n})$. Since $X$ is a derivation

$\begin{split}X(g)=X(g_{C^{\infty}(\mathbb{R}^{n})}(x_{1},\ldots,x_{n}))&=\sum(\partial_{i}g)_{C^{\infty}(\mathbb{R}^{n})}(x_{1},\ldots,x_{n})\,X(x_{i})\\\
&=\sum\partial_{i}g\,X(x_{i})=\varphi_{X}(\sum\partial_{i}g\,dx_{i})=\varphi_{X}(dg).\end{split}$

Thus there is a unique map of $C^{\infty}(\mathbb{R}^{n})$-modules
$\varphi_{X}:\bm{\Upomega}_{dR}^{1}(\mathbb{R}^{n})\to\mathscr{N}$ which
satisfies $\varphi_{X}\circ d=X$. It is given by $\varphi_{X}(\sum
a_{i}dx_{i})=\sum a_{i}X(x_{i})$ and ∎

######  Example 6.12.

Let $U\subset\mathbb{R}^{n}$ be an open set. The module
$\bm{\Upomega}_{dR}^{1}(U)$ of 1-forms together with the exterior derivative
$d:C^{\infty}(U)\to\bm{\Upomega}_{dR}^{1}(U)$ is a module of Kähler
differentials over the $C^{\infty}$-ring $C^{\infty}(U)$:
$\Omega^{1}_{C^{\infty}(U)}=\bm{\Upomega}_{dR}^{1}(U)$.

###### Proof.

As in the case of $\mathbb{R}^{n}$, the differentials of the standard
coordinate functions $x_{1}|_{U},\ldots,x_{n}|_{U}:U\to\mathbb{R}^{n}$
generate freely the $C^{\infty}(U)$-module $\bm{\Upomega}_{dR}^{1}(U)$ of one
forms. Therefore for any $C^{\infty}(U)$-module $\mathscr{N}$ and any
derivation $X:C^{\infty}(U)\to\mathscr{N}$ there is a $C^{\infty}(U)$-module
map $\varphi_{X}:\bm{\Upomega}_{dR}^{1}(U)\to\mathscr{N}$ defined uniquely by

$\varphi_{X}(\sum a_{i}dx_{i}|_{U})=\sum a_{i}\,X(x_{i}|_{U}).$

Using the fact that

$df=\sum(\partial_{i}f)\,dx_{i}|_{U}$

for all $f\in C^{\infty}(U)$ it is not hard to show that the exterior
derivative $d:C^{\infty}(U)\to\bm{\Upomega}_{dR}^{1}(U)$ is a
$C^{\infty}$-derivation. It remains to check that

$\varphi_{X}\circ d=X.$

Since for any $g\in C^{\infty}(\mathbb{R}^{n})$

$g|_{U}=g\circ(x_{1}|_{U},\ldots,x_{n}|_{U})=g_{C^{\infty}(U)}(x_{1}|_{U},\ldots,x_{n}|_{U})$
$d(g|_{U})=d\left(g_{C^{\infty}(U)}(x_{1}|_{U},\ldots,x_{n}|_{U})\right)=\sum(\partial_{i}g)_{C^{\infty}(U)}(x_{1}|_{U},\ldots,x_{n}|_{U})dx_{i}|_{U}=\sum_{i}\partial_{i}g|_{U}dx_{i}|_{U}.$

Therefore

$\varphi_{X}(d(g|_{U}))=\sum\partial_{i}g|_{U}\,X(x_{i}|_{U})=X(g|_{U}),$

where in the last equality we used the fact that $X$ is a
$C^{\infty}$-derivation. However not all functions in $C^{\infty}(U)$ are
restrictions of smooth functions on $\mathbb{R}^{n}$. For arbitrary functions
in $C^{\infty}(U)$ we argue as follows. By the Localization Theorem of Muñoz
Díaz and J. Ortega [MO] (see also [NGSS, p. 28]) given a function $f\in
C^{\infty}(U)$ there exist $g,h\in C^{\infty}(\mathbb{R}^{n})$ so that

$h|_{U}f=g|_{U}$

and $h|_{U}$ is invertible in $C^{\infty}(U)$. Then

$df=d\frac{g|_{U}}{h|_{U}}=\frac{1}{h|_{U}}d(g|_{U})-\frac{g|_{U}}{(h|_{U})^{2}}d(h|_{U}).$

We now write $g$ and $h$ for $g|_{U}$ and $h|_{U}$ to reduce the clutter. We
have

$\varphi_{X}(df)=\varphi_{X}(\frac{1}{h}dg-\frac{g}{h^{2}}dh)=\frac{1}{h}X(g)-\frac{g}{h^{2}}X(h)\overset{\textrm{by
Lemma~{}\ref{lem:B.2}}}{=}X(\frac{g}{h})=X(f).$

And we are done. ∎

######  Remark 6.13.

It follows from the definition of the module $\Omega^{1}_{\mathscr{A}}$ of
Kähler differentials of a $C^{\infty}$-ring $\mathscr{A}$ that the module has
to be unique in the following sense: if $\mathscr{N}$ is another $\mathscr{A}$
module and $d:\mathscr{A}\to\mathscr{N}$ is another derivation with the same
universal property as $d_{\mathscr{A}}:\mathscr{A}\to\Omega_{\mathscr{A}}^{1}$
then there is a unique isomorphism
$\psi:\mathscr{N}\to\Omega_{\mathscr{A}}^{1}$ of $\mathscr{A}$-modules so that

$\psi\circ d=d_{\mathscr{A}}.$

Following a common practice we may refer to any $\mathscr{A}$-module
$\mathscr{N}$ with a derivation $\partial:\mathscr{A}\to\mathscr{N}$ that has
the appropriate universal property as “the” module of Kähler differentials of