the $C^{\infty}$-ring $\mathscr{A}$.

We now prove existence of $C^{\infty}$-Kähler differentials.

######  Theorem 6.14 (Dubuc and Kock[DK]).

For any $C^{\infty}$-ring $\mathscr{A}$ there exists a module of Kähler
differentials $\Omega_{\mathscr{A}}^{1}$ together with the structure map
$d_{\mathscr{A}}:\mathscr{A}\to\Omega_{\mathscr{A}}^{1}$.

Before we proceed with a proof of this theorem, we note that for any
$C^{\infty}$-ring $\mathscr{A}$ and any set $S$ there exists a free
$\mathscr{A}$-module $F[S]$ on the set $S$. The elements of $F[S]$ are finite
formal linear combination $a_{1}s_{1}+\cdots+a_{n}s_{n}$ for $n>0$,
$a_{1},\ldots,a_{n}\in\mathscr{A}$, $s_{1},\ldots,s_{n}\in S$.

###### Proof of Theorem 6.14.

In commutative algebra there are two equivalent constructions of Kähler
differentials of a commutative ring. Dubuc and Kock adapted one of them to the
$C^{\infty}$-ring setting (strictly speaking they adapted it to the setting of
Fermat theories, which include $C^{\infty}$-rings). Joyce [J] adapts the
other. It will be convenient for us to follow Joyce’s approach.

Consider the free $\mathscr{A}$-module $F[\hat{d}\mathscr{A}]$ where
$\hat{d}\mathscr{A}$ is a set of formal symbols $\hat{d}b$, one symbol for
each $b\in\mathscr{A}$. Thus

$F[\hat{d}\mathscr{A}]=\\{a_{1}\hat{d}b_{1}+\cdots+a_{n}\hat{d}b_{n}\mid
n>0,a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}\in\mathscr{A}\\}.$

Let $I$ denote the submodule of $F[\hat{d}\mathscr{A}]$ generated by the set

$\mathscr{R}:=\\{\hat{d}(f_{\mathscr{A}}(a_{1},\ldots,a_{n}))-\sum(\partial_{i}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\,\hat{d}a_{i}\mid
n>0,f\in C^{\infty}(\mathbb{R}^{n}),a_{1},\ldots,a_{n}\in\mathscr{A}\\},$

that is,

$I=\langle\mathscr{R}\rangle.$

Let

$\Omega^{1}_{\mathscr{A}}:=F[\hat{d}\mathscr{A}]/I.$

Then $\Omega^{1}_{\mathscr{A}}$ is an $\mathscr{A}$-module. Define
$d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ by

$d_{\mathscr{A}}(a)=\hat{d}a+I.$

It is easy to check that $d_{\mathscr{A}}$ is a $C^{\infty}$-derivation. Note
also that the module $\Omega^{1}_{\mathscr{A}}$ is generated by the set
$\\{d_{\mathscr{A}}(a)\\}_{a\in\mathscr{A}}$.

It remain to check the universal property of $d_{\mathscr{A}}$. Let
$\mathscr{N}$ be an $\mathscr{A}$-module and $X:\mathscr{A}\to\mathscr{N}$ a
$C^{\infty}$-derivation. Since $F[\hat{d}\mathscr{A}]$ is a free
$\mathscr{A}$-module on the set $\\{\hat{d}b\\}_{b\in\mathscr{A}}$, there is a
unique map of $\mathscr{A}$-modules
$\hat{\varphi}:F[\hat{d}\mathscr{A}]\to\mathscr{N}$ with

$\hat{\varphi}(\hat{d}b)=X(b).$

for all $b\in\mathscr{A}$. Since $X$ is a derivation,
$\hat{\varphi}(\alpha)=0$ for all $\alpha$ in the submodule $I$. Consequently
$\hat{\varphi}$ descends to a map of $\mathscr{A}$-modules

$\varphi_{X}:\Omega^{1}_{\mathscr{A}}\to\mathscr{N}.$

Then by construction of $\varphi_{X}$ and of $\Omega^{1}_{\mathscr{A}}$

$\varphi_{X}(d_{\mathscr{A}}b)=X(b)$

for all $b\in\mathscr{A}$. Since the set
$\\{d_{\mathscr{A}}b\\}_{b\in\mathscr{A}}$ generates the module
$\Omega^{1}_{\mathscr{A}}$ of Kähler differentials, the map $\varphi_{X}$ is
unique. ∎

######  Remark 6.15.

If $M$ is a manifold then the $C^{\infty}(M)$-module
$\bm{\Upomega}^{1}_{dR}(M)$ of ordinary one-forms on $M$ is the module of
Kähler differentials $\Omega_{C^{\infty}(M)}^{1}$. This fact is stated in [J,
Example 5.4] (without proof). In the more general case where $M$ is a manifold
with corners this fact is proved in [FS, Proposition 4.7.5].

######  Remark 6.16.

Kähler differentials coming from the $C^{\infty}$-algebra of smooth functions
on a differential space $(M,C^{\infty}(M))$ can be evaluated on the Zariski
tangent vectors. Let $p\in M$ be a point. Then, as we have seen before, the
evaluation map $ev_{p}:C^{\infty}(M)\to\mathbb{R}$ makes $\mathbb{R}$ into a
$C^{\infty}(M)$-module. A $C^{\infty}(M)$-derivation
$v:C^{\infty}(M)\to\mathbb{R}$ is, by definition, a Zariski tangent vector to
$M$ at $p$. for all $f\in C^{\infty}(M)$. Given
$\alpha\in\Omega^{1}_{C^{\infty}(M)}$ we think of $\imath_{v}\alpha$ as an
evaluation of the Kähler differential $\alpha$ on the tangent vector $v$ at
the point $p$:

$\alpha_{p}(v):=\imath_{v}\alpha$

(this forumla simultaneously defines $\alpha_{p}$ and $\alpha_{p}(v)$).

######  Remark 6.17.

Let $(M,C^{\infty}(M))$ be a differential space and $X:C^{\infty}(M)\to
C^{\infty}(M)$ a $C^{\infty}$-derivation. By the universal property of
$d:C^{\infty}(M)\to\Omega^{1}_{C^{\infty}(M)}$ there is a map of
$C^{\infty}(M)$-modules $\imath_{X}:\Omega^{1}_{C^{\infty}(M)}\to
C^{\infty}(M)$ so that

$\imath_{X}df=X(f)$

for all $f\in C^{\infty}(M)$. Given $\alpha\in\Omega^{1}_{C^{\infty}(M)}$ we
think of $\imath_{X}\alpha$ as a contraction of $X$ and $\alpha$. We will come
back to this point in the next section.

######  Example 6.18.

Consider the standard Cantor set $C\subset\mathbb{R}$ as in Example 3.33. Then
$dx|_{C}\not=0$ because

  1. (6.18.i)

$\imath_{\frac{d}{dx}}dx=1$,

  2. (6.18.ii)

$\frac{d}{dx}$ restricts to a derivation $\frac{d}{dx}|_{C}$ on $C$ and

  3. (6.18.iii)

$1|_{C}=\left(\imath_{\frac{d}{dx}}dx\right)|_{C}=\imath_{\frac{d}{dx}|_{C}}(dx|_{C})$.

We next check that the module of Kähler differentials
$\Omega^{1}_{\mathscr{A}}$ is natural in the $C^{\infty}$-ring $\mathscr{A}$.

######  Lemma 6.19.

Let $f:\mathscr{A}\to\mathscr{B}$ be a map of $C^{\infty}$-rings. Then there
is a unique map
$\Omega_{f}:\Omega_{\mathscr{A}}^{1}\to\Omega^{1}_{\mathscr{B}}$ of
$\mathscr{A}$-modules making the diagram

(6.2)
$\textstyle{\mathscr{A}}$$\textstyle{\Omega^{1}_{\mathscr{A}}}$$\textstyle{\mathscr{B}}$$\textstyle{\Omega^{1}_{\mathscr{B}}}$$\scriptstyle{d_{\mathscr{A}}}$$\scriptstyle{f}$$\scriptstyle{\Omega_{f}}$$\scriptstyle{d_{\mathscr{B}}}$

commute.

###### Proof.

The composite $d_{\mathscr{B}}\circ f:\mathscr{A}\to\Omega^{1}_{\mathscr{B}}$
is a $C^{\infty}$-derivation. By the universal property of $d_{\mathscr{A}}$
there is a unique map of $\mathscr{A}$-modules
$\Omega_{f}:\Omega^{1}_{\mathscr{A}}\to\Omega^{1}_{\mathscr{B}}$ making the
diagram (6.2) commute. ∎

The next lemma shows that just like in the case of commutative rings and
Kähler differentials, the Kähler differential of a quotient $C^{\infty}$-ring
$\mathscr{A}/I$ is a quotient of $\Omega^{1}_{\mathscr{A}}$ by an appropriate
submodule.

######  Lemma 6.20.

Let $\pi:\mathscr{A}\xrightarrow{\pi}\mathscr{B}$ be a surjective map of
$C^{\infty}$-rings, $I=\ker\pi$ and $0\to
J\to\Omega^{1}_{\mathscr{A}}\xrightarrow{p}\mathscr{M}\to 0$ a short exact
sequence of $\mathscr{A}$-modules. Then $\mathscr{M}=\Omega^{1}_{\mathscr{B}}$
if and only if $J=Id\mathscr{A}+\mathscr{A}dI$.

Moreover if $J=Id\mathscr{A}+\mathscr{A}dI$ then the universal derivation
$d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ descends to a unique
map $\bar{d}:\mathscr{B}\to\mathscr{M}$; this map is the universal derivation
$d_{\mathscr{B}}:\mathscr{B}\to\Omega^{1}_{\mathscr{B}}$ for the
$C^{\infty}$-ring $\mathscr{B}$.

###### Proof.

We argue first that

(6.3) $I\,d\mathscr{A}+\mathscr{A}\,dI=\mathscr{A}dI.$

For any $a\in\mathscr{A}$ and any $i\in I$, $d(ia)\in dI$. On the other hand
$d(ia)=a\,di+i\,da$. Hence $i\,da=d(ia)-ia\,di\in\mathscr{A}\,dI$ and
therefore $I\,d\mathscr{A}\subseteq\mathscr{A}dI$. We conclude that (6.3)
holds.

Now suppose $J=Id\mathscr{A}+\mathscr{A}dI$. By assumption
$\mathscr{B}=\mathscr{A}/I$ and $\mathscr{M}=\Omega^{1}_{\mathscr{A}}/J$. For
any $i\in I$, $a,b\in\mathscr{A}$

$i(a\,db)=(ia)\,db\in I\,d\mathscr{A}\subseteq J.$

Since $\Omega^{1}_{\mathscr{A}}$ is generated by $d(\mathscr{A})$ as an
$\mathscr{A}$-module, it follows that $i\alpha\in J$ for all $i\in I$,
$\alpha\in\Omega^{1}_{\mathscr{A}}$. Consequently there is a well-defined
action of $\mathscr{B}=\mathscr{A}/I$ on $\mathscr{M}=\Omega^{1}/J$ given by

$(a+I)\cdot(\alpha+J):=a\cdot\alpha+J.$

Since $p:\Omega^{1}_{\mathscr{A}}\to\mathscr{M}$ is a map of
$\mathscr{A}$-modules and
$d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ is a
$C^{\infty}$-derivation, the composite $p\circ
d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}/J$ is a
$C^{\infty}$-derivation. Since $d_{\mathscr{A}}i\in\mathscr{A}dI\subseteq J$
for any $i\in I$,

$(p\circ d_{\mathscr{A}})(i)=p(d_{\mathscr{A}}i)=0.$

Consequently $p\circ d_{\mathscr{A}}$ factors through
$\pi:\mathscr{A}\to\mathscr{A}/I=\mathscr{B}$: there is a map
$\bar{d}:\mathscr{A}/I\to\Omega^{1}_{\mathscr{A}}/I$ making the diagram

$\textstyle{\mathscr{A}}$$\textstyle{\Omega^{1}_{\mathscr{A}}}$$\textstyle{\mathscr{A}/I}$$\textstyle{\Omega^{1}_{\mathscr{A}}/J}$$\scriptstyle{d_{\mathscr{A}}}$$\scriptstyle{\pi}$$\scriptstyle{p}$$\scriptstyle{\bar{d}}$

commute. Since $\pi$ is surjective the map $\bar{d}$ is unique. Since $\pi$ is
a $C^{\infty}$-ring homomorphism and $\bar{d}\circ\pi=p\circ d_{\mathscr{A}}$
is a $C^{\infty}$-derivation, the map $\bar{d}$ is a $C^{\infty}$-derivation.

It remains to check that $\bar{d}$ is a universal derivation. To this end
consider a $C^{\infty}$-derivation $X:\mathscr{A}/I\to\mathscr{N}$ with values
in some $\mathscr{A}/I$-module $\mathscr{N}$. Then
$X\circ\pi:\mathscr{A}\to\mathscr{N}$ is a $C^{\infty}$-derivation as well
(where we now think of $\mathscr{N}$ as an $\mathscr{A}$-module). Hence there
exists a unique $\mathscr{A}$-module map
$\varphi_{X\circ\pi}:\mathscr{A}\to\mathscr{N}$ so that

$\varphi_{X\circ\pi}\circ d_{\mathscr{A}}=X\circ\pi.$

Moreover for all $i\in I$ and all $a\in\mathscr{A}$

$\varphi_{X\circ\pi}(a\,d_{\mathscr{A}}i)=a\cdot(X\circ\pi)(i)=a\cdot X(0)=0.$

Hence $\varphi_{X\circ\pi}$ vanishes on
$\mathscr{A}\,dI\stackrel{{\scriptstyle\eqref{eq:5.3}}}{{=}}\mathscr{A}\,dI+I\,d\mathscr{A}=J$.
It follows that there is a unique $\mathscr{A}/I$-module map
$\varphi_{X}:\Omega^{1}_{\mathscr{A}}/J\to\mathscr{N}$ so that
$\varphi_{X}\circ p=\varphi_{X\circ\pi}$. Finally for any
$a+I\in\mathscr{A}/I$

$\varphi_{X}(\bar{d}(a+I))=\varphi_{X}(d_{\mathscr{A}}(a)+J)=\varphi_{X\circ\pi}(d_{\mathscr{A}}(a))=(X\circ\pi)(a)=X(a+I).$

Hence

$\varphi_{X}\circ d_{\mathscr{A}/I}=X.$

This proves that if $J=Id\mathscr{A}+\mathscr{A}dI$ then
$\bar{d}:\mathscr{B}\to\mathscr{M}$ is
$d_{\mathscr{B}}:B\to\Omega^{1}_{\mathscr{B}}$.

The converse follows from the uniqueness of $\Omega^{1}_{\mathscr{B}}$ (and of
$d_{\mathscr{B}}:\mathscr{B}\to\Omega^{1}_{\mathscr{B}}$) up to a unique
isomorphism. ∎

######  Corollary 6.21.

Let $F:\mathscr{A}\to\mathscr{B}$ be a surjective map of $C^{\infty}$-rings,
$I=\ker F$ and
$J=\ker\\{\Omega_{F}:\Omega_{\mathscr{A}}^{1}\to\Omega_{\mathscr{B}}^{1}\\}$.
Then

$J=\mathscr{A}\,dI.$

######  Example 6.22.

We compute the module of global Kähler differential
$\Omega^{1}_{C^{\infty}(M)}$ on a singular space $M$: the closed subset
$M\subset\mathbb{R}^{2}$ consisting of the union of two coordinate axes:

$M=\\{(x,y)\in\mathbb{R}^{2}\mid xy=0\\}.$

(Since the $C^{\infty}$-ring $C^{\infty}(M)$ happens to be finitely presented
it follows from the results in the appendix that the assignment
$U\mapsto\Omega^{1}_{C^{\infty}(U)}$ is actually a sheaf of Kähler
differentials hence $\Omega^{1}_{C^{\infty}(M)}$ is a module of global
sections of a sheaf.)

Let $I$ denote the ideal in $C^{\infty}(\mathbb{R}^{2})$ consisting of all the
functions that vanish on $M$. Using Hadamard’s lemma one can show that $I$ is
generated by the function $xy$. [Here is a quick argument. Let $f(x,y)$ be in
$I$. Then $f(0,y)=0$ for all $y$ hence by Hadamard

$f(x,y)=f(x,y)-f(0,y)=xg(x,y)$

for some $g\in C^{\infty}(\mathbb{R}^{2})$. Since $f(x,0)=0$ for all $x$,
$g(x,0)=0$ for all $x\not=0$ and, by continuity, $g(x,0)=0$ for all $x$.
Therefore

$g(x,y)=g(x,y)-g(x,0)=yh(x,y)$

for some $h\in C^{\infty}(\mathbb{R}^{2})$. Hence $f(x,y)=xyh(x,y)$ for some
$h\in C^{\infty}(\mathbb{R}^{2})$.] Thus
$C^{\infty}(M)=C^{\infty}(\mathbb{R}^{2})/\langle xy\rangle$. By Lemma 6.20

$\Omega^{1}_{C^{\infty}(M)}=\Omega^{1}_{C^{\infty}(\mathbb{R}^{2})}/J$

where

$J=C^{\infty}(\mathbb{R}^{2})\,dI+I\,d(C^{\infty}(R^{2}))=\\{fd(xy)+xyg\,dx+xyh\,dy\mid
f,g,h\in C^{\infty}(\mathbb{R}^{2})\\}.$

Since
$\Omega^{1}_{C^{\infty}(\mathbb{R}^{2})}=\bm{\Upomega}^{1}_{dR}(\mathbb{R}^{2})$
is freely generated by $dx,dy$, the module
$\Omega^{1}_{C^{\infty}(M)}=\Omega^{1}_{C^{\infty}(\mathbb{R}^{2})}/J$ is
generated by $dx+J$ and $dy+J$ which we suggestively write as $dx|_{M}$ and
$dy|_{M}$, respectively. Alternatively we have the inclusion map
$\imath:M\hookrightarrow\mathbb{R}^{2}$, which is a map of differential
spaces. Hence we have a map of $C^{\infty}$-rings
$\imath^{*}:C^{\infty}(\mathbb{R}^{2})\to C^{\infty}(M)$ with
$\imath^{*}f=f|_{M}$. By Lemma 6.19 the map $\imath^{*}$ induces the
$\Omega_{\imath^{*}}:\Omega^{1}_{C^{\infty}(\mathbb{R}^{2})}\to\Omega^{1}_{C^{\infty}(M)}$
which commutes with $d$. We set

$\alpha|_{M}:=\Omega_{\imath^{*}}\alpha$

for all
$\alpha\in\Omega^{1}_{C^{\infty}(\mathbb{R}^{2})}=\Omega^{1}(\mathbb{R}^{2})$
to reduce clutter. Note that

$xdy|_{M}+ydx|_{M}=d(xy)|_{M}=d(xy|_{M})=d0=0,$

so the module $\Omega^{1}_{C^{\infty}(M)}$ is not freely generated by
$dx|_{M}$, $dy|_{M}$. On the other hand $xdy\not\in J$ so $xdy|_{M}\not=0$.
None the less one can show that the Kähler differential $xdy|_{M}$ vanishes on
all the Zariski tangent vectors to $M$. Recall that those are derivations with
values in the $C^{\infty}(M)$ modules $\mathbb{R}$ where the module structure
comes from the evaluations maps ( c.f. Example 6.7). Additionally $xdy|_{M}$
vanishes on all vector fields on $M$, i.e., on all derivations of
$C^{\infty}(M)$ with values in $C^{\infty}(M)$.

######  Remark 6.23.

The example above shows that for a differential space $(M,C^{\infty}(M))$ the
canonical $C^{\infty}(M)$-linear map

$\Omega^{1}_{C^{\infty}(M)}\to\operatorname{Hom}_{C^{\infty}(M)}(\operatorname{Der}(C^{\infty}(M)),C^{\infty}(M)),\qquad\alpha\mapsto(X\mapsto\imath_{X}\alpha)$

need not be injective. Consequently if we were to define the module of
differential 1-forms on a differential space $(M,C^{\infty}(M))$ to be
$\operatorname{Hom}_{C^{\infty}(M)}(\operatorname{Der}(C^{\infty}(M)),C^{\infty}(M))$
we would lose information.

Note that no loss of information occurs when $M$ is a manifold because for a
vector bundle $E\to M$ there is a canonical isomorphism
$\Gamma(E^{*})\to\operatorname{Hom}_{C^{\infty}(M)}(\Gamma(E),C^{\infty}(M))$
of $C^{\infty}(M)$-modules (here as before $\Gamma(E)$ is the module of global
sections of $E$). See [Co, Proposition 7.5.4].

##  7\. $C^{\infty}$-algebraic de Rham complex

In this section by analogy with the algebraic de Rham complex [Gr66, Gr67] we
construct the $C^{\infty}$-algebraic de Rham complex
$(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}},d)$ of a $C^{\infty}$-ring
$\mathscr{A}$.

We start with by adapting and modifying a number of definitions from graded
commutative algebra. In algebra, graded objects are often viewed as direct
sum. For example a vector space graded by $\mathbb{Z}$ is a direct sum of
vector spaces $V^{\bullet}=\oplus_{i\in\mathbb{Z}}V^{i}$. However, viewing
graded sheaves as direct sums leads to problems. The issue is that a direct
sum of sheaves of abelian groups is not a sheaf. More precisely, let $X$ be a
space and $\\{P_{k}\\}_{k\in\mathbb{Z}}$ be a collection of presheaves of
abelian groups. Recall that the category of presheaves of abelian groups is
complete and cocomplete with limits and colimits computed “obeject-wise.” In
particular the direct sum $\oplus_{k\in Z}P_{k}$ is given by

$\left(\oplus_{k\in Z}P_{k}\right)\,(U):=\oplus_{k\in\mathbb{Z}}P_{k}(U)$

for all open subsets $U\subseteq X$ (the direct sum on the right is computed
in the category $\mathsf{Ab}$ of abelian groups). However, even if all
$P_{k}$’s are sheaves, their direct sum $\oplus_{k\in Z}P_{k}$ may only be a
presheaf and not a sheaf (see Example 7.1 below). It may be confusing (it was
confusing to me) that the sheafification functor
$\mathsf{sh}:\mathsf{Psh}_{X}\to\mathsf{Sh}_{X}$ from presheaves to sheaves is
left adjoint to the inclusion $\imath:\mathsf{Sh}_{X}\to\mathsf{Psh}_{X}$,
hence has to preserves direct sums (which are coproducts, hence colimits).
Unfortunately the fact that $\mathsf{sh}$ preserves coproducts only says that
for a collection $\\{P_{k}\\}_{k\in\mathbb{Z}}$ of sheaves

(7.1)
$\mathsf{sh}(\oplus_{k\in\mathbb{Z}}\imath(P_{k}))=\oplus_{k\in\mathbb{Z}}P_{k},$

where on the right the coproduct $\oplus_{k\in\mathbb{Z}}P_{k}$ is computed in
the category of sheaves, and hence need not be given by an object-wise
formula. In fact (7.1) simply says: in order to compute the direct sum of
sheaves, take their direct sum as presheaves (that is, object-wise) and
sheafify. Of course, it is not always easy to figure out what sheafification
actually leaves you with.

######  Example 7.1.

Let $B^{n}$ denote the open $n$-ball in $\mathbb{R}^{n}$ of radius 1 centered
at 0\. Let $M=\bigsqcup_{n\geq 0}B^{n}$, the coproduct of the balls $B^{n}$’s.
The space $M$ is a second countable Hausdorff manifold of unbounded dimension.
Let $\bm{\Upomega}^{k}_{dR,M}$ denote the sheaf of ordinary differential
$k$-forms on $M$. Recall that in the category of presheaves countable direct
sums (coproducts) always exist and are computed “object-wise.” So we can form
a direct sum (in the category of presheaves)

$\bm{\Upomega}^{\bullet}_{dR,M}:=\oplus_{k\geq 0}\bm{\Upomega}^{k}_{dR,M}.$

However this presheaf is not a sheaf. Here is one way to see it: for each $n$
pick a volume form $\alpha_{n}\in\bm{\Upomega}_{dR}^{n}(B^{n})$. Since the
balls $B^{n}$ are all disjoint, the sequence
$(\alpha_{0},\alpha_{1},\ldots,\alpha_{n},\ldots)$ of forms should define a
global section of the sheafification $\mathsf{sh}(\oplus_{k\geq
0}\bm{\Upomega}^{k}_{dR,M})$ but such a section is not in the direct sum
$\oplus_{k\geq 0}\bm{\Upomega}^{k}_{dR,M}(M)$. Hence the direct sum
$\oplus_{k\geq 0}\bm{\Upomega}^{k}_{dR,M}$ of sheaves (in the category of
presheaves on $M$) is not a sheaf.

######  Remark 7.2.

Throughout the paper we follow the Koszul sign convention: given two objects
$a$ and $b$ of degrees $|a|$ and $|b|$ respectively moving $a$ past $b$
introduces the sign $(-1)^{|a||b|}$.

######  Definition 7.3.

A commutative graded algebra (CGA) over a $C^{\infty}$-ring $\mathscr{A}$ is a
sequence of $\mathscr{A}$-modules $\\{B^{k}\\}_{k\in\mathbb{Z}}$ together with
a sequence of $\mathscr{A}$-bilinear maps $\wedge:B^{k}\times B^{\ell}\to
B^{k+\ell}$ so that $\wedge$ is associative and graded commutative. That is
for all $k,\ell,m\in\mathbb{Z}$ and all $a\in B^{k}$, $b\in B^{\ell}$ and
$c\in B^{m}$

  1. (7.3.i)

$(a\wedge b)\wedge c=a\wedge(b\wedge c)$ and

  2. (7.3.ii)

$b\wedge a=(-1)^{k\ell}a\wedge b$ (Koszul sign convention).

We denote such a CGA by $(B^{\bullet},\wedge)$. We write $b\in B^{\bullet}$ if
$b\in B^{k}$ for some $k\in\mathbb{Z}$. We refer to elements of $B^{k}$ as
homogeneous elements of degree k and write $|b|=k$ to indicate that $b\in
B^{k}$.

######  Example 7.4.

Let $\mathscr{A}$ be a $C^{\infty}$-ring and $\mathscr{M}$ an
$\mathscr{A}$-module. Then the exterior algebra
$\Lambda^{\bullet}\mathscr{M}=\\{\Lambda^{k}\mathscr{M}\\}_{k\in\mathbb{Z}}$
together with the usual wedge product $\wedge$ is a CGA over $\mathscr{A}$.

######  Remark 7.5.

Commutative graded algebras (CGAs) are really graded-commutative algebras:
that is, they are graded and are commutative up to the appropriate sign, and
they are not commutative and graded. This is why they are also called GCAs.

######  Definition 7.6.

A map of CGA’s $f:(B^{\bullet},\wedge)\to(C^{\bullet},\wedge)$ is a collection
of maps $f_{k}:B^{k}\to C^{k}$ so that $f(\alpha\wedge\beta)=f(\alpha)\wedge
f(\beta)$ for all $\alpha,\beta\in B^{\bullet}$.

######  Remark 7.7.

If $\varphi:\mathscr{A}\to\mathscr{A}^{\prime}$ is a map of $C^{\infty}$-rings
and $C$ is a $\mathscr{A}^{\prime}$ module then $C$ is also a
$\mathscr{A}$-module with the action of $\mathscr{A}$ on $C$ given by $a\cdot
c=\varphi(a)c$ for all $a\in\mathscr{A}$, $c\in C$.

Similarly if $\varphi:\mathscr{A}\to\mathscr{A}^{\prime}$ is a map of
$C^{\infty}$-rings and $(C^{\bullet},\wedge)$ is a CGA over
$\mathscr{A}^{\prime}$, then it is also an CGA over $\mathscr{A}$ by way of
$\varphi$.

Consequently given a CGA $(B^{\bullet},\wedge)$ over $\mathscr{A}$, a CGA
$(C^{\bullet},\wedge)$ over $\mathscr{A}^{\prime}$ and a map
$\mathscr{A}\to\mathscr{A}^{\prime}$ it makes sense to define a map from
$(B^{\bullet},\wedge)$ to $(C^{\bullet},\wedge)$, by regarding
$(C^{\bullet},\wedge)$ as a CGA over $\mathscr{A}$. This allows us to define
presheaves of CGAs over a $C^{\infty}$-ringed space; see Definition 8.1 below.

######  Definition 7.8.

Let $(B^{\bullet},\wedge)$ be a commutative graded algebra over a
$C^{\infty}$-ring $\mathscr{A}$. A graded derivation of degree
$k\in\mathbb{Z}$ is a sequence of $\mathbb{R}$-linear maps
$X=\\{X^{\ell}:B^{\ell}\to B^{\ell+k}\\}_{\ell\in\mathbb{Z}}$ so that for all
homogeneous elements $x,y\in B^{\bullet}$

$X(x\wedge y)=X(x)\wedge y+(-1)^{|x|k}x\wedge X(y),$

where $|x|$ is the degree of $x$, i.e., $x\in B^{|x|}$ (to reduced the clutter
the $X$ on the left stands for $X^{|x|+|y|}$ and so on). Again our signs
follow the Koszul sign convention.

######  Example 7.9.

Let $M$ be a smooth manifold, $\mathscr{A}=C^{\infty}(M)$, the
$C^{\infty}$-ring of smooth functions. Then the exterior derivative
$d:\bm{\Upomega}^{\bullet}(M)\to\bm{\Upomega}^{\bullet+1}$ is a graded
derivation of degree +1 of the CGA $(\bm{\Upomega}^{\bullet}(M),\wedge)$ of
global de Rham differential forms.

Any vector field $v:M\to TM$ give rise to a derivation
$\imath(v):\bm{\Upomega}^{\bullet}(M)\to\bm{\Upomega}^{\bullet-1}$ of degree
-1: it’s contraction with $v$. The ordinary Lie derivative
${\mathcal{L}}_{v}:\bm{\Upomega}^{\bullet}(M)\to\bm{\Upomega}^{\bullet}$ with
respect to $v$ is a degree 0 derivation.

######  Remark 7.10.

The set of all graded derivations of a CGA $(B^{\bullet},\wedge)$ over
$\mathscr{A}$ of degree $k$ is a vector space over $\mathbb{R}$ and an
$\mathscr{A}$-module. The operations are defined pointwise. For example if
$X=\\{X^{k}\\}_{k\in\mathbb{Z}}$ is a derivation and $a\in\mathscr{A}$ then
$aX=\\{aX^{k}\\}_{k\in\mathbb{Z}}$ and so on.

We denote it by $\operatorname{Der}^{k}(B^{\bullet})$. We write
$X:B^{\bullet}\to B^{\bullet+k}$ if $X\in\operatorname{Der}^{k}(B^{\bullet})$

######  Remark 7.11.

We use the same symbol $\operatorname{Der}$ to denote both the
$C^{\infty}$-derivations of $C^{\infty}$-rings and graded derivations of
graded commutative algebras. It should be clear from the context which one is
which.

######  Definition 7.12.

A commutative differential  graded algebra (CDGA) over a $C^{\infty}$-ring
$\mathscr{A}$ is a commutative graded algebra $(\mathscr{B}^{\bullet},\wedge)$
together with a degree +1 derivation
$d\in\operatorname{Der}^{1}(\mathscr{B}^{\bullet})$ called a differential,
which squares to 0. That is, $d\circ d=0$.

We write $(\mathscr{B}^{\bullet},\wedge,d)$ to denote such a CDGA.

######  Definition 7.13.

Let $\Omega^{1}_{\mathscr{A}}$ be the module of $C^{\infty}$-Kähler
differentials of a $C^{\infty}$-ring $\mathscr{A}$. For $k>0$ we denote the
$k$-th exterior power of the $\mathscr{A}$-module $\Omega^{1}_{\mathscr{A}}$
by $\Lambda^{k}(\Omega^{1}_{\mathscr{A}})$. Note that the wedge is taken over
the $\mathbb{R}$-algebra $\mathscr{A}$, not over $\mathbb{R}$. By convention

$\Lambda^{0}(\Omega^{1}_{\mathscr{A}}):=\mathscr{A}.$

We set

$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}:=\\{\Lambda^{k}\Omega^{1}_{\mathscr{A}}\\}_{k\geq
0}$

which is a commutative graded algebra over the $C^{\infty}$-ring
$\mathscr{A}$.

We now turn $\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$ into a commutative
differential graded algebra (a CDGA).

######  Theorem 7.14.

Let $\mathscr{A}$ be a $C^{\infty}$-ring. The universal differential
$d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ extends to unique
degree 1 $\mathbb{R}$-linear map

$d:\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet+1}\Omega^{1}_{\mathscr{A}}$

so that for all $k>0$ and all $a,b_{1},\ldots,b_{k}\in\mathscr{A}$

(7.2) $d(a\,d_{\mathscr{A}}b_{1}\wedge\ldots\wedge
d_{\mathscr{A}}b_{k})=d_{\mathscr{A}}a\wedge
d_{\mathscr{A}}b_{1}\wedge\ldots\wedge d_{\mathscr{A}}b_{k}.$

Consequently $d\circ d=0$.

We mimic [SPr, Tag 0FKL]. As a first step we prove

######  Lemma 7.15.

Let $\mathscr{A}$ be a $C^{\infty}$ ring. The universal differential
$d_{\mathscr{A}}:\mathscr{A}\to\Omega^{1}_{\mathscr{A}}$ gives rise to a
unique $\mathbb{R}$-linear map
$d:\Omega^{1}_{\mathscr{A}}\to\Omega^{1}_{\mathscr{A}}\wedge\Omega^{1}_{\mathscr{A}}=\Lambda^{2}\Omega^{1}_{\mathscr{A}}$
with

  * •

$d(\sum_{i=1}^{k}a_{i}d_{\mathscr{A}}b_{i})=\sum_{i}d_{\mathscr{A}}a_{i}\wedge
d_{\mathscr{A}}b_{i}$ for all $k>0$, all
$a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}\in\mathscr{A}$ and

  * •

$d(d_{\mathscr{A}}a)=0$ for all $a\in\mathscr{A}$.

###### Proof.

It is easy to see that if such map $d$ exists, then it has to be unique. To
prove existence of $d$ consider the $\mathbb{R}$-linear map

$\varphi:F(\hat{d}\mathscr{A})\to\Omega^{1}_{\mathscr{A}}\wedge\Omega^{1}_{\mathscr{A}},\qquad\varphi(\sum_{i=1}^{k}a_{i}\hat{d}b_{i})=\sum_{i}d_{\mathscr{A}}a_{i}\wedge
d_{\mathscr{A}}b_{i},$

where, as in the proof of Theorem 6.14, $F(\hat{d}\mathscr{A})$ is a free
$\mathscr{A}$-module on the set $\\{\hat{d}b\\}_{b\in\mathscr{A}}$. Since
$d_{\mathscr{A}}$ is a derivation, $d_{\mathscr{A}}(1)=0$ and hence
$\varphi(\hat{d}b)=\varphi(1\,\hat{d}b)=d_{\mathscr{A}}(1)\wedge
d_{\mathscr{A}}b=0$. We check that $\varphi$ vanishes on the submodule
$I=\langle\mathscr{R}\rangle$ of $F(\hat{d}\mathscr{A})$ (see the proof of
Theorem 6.14 for notation) hence descends to a map
$d:\Omega^{1}_{\mathscr{A}}=F(\hat{d}\mathscr{A})/I\to\Omega^{1}_{\mathscr{A}}\wedge\Omega^{1}_{\mathscr{A}}$.
This is a computation that uses the fact that mixed partials commute. Given
$n>0$, $f\in C^{\infty}(\mathbb{R}^{n})$ and
$a_{1},\ldots,a_{n}\in\mathscr{A}$

$\begin{split}\varphi&\left(\hat{d}(f_{\mathscr{A}}(a_{1},\ldots,a_{n})-\sum(\partial_{i}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\,\hat{d}a_{i}\right)\\\
=&0-\sum_{i}d_{\mathscr{A}}\left((\partial_{i}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\right)\wedge
d_{\mathscr{A}}a_{i}\\\
=&-\sum_{j,i}(\partial_{ji}^{2}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})d_{\mathscr{A}}a_{j}\wedge
d_{\mathscr{A}}a_{i}\\\
=&\sum_{i<j}\left((\partial_{ij}^{2}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})-(\partial_{ji}^{2}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\right)\,d_{\mathscr{A}}a_{i}\wedge
d_{\mathscr{A}}a_{j}\\\
=&\sum_{i<j}(\partial_{ij}^{2}f-\partial_{ji}^{2}f)_{\mathscr{A}}(a_{1},\ldots,a_{n})\,d_{\mathscr{A}}a_{i}\wedge
d_{\mathscr{A}}a_{j}=0.\end{split}$

∎

###### Proof of Theorem 7.14.

It remains to construct

$d:\Lambda^{p}\Omega^{1}_{\mathscr{A}}\to\Lambda^{p+1}\Omega^{1}_{\mathscr{A}}$

for $p>1$ and show that $d\circ d=0$. The construction is identical to the one
in [SPr]. Consider the map

$\beta:\overbrace{\Omega_{\mathscr{A}}^{1}\times\cdots\times\Omega_{\mathscr{A}}^{1}}^{p}\to\Lambda^{p+1}\Omega^{1}_{\mathscr{A}},\qquad\beta(\alpha_{1},\ldots,\alpha_{p})=\sum_{i}(-1)^{i+1}\alpha_{1}\wedge\ldots\wedge
d_{\mathscr{A}}\alpha_{i}\wedge\ldots\wedge\alpha_{p}.$

Since $\beta$ is $\mathbb{R}$-multilinear it descends to an
$\mathbb{R}$-linear map

$\gamma:\Omega_{\mathscr{A}}^{1}\otimes_{\mathbb{R}}\cdots\otimes_{\mathbb{R}}\Omega_{\mathscr{A}}^{1}\to\Omega^{p+1}_{\mathscr{A}},\qquad\gamma(\alpha_{1}\otimes\cdots\otimes\alpha_{p})=\sum_{i}(-1)^{i+1}\alpha_{1}\wedge\ldots\wedge
d_{\mathscr{A}}\alpha_{i}\wedge\ldots\wedge\alpha_{p}.$

We will show that $\gamma$ factors through the natural surjection

$\Omega_{\mathscr{A}}^{1}\otimes_{\mathbb{R}}\cdots\otimes_{\mathbb{R}}\Omega_{\mathscr{A}}^{1}\to\Lambda^{p}\Omega^{1}_{\mathscr{A}}$

to give the desired map
$d:\Lambda^{p}\Omega^{1}_{\mathscr{A}}\to\Lambda^{p+1}\Omega^{1}_{\mathscr{A}}$
with

$d(\alpha_{1}\wedge\ldots\wedge\alpha_{p})=\sum_{i}(-1)^{i+1}\alpha_{1}\wedge\ldots\wedge
d_{\mathscr{A}}\alpha_{i}\wedge\ldots\wedge\alpha_{p}.$

It is easy to see that $\gamma$ is alternating. It remains to check that for
any two indices $i\not=j$, any $a\in\mathscr{A}$ and any
$\alpha_{1},\ldots,\alpha_{p}\in\Omega^{p}_{\mathscr{A}}$

(7.3) $\gamma(\alpha_{1}\otimes\cdots\otimes
a\alpha_{i}\otimes\cdots\otimes\alpha_{p}-\alpha_{1}\otimes\cdots\otimes
a\alpha_{j}\otimes\cdots\otimes\alpha_{p})=0.$

Now

$\begin{split}\gamma(\alpha_{1}\otimes\cdots\otimes
a\alpha_{i}\otimes\cdots\otimes\alpha_{p})&=a\gamma(\alpha_{1}\otimes\cdots\otimes\alpha_{p})+(-1)^{i+1}\alpha_{1}\wedge\ldots\wedge\alpha_{i-1}\wedge
da\wedge\alpha_{i}\wedge\ldots\wedge\alpha_{p}\\\
&=a\gamma(\alpha_{1}\otimes\cdots\otimes\alpha_{p})+da\wedge\alpha_{1}\wedge\ldots\wedge\alpha_{p}.\end{split}$

Hence (7.3) holds.

Since $d_{\mathscr{A}}(1)=0$,

$d(d_{\mathscr{A}}b_{1}\wedge\ldots\wedge
d_{\mathscr{A}}b_{k})=d(1d_{\mathscr{A}}b_{1}\wedge\ldots\wedge
d_{\mathscr{A}}b_{k})=d(1)\wedge d_{\mathscr{A}}b_{1}\wedge\ldots\wedge
d_{\mathscr{A}}b_{k}=0.$

Consequently

$(d\circ d)\,(a\,\,d_{\mathscr{A}}b_{1}\wedge\ldots\wedge
d_{\mathscr{A}}b_{k})=d(d_{\mathscr{A}}a\wedge
d_{\mathscr{A}}b_{1}\wedge\ldots\wedge d_{\mathscr{A}}b_{k})=0.$

Hence $d\circ d=0$ and we are done. ∎

######  Corollary 7.16.

For any $C^{\infty}$-ring $\mathscr{A}$ the exterior derivative
$d:\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet+1}\Omega^{1}_{\mathscr{A}}$
is a derivation of the graded commutative algebra
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$ of degree +1: for any $k$, any
$\alpha\in\Lambda^{k}\Omega^{1}_{\mathscr{A}}$ and any
$\beta\in\Lambda^{\ell}\Omega^{1}_{\mathscr{A}}$

(7.4) $d(\alpha\wedge\beta)=(d\alpha)\wedge\beta+(-1)^{k}\alpha\wedge d\beta.$

###### Proof.

It is no loss of generality to assume that
$\alpha=a_{0}\,da_{1}\wedge\ldots\wedge a_{k}$ and
$\beta=b_{0}\,db_{1}\wedge\ldots\wedge db_{\ell}$ for some $a_{0},\ldots
b_{\ell}\in\mathscr{A}$. Then

$\alpha\wedge\beta=a_{0}b_{0}\,da_{1}\wedge\ldots db_{\ell}$

and (7.4) follows easily from (7.2). ∎

The $C^{\infty}$-algebraic de Rham complex
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$ is functorial in the
$C^{\infty}$-ring $\mathscr{A}$:

######  Proposition 7.17.

A map $f:\mathscr{A}\to\mathscr{B}$ of $C^{\infty}$-rings induces a unique map
$\Lambda^{\bullet}(f):\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet}\Omega^{1}_{\mathscr{B}}$
of commutative differential graded algebras. Explicitly

$\Lambda^{\bullet}(f)(a_{0}d_{\mathscr{A}}a_{1}\wedge\cdots\wedge
d_{\mathscr{A}}a_{k})=f(a_{0})d_{\mathscr{B}}f(a_{1})\wedge\cdots\wedge
d_{\mathscr{B}}f(a_{k})$

for all $k>0$ and all $a_{0},\ldots,a_{k}\in\mathscr{A}$.

###### Proof.

By Lemma 6.19 the map $f:\mathscr{A}\to\mathscr{B}$ induces a map
$\Omega_{f}:\Omega^{1}_{\mathscr{A}}\to\Omega^{1}_{\mathscr{B}}$. By taking
exterior powers we get
$\Lambda^{k}\Omega_{f}:\Lambda^{k}\Omega^{1}_{\mathscr{A}}\to\Lambda^{k}\Omega^{1}_{\mathscr{B}}$
for each $k\geq 1$. Together these maps give us a map of commutative graded
algebras

$\Lambda^{\bullet}(f):=\Lambda^{\bullet}Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet}\Omega^{1}_{\mathscr{B}}$

with the property that

$\Lambda^{\bullet}(f)(a_{0}d_{\mathscr{A}}a_{1}\wedge\cdots\wedge
d_{\mathscr{A}}a_{k})=f(a_{0})d_{\mathscr{B}}f(a_{1})\wedge\cdots\wedge
d_{\mathscr{B}}f(a_{k})$

for all $k>0$ and all $a_{0},\ldots,a_{k}\in\mathscr{A}$ (since
$d_{\mathscr{B}}(f(a))=\Omega_{f}(d_{\mathscr{A}}a)$ for all
$a\in\mathscr{A}$). Now

$\begin{split}d\left(\Lambda^{\bullet}(f)\,(a_{0}d_{\mathscr{A}}a_{1}\wedge\cdots\wedge
d_{\mathscr{A}}a_{k})\right)&=d\left(f(a_{0})\wedge
d_{\mathscr{B}}f(a_{1})\wedge\cdots\wedge d_{\mathscr{B}}f(a_{k})\right)\\\
&=d_{\mathscr{B}}f(a_{0})\wedge d_{\mathscr{B}}f(a_{1})\wedge\cdots\wedge
d_{\mathscr{B}}f(a_{k})\\\
&=\Lambda^{\bullet}(f)\left(d_{\mathscr{A}}a_{0}\wedge
d_{\mathscr{A}}a_{1}\wedge\cdots\wedge d_{\mathscr{A}}a_{k}\right)\\\
&=\Lambda^{\bullet}(f)\left(d\left(a_{0}\,d_{\mathscr{A}}a_{1}\wedge\cdots\wedge
d_{\mathscr{A}}a_{k}\right)\right).\end{split}$

Hence $\Lambda^{\bullet}(f)$ also commutes with $d$ and we are done. ∎

We end the section by constructing contractions. Contractions are important
for constructing Cartan calculus for $C^{\infty}$-rings, for differential
spaces and, more generally, for local $C^{\infty}$-ringed spaces. A
construction of Cartan calculus will be carried out elsewhere, so the reader
may safely skip the rest of the section. We start by proving a well-known
algebraic fact.

######  Lemma 7.18.

Let $\mathscr{A}$ be an $\mathbb{R}$-algebra, $\mathscr{M}$ an
$\mathscr{A}$-module, $\Lambda^{\bullet}\mathscr{M}$ the exterior algebra of
$\mathscr{M}$ (wedge over $\mathscr{A}$, not $\mathbb{R}$) and
$\varphi:\mathscr{M}\to\mathscr{A}$ a map of $\mathscr{A}$-modules. Then
$\varphi$ extends uniquely to a degree -1 $\mathscr{A}$-linear map

$\tilde{\varphi}:\Lambda^{\bullet}\mathscr{M}\to\Lambda^{\bullet-1}\mathscr{M}$

with

$\tilde{\varphi}(\alpha_{1}\wedge\ldots\wedge\alpha_{p})=\sum_{i}(-1)^{i+1}\varphi(\alpha_{i})\alpha_{1}\wedge\ldots\wedge\widehat{\alpha_{i}}\wedge\ldots\wedge\alpha_{p}$

for all $p\geq 1$ and all $\alpha_{1},\ldots,\alpha_{p}\in\mathscr{M}$.

###### Proof.

Fix $p>0$ and consider the map
$\psi:\overbrace{\mathscr{M}\times\cdots\times\mathscr{M}}^{p}\to\Lambda^{p-1}\mathscr{M}$
given by

$\psi(\alpha_{1},\ldots,\alpha_{p})=\sum_{i}(-1)^{i+1}\varphi(\alpha_{i})\alpha_{1}\wedge\ldots\wedge\widehat{\alpha_{i}}\wedge\ldots\wedge\alpha_{p}$

The map $\psi$ is $\mathscr{A}$-multilinear and alternating hence gives rise
to the desired map
$\tilde{\varphi}:\Lambda^{p}\mathscr{M}\to\Lambda^{p-1}\mathscr{M}$. ∎

######  Lemma 7.19.

Let $\mathscr{A}$ be a $C^{\infty}$-ring. There is a map

$\imath:\operatorname{Hom}(\Omega^{1}_{\mathscr{A}},\mathscr{A})\to\operatorname{Der}^{-1}(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}})$

of $\mathscr{A}$ modules which is uniquely defined by

(7.5) $\imath(\varphi)(adb_{1}\wedge\cdots\wedge
db_{k})=\sum_{i}(-1)^{i+1}a\varphi(db_{i})db_{1}\wedge\ldots\wedge\widehat{db_{i}}\wedge\ldots\wedge
db_{k}$

for all $\varphi\in\operatorname{Hom}(\Omega^{1}_{\mathscr{A}},\mathscr{A})$,
all $k>0$ and all $a,b_{1},\ldots,b_{k}\in\mathscr{A}$. Additionally, for all
$\varphi\in\operatorname{Hom}(\Omega^{1}_{\mathscr{A}},\mathscr{A})$ the map

$\imath(\varphi):\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet-1}\Omega^{1}_{\mathscr{A}}$

is $\mathscr{A}$-linear.

###### Proof.

For any $\varphi:\Omega^{1}_{\mathscr{A}}\to\mathscr{A}$ the existence and
uniqueness of
$\imath{\varphi}:\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet-1}\Omega^{1}_{\mathscr{A}}$
satisfying (7.5) follows from Lemma lem:7.711. A computation using (7.5) and
the fact that $\Lambda^{k}\Omega^{1}_{\mathscr{A}}$ is generated by the terms
of the form $adb_{1}\wedge\cdots\wedge db_{k}$ shows that

(7.6)
$\imath(\varphi)(\alpha\wedge\beta)=(\imath(\varphi)\alpha)\wedge\beta+(-1)^{(-1)|\alpha|}\alpha\wedge\imath(\varphi)\beta$

for any homogeneous
$\alpha,\beta\in\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$, i.e.,
$\imath(\varphi)$ is a degree -1 derivation of the commutative graded algebra
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$. Finally it is easy to check that
$\imath$ is a map of $\mathscr{A}$-modules. ∎

Recall that for any $C^{\infty}$-ring $\mathscr{A}$ there is a canonical
isomorphism of $\mathscr{A}$-modules

(7.7)
$d^{*}:\operatorname{Hom}(\Omega^{1}_{\mathscr{A}},\mathscr{A})\to\operatorname{Der}(\mathscr{A}),\qquad
d^{*}(\varphi)=\varphi\circ d.$

Consequently Lemma 7.19 can be restated as follows:

######  Lemma 7.20.

Let $\mathscr{A}$ be a $C^{\infty}$-ring. There is a map

$\imath:\operatorname{Der}(\mathscr{A})\to\operatorname{Der}^{-1}(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}),\qquad
X\mapsto\imath_{X}$

of $\mathscr{A}$ modules which is uniquely defined by

(7.8) $\imath_{X}(adb_{1}\wedge\cdots\wedge
db_{k})=\sum_{i}(-1)^{i+1}aX(b_{i})db_{1}\wedge\ldots\wedge\widehat{db_{i}}\wedge\ldots\wedge
db_{k}$

for all $X\in\operatorname{Der}(\mathscr{A})$, all $k>0$ and all
$a,b_{1},\ldots,b_{k}\in\mathscr{A}$. Additionally, for all
$X\in\operatorname{Der}(\mathscr{A})$ the map

$\imath_{X}:\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}\to\Lambda^{\bullet-1}\Omega^{1}_{\mathscr{A}}$

is $\mathscr{A}$-linear.

######  Lemma 7.21 (contractions are functorial).

Let $f:\mathscr{A}\to\mathscr{B}$ be a map of $C^{\infty}$-rings,
$X:\mathscr{A}\to\mathscr{A}$, $Y:\mathscr{B}\to\mathscr{B}$ two derivations
with

$f\circ X=Y\circ f.$

Then

(7.9)
$\Lambda^{\bullet}(f)(\imath_{X}\omega)=\imath_{Y}(\Lambda^{\bullet}(f)(\omega))$

for all $\omega\in\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$.

###### Proof.

It is enough to check that (7.9) holds on generators of
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$. For any $k\geq 1$ and any
$a_{0},\ldots,a_{k}\in\mathscr{A}$

$\begin{split}\Lambda^{\bullet}(f)\left(\imath_{X}(a_{0}\,d_{\mathscr{A}}a_{1}\wedge\ldots\wedge
d_{\mathscr{A}}a_{k})\right)&=\Lambda^{\bullet}(f)\left(a_{0}\sum_{i}(-1)^{i+1}X(a_{i})d_{\mathscr{A}}a_{1}\wedge\ldots\wedge\widehat{d_{\mathscr{A}}a_{i}}\wedge\ldots\wedge
d_{\mathscr{A}}a_{k}\right)\\\
&=f(a_{0})\sum_{i}(-1)^{i+1}f(X(a_{i}))d_{\mathscr{B}}f(a_{1})\wedge\ldots\wedge\widehat{d_{\mathscr{B}}f(a_{i})}\wedge\ldots\wedge
d_{\mathscr{B}}f(a_{k})\\\
&=f(a_{0})\sum_{i}(-1)^{i+1}Y(f(a_{i}))d_{\mathscr{B}}f(a_{1})\wedge\ldots\wedge\widehat{d_{\mathscr{B}}f(a_{i})}\wedge\ldots\wedge
d_{\mathscr{B}}f(a_{k})\\\
&=\imath_{Y}\left(\Lambda^{\bullet}(f)(a_{0}\,d_{\mathscr{A}}a_{1}\wedge\ldots\wedge
d_{\mathscr{A}}a_{k})\right)\end{split}$

and we are done. ∎

##  8\. Differential forms on $C^{\infty}$-ringed spaces, integration and
Stokes’ theorem

In this section we construct a sheaf of differential forms on a
$C^{\infty}$-ringed space. (Of course, we are only interested in differential
forms on local $C^{\infty}$-ringed spaces, but the locality condition on the
stalk doesn’t seem to matter in the constructions.) We show that our
differential forms pull back, that pull back commutes with the exterior
derivative and exterior multiplication and that our differential forms can be
integrated over simplices. Consequently a version of Stokes’ theorem holds for
these forms.

######  Definition 8.1.

A presheaf of CGAs $(\mathscr{M}^{\bullet},\wedge)$ over a $C^{\infty}$-ringed
space $(M,\mathscr{A})$ is a sequence of $\mathscr{A}$-modules
$\\{\mathscr{M}^{k}\\}_{k\in\mathbb{Z}}$ together with a sequence of
$\mathscr{A}$-bilinear maps of presheaves
$\wedge:\mathscr{M}^{k}\times\mathscr{M}^{\ell}\to\mathscr{M}^{k+\ell}$ so
that $\wedge$ is associative and graded commutative. That is, for all open
sets $U\subset M$ the pair
$(\\{\mathscr{M}^{k}(U)\\}_{k\in\mathbb{Z}},\wedge_{U})$ is CGA and the
restriction maps
$r^{U}_{V}:(\mathscr{M}^{\bullet}(U),\wedge_{U})\to(\mathscr{M}^{\bullet}(V),\wedge_{V})$
are maps of CGAs (cf. Remark 7.7).

Equivalently we require that the appropriate diagrams in the category of
presheaves of $\mathscr{A}$-modules commute.

######  Definition 8.2.

A graded derivation $X$ of degree $k$ of a presheaf
$(\mathscr{M}^{\bullet}=\\{\mathscr{M}^{k}\\}_{k\in\mathbb{Z}},\wedge)$ of
CGAs is a collection of maps of presheaves
$\\{X^{\ell}:\mathscr{M}^{\ell}\to\mathscr{M}^{\ell+k}\\}_{\ell\in\mathbb{Z}}$
so that

$X_{U}:\mathscr{M}^{\bullet}(U)\to\mathscr{M}^{\bullet+k}(U)$

is a degree $k$ derivation of the CGA $(\mathscr{M}^{\bullet}(U),\wedge)$ of
$\mathscr{A}(U)$-modules for each open set $U\subset X$. Equivalently $X$
makes the appropriate diagrams in the category of presheaves of
$\mathscr{A}$-modules commute.

######  Definition 8.3.

A presheaf of CDGAs $(\mathscr{M}^{\bullet},\wedge,d)$ over a
$C^{\infty}$-ringed space $(M,\mathscr{A})$ is a presheaf
$(\mathscr{M}^{\bullet},\wedge)$ of CGAs over $(M,\mathscr{A})$ together with
a sequence of maps
$\\{d:\mathscr{M}^{k}\to\mathscr{M}^{k+1}\\}_{k\in\mathbb{Z}}$ of presheaves
of real vector spaces so that for each open set $U\subset M$ the triple
$(\mathscr{M}^{\bullet}(U),\wedge_{U},d_{U})$ is a CDGA over the
$C^{\infty}$-ring $\mathscr{A}(U)$.

Equivalently a presheaf CDGAs over a $C^{\infty}$-ringed space
$(M,\mathscr{A})$ is a triple $(\mathscr{M}^{\bullet},\wedge,d)$, where
$(\mathscr{M}^{\bullet},\wedge)$ is a CGA over $(M,\mathscr{A})$ and
$d:\mathscr{M}^{\bullet}\to\mathscr{M}^{\bullet+1}$ is a graded derivation of
degree 1 with $d\circ d=0$.

A sheaf of CDGAs is a presheaf of CDGAs $(\mathscr{S}^{\bullet},\wedge,d)$ so
that $\mathscr{S}^{k}$ is a sheaf for all $k\in\mathbb{Z}$.

######  Construction 8.4.

Let $(M,\mathscr{A})$ be a $C^{\infty}$-ringed space. We construct a presheaf

$(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}=\\{\Lambda^{k}\Omega^{1}_{\mathscr{A}}\\}_{k\geq
0},\\{\wedge:\Lambda^{k}\Omega^{1}_{\mathscr{A}}\times\Lambda^{\ell}\Omega^{1}_{\mathscr{A}}\to\Lambda^{k+\ell}\Omega^{1}_{\mathscr{A}}\\}_{k,\ell\geq
0},\\{d:\Lambda^{k}\Omega^{1}_{\mathscr{A}}\to\Lambda^{k+1}\Omega^{1}_{\mathscr{A}}\\}_{k\geq
0})$

of CDGAs over the sheaf $\mathscr{A}$ as follows:

For every open set $U\subset M$ we have an $\mathscr{A}(U)$-module
$\Omega^{1}_{\mathscr{A}(U)}$ of Kähler differentials, which comes with the
universal derivation $d:\mathscr{A}(U)\to\Omega^{1}_{\mathscr{A}(U)}$. By
Lemma 6.19 the assignment $U\mapsto\Omega^{1}_{\mathscr{A}(U)}$ is a presheaf
of $\mathscr{A}$-modules. We denote this presheaf by
$\Omega^{1}_{\mathscr{A}}$. Consequently for all $k\geq 0$,
$U\mapsto\Lambda^{k}\Omega^{1}_{\mathscr{A}(U)}$ is also a presheaf of
$\mathscr{A}$-modules. We denote it by $\Lambda^{k}\Omega^{1}_{\mathscr{A}}$.
Proposition 7.17 implies that we have maps

$\wedge:\Lambda^{k}\Omega^{1}_{\mathscr{A}}\times\Lambda^{\ell}\Omega^{1}_{\mathscr{A}}\to\Lambda^{k+\ell}\Omega^{1}_{\mathscr{A}}$

of presheaves of $\mathscr{A}$-modules and that these maps turn the sequence
$\\{\Lambda^{k}\Omega^{1}_{\mathscr{A}}\\}_{k\geq 0}$ of presheaves into a
presheaf of commutative graded $\mathscr{A}$-algebras. Moreover, by
Proposition 7.17 for any pair $U,V$ of open sets in $M$ with $V\subset U$ the
restrictions $\Omega^{k}_{\mathscr{A}}(U)\to\Omega^{k}_{\mathscr{A}}(V)$
commute with $d$. We thus get a presheaf of CDGAs
$(\\{\Lambda^{k}\Omega^{1}_{\mathscr{A}}\\}_{k\geq 0},\wedge,d)$ over the
sheaf $\mathscr{A}$.

######  Notation 8.5.

Given a $C^{\infty}$-ringed space $(M,\mathscr{A})$ we denote the presheaf of
CDGAs produced by Construction 8.4 either by
$(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}},d,\wedge)$ or simply by
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$ with $d$ and $\wedge$ understood.

As was mentioned above, if $M$ is a manifold with corners then the presheaf
$\Omega^{1}_{C^{\infty}_{M}}$ of Kähler differentials on $M$ agrees with the
sheaf of sections $\bm{\Upomega}^{1}_{dR,M}$ of the cotangent bundle
$T^{*}M\to M$; this is a result of Francis-Staite [FS, Proposition 4.7.5] (cf.
Remark 6.15), hence in particular is a sheaf. This isomorphism of presheaves
extends to an isomorphism of the corresponding CDGAs.

######  Lemma 8.6.

Let $M$ be a manifold with corners. Then the presheaf
$(\Lambda^{\bullet}\Omega^{1}_{C^{\infty}_{M}},d,\wedge)$ of
$C^{\infty}$-algebraic de Rham forms is isomorphic to the sheaf of CDGAs
$(\bm{\Upomega}_{dR,M}^{\bullet},d,\wedge)$ of ordinary de Rham differential
forms as we show next.

###### Proof.

Given a vector bundle $E\to M$ denote temporarily its sheaf of sections by
$\mathscr{S}_{E}$.

For a vector bundle $E\to M$ over a (second countable Hausdorff) manifold with
corners $M$ taking $k$th exterior power of the bundle commutes with taking
global sections: there is an isomorphism of $C^{\infty}_{M}(M)$-modules

(8.1) $\varphi_{M}:\Lambda^{k}\left(\Gamma(E)\right)\to\Gamma(\Lambda^{k}E)$

with the property that for any $k$ global sections $s_{1},\ldots,s_{k}$ the
section $\varphi(s_{1}\wedge\cdots\wedge s_{k})$ is the section defined by

$q\mapsto s_{1}(q)\wedge\cdots\wedge s_{k}(q)$

(the wedge on the right is in the exterior algebra $\Lambda^{\bullet}E_{q}$ of
the fiber $E_{q}$ of $E$). See, for example, [Co, Section 7.5]. if $V\subset
U\subset M$ are two open sets in $M$ then the diagram

$\textstyle{\Lambda^{k}\Gamma(E|_{U})}$$\textstyle{\Gamma(\Lambda^{k}(E|_{U}))}$$\textstyle{\Lambda^{k}\Gamma(E|_{V})}$$\textstyle{\Gamma(\Lambda^{k}(E|_{V}))}$$\scriptstyle{\varphi_{U}}$$\scriptstyle{|_{V}}$$\scriptstyle{|_{V}}$$\scriptstyle{\varphi_{V}}$

commutes, where the vertical maps are restrictions. Hence we get an
isomorphism of presheaves

$\varphi:\Lambda^{k}\mathscr{S}_{E}\to\mathscr{S}_{\Lambda^{k}E}.$

Since the right hand side is a sheaf, the exterior power
$\Lambda^{k}\mathscr{S}_{E}$ is a sheaf as well.

Now consider $E=T^{*}M$. It follow from the paragraph above that

$\bm{\Upomega}^{k}_{dR,M}=\mathscr{S}_{\Lambda^{k}T^{*}M}\simeq\Lambda^{k}\mathscr{S}_{T^{*}M}=\Lambda^{k}\bm{\Upomega}_{dR,M}^{1}.$

Since the presheaf $\Omega^{1}_{C^{\infty}_{M}}$ is isomorphic to
$\bm{\Upomega}_{dR,M}^{1}$ [FS, Proposition 4.7.5], the $k$th exterior power
$\Lambda^{k}\Omega^{1}_{C^{\infty}_{M}}$ of $\Omega^{1}_{C^{\infty}_{M}}$ is
isomorphic to $\Lambda^{k}\bm{\Upomega}_{dR,M}^{1}$. The result follows. ∎

######  Lemma 8.7.

A map $(f,f_{\\#}):(M,\mathscr{A})\to(N,\mathscr{B})$ of $C^{\infty}$-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.

###### Proof.

The proof is another application of functoriality of
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{C}}$ in the $C^{\infty}$-ring
$\mathscr{C}$, which is Proposition 7.17.

For any sheaf $\mathscr{C}$ on $N$, any $k\geq 0$ and any open set $W\subset
N$

$(\Lambda^{k}\Omega^{1}_{\mathscr{C}})(W)=\Lambda^{k}(\Omega^{1}_{\mathscr{C}(W)}),$

where on the left we evaluate the presheaf on an open set, and on the right we
take the $k$th exterior power of an $\mathscr{C}(W)$ module
$\Omega^{1}_{\mathscr{C}(W)}$. It follows that for an open set $U\subset N$

$\left(\Lambda^{k}\Omega^{1}_{f_{*}\mathscr{A}}\right)(U)=\Lambda^{k}\left(\Omega^{1}_{(f_{*}\mathscr{A})(U)}\right)=(\Lambda^{k}\Omega^{1}_{\mathscr{A}})(f^{-1}(U))=\left(f_{*}(\Lambda^{k}\Omega^{1}_{\mathscr{A}})\right)\,(U).$

Consequently,

(8.2)
$(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}(f^{-1}(U))},\wedge,d)=\left[f_{*}(\Lambda^{\bullet}_{\mathscr{A}},\wedge,d)\right](U),$

an equality of CDGAs of modules. For any pair of open subsets $V\subset
U\subset N$ of $N$ the diagram

$\textstyle{\mathscr{B}(U)}$$\textstyle{\mathscr{A}(f^{-1}(U))}$$\textstyle{\mathscr{B}(V)}$$\textstyle{\mathscr{A}(f^{-1}(V))}$$\scriptstyle{f_{\\#}}$$\scriptstyle{f_{\\#}}$

commutes (the vertical maps are the restrictions). By Proposition 7.17 the
diagram of CDGA’s of modules

$\textstyle{\Lambda^{\bullet}\Omega^{1}_{\mathscr{B}(U)}}$$\textstyle{\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}(f^{-1}(U))}}$$\textstyle{\Lambda^{\bullet}\Omega^{1}_{\mathscr{B}(V)}}$$\textstyle{\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}(f^{-1}(V))}}$$\scriptstyle{\Lambda^{\bullet}(f_{\\#})}$$\scriptstyle{\Lambda^{\bullet}(f_{\\#})}$

commutes. Equation (8.2) now implies that the maps
$\Lambda^{\bullet}(f_{\\#})$ of modules assemble into a desired map
$\tilde{f}$ of presheaves of CDGAs over the sheaf $\mathscr{B}$. ∎

In general there is no reason to expect that the presheaf
$\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}}$ of CDGAs associated to
$C^{\infty}$-ringed differential space $(M,\mathscr{A})$ is a sheaf. So we
need to sheafify.

######  Lemma 8.8.

Let $(\\{{\mathcal{M}}^{k}\\}_{k=0}^{\infty},d,\wedge)$ be a presheaf of CDGAs
on a $C^{\infty}$-ringed space $(M,\mathscr{A})$. Then
$(\\{\mathsf{sh}({\mathcal{M}}^{k})\\}_{k=0}^{\infty},\mathsf{sh}(d),\mathsf{sh}(\wedge))$
is a sheaf of CDGAs on the $C^{\infty}$-ringed space $(M,\mathscr{A})$. Here
as before $\mathsf{sh}:\mathsf{Psh}_{M}\to\mathsf{Sh}_{M}$ is the
sheafification functor.

###### Proof.

Note that since the sheafification of the zero (pre)sheaf is zero, the
sheafification of any zero morphism is zero as well: $\mathsf{sh}(0)=0$. Then
$0=\mathsf{sh}(0)=\mathsf{sh}(d^{2})=(\mathsf{sh}(d))^{2}$. Hence
$(\\{\mathsf{sh}({\mathcal{M}}^{k})\\}_{k=0}^{\infty},\mathsf{sh}(d))$ is a
cochain complex of sheaves. Since sheafification preserves finite products we
get collection of morphisms

$\mathsf{sh}(\wedge):\mathsf{sh}({\mathcal{M}}^{k})\times\mathsf{sh}({\mathcal{M}}^{\ell})\to\mathsf{sh}({\mathcal{M}}^{k+\ell})$

We can express the fact that $\wedge$ is graded commutative and the fact that
$d$ is a derivation of degree 1 with respect to $\wedge$ in terms of
commutative diagrams. Since $\mathsf{sh}$ is a functor, it follows that
$(\\{\mathsf{sh}({\mathcal{M}}^{k})\\}_{k=0}^{\infty},\mathsf{sh}(d),\mathsf{sh}(\wedge))$
is a sheaf of CDGAs. ∎

######  Definition 8.9.

We define a $C^{\infty}$-algebraic de Rham complex
$(\bm{\Upomega}^{\bullet}_{\mathscr{A}},\mathsf{sh}(\wedge),\mathsf{sh}(d)\\}$
of sheaves on a $C^{\infty}$-ringed space $(M,\mathscr{A})$ to be the
sheafification of the presheaf of CDGAs
$(\\{\Lambda^{k}\Omega^{1}_{\mathscr{A}}\\}_{k\geq 0},\wedge,d)$ (c.f.
Construction 8.4).

######  Remark 8.10.

Given a manifold $M$ we have the de Rham complex
$\bm{\Upomega}^{\bullet}_{dR,M}$ of sheaves of ordinary differential forms and
the $C^{\infty}$-algebraic de Rham complex
$\bm{\Upomega}_{C^{\infty}_{M}}^{\bullet}$ of the manifold $M$ viewed as local
$C^{\infty}$-ringed space $(M,C^{\infty}_{M})$. By Lemma 8.6 the two sheaves
of CDGAs agree.

Differential forms pull back. Our $C^{\infty}$-algebraic differential forms on
$C^{\infty}$-ringed spaces had better pull back as well: given a map
$(f,f_{\\#}):(M,\mathscr{A})\to(N,\mathscr{B})$ of $C^{\infty}$-ringed spaces
we expect to have a map of sheaves of CDGAs
$\tilde{f}:\bm{\Upomega}^{\bullet}_{\mathscr{A}}\to
f_{*}(\bm{\Upomega}^{\bullet}_{\mathscr{B}})$. Moreover when $M$ and $N$ are
manifolds we expect the map on global sections
$\tilde{f}:\bm{\Upomega}^{\bullet}_{C^{\infty}_{M}}(M)\to
f_{*}(\bm{\Upomega}^{\bullet}_{C^{\infty}_{N}})(M)=\bm{\Upomega}^{\bullet}_{C^{\infty}_{N}}(N)$
to be the pullback map
$f^{*}:\bm{\Upomega}_{dR,M}^{\bullet}(M)\to\bm{\Upomega}_{dR,M}^{\bullet}(N)$.
We start by doing a category-theoretic exercise.

######  Lemma 8.11.

Let
$\textstyle{\mathscr{C}}$$\textstyle{\mathscr{D}}$$\textstyle{\perp}$$\scriptstyle{L}$$\scriptstyle{R}$
be a pair of adjoint functors,
$\eta:\operatorname{id}_{\mathscr{C}}\Rightarrow RL$ the unit of adjunction,
$h:d\to d^{\prime}$ a morphism in $\mathscr{D}$, $g:c\to c^{\prime}$, $k:c\to
Rd$, $k^{\prime}:c^{\prime}\to Rd^{\prime}$ morphisms in $\mathscr{C}$ so that
the diagram

(8.3)
$\textstyle{c}$$\textstyle{Rd}$$\textstyle{c^{\prime}}$$\textstyle{Rd^{\prime}}$$\scriptstyle{k}$$\scriptstyle{g}$$\scriptstyle{Rh}$$\scriptstyle{k^{\prime}}$

commutes in $\mathscr{C}$. Let $\tilde{k}:Lc\to d$,
$\tilde{k^{\prime}}:Lc^{\prime}\to d^{\prime}$ be the unique morphisms with
$R\tilde{k}\circ\eta_{c}=k$,
$R\tilde{k^{\prime}}\circ\eta_{c^{\prime}}=k^{\prime}$. Then the diagram

$\textstyle{Lc}$$\textstyle{d}$$\textstyle{Lc^{\prime}}$$\textstyle{d^{\prime}}$$\scriptstyle{\tilde{k}}$$\scriptstyle{Lg}$$\scriptstyle{h}$$\scriptstyle{\tilde{k^{\prime}}}$

commutes in $\mathscr{D}$.

###### Proof.

Since $L$ is left adjoint to $R$, for all $a\in\mathscr{C}$, $b\in\mathscr{C}$
there are bijections
$\theta_{a,b}:\operatorname{Hom}_{\mathscr{C}}(a,Rb)\to\operatorname{Hom}_{\mathscr{D}}(La,b)$
which are natural in $a$ and $b$. Moreover

$(\theta_{a,b})^{-1}(\tilde{\ell})=R(\tilde{\ell})\circ\eta_{a}$

for all $\tilde{l}\in\operatorname{Hom}_{\mathscr{D}}(La,b)$. Hence
$\tilde{k}=\theta_{c,d}(k)$ and
$\tilde{k^{\prime}}=\theta_{c^{\prime},d^{\prime}}(k^{\prime})$. Since
$\theta_{a,b}$ is natural in $a$ and $b$ the diagram

$\textstyle{\operatorname{Hom}(c,Rd)}$$\textstyle{\operatorname{Hom}(Lc,d)}$$\textstyle{\operatorname{Hom}(c,Rd^{\prime})}$$\textstyle{\operatorname{Hom}(Lc,d^{\prime}}$$\textstyle{\operatorname{Hom}(c^{\prime},Rd^{\prime})}$$\textstyle{\operatorname{Hom}(Lc^{\prime},d^{\prime})}$$\scriptstyle{\theta_{c,d}}$$\scriptstyle{Rh_{*}}$$\scriptstyle{h_{*}}$$\scriptstyle{\theta_{c,d^{\prime}}}$$\scriptstyle{g^{*}}$$\scriptstyle{Lg^{*}}$$\scriptstyle{\theta_{c^{\prime},d^{\prime}}}$

commutes. We now compute:

$h\circ\tilde{k}=h_{*}\theta_{c,d}(k)=\theta_{c,d^{\prime}}(Rh_{*}k)\overset{\textrm{
since \eqref{eq:9.2} commutes}}{=}\theta_{c,d^{\prime}}(k^{\prime}\circ
g)=Lg^{*}\theta_{c^{\prime},d^{\prime}}(k^{\prime})=\tilde{k^{\prime}}\circ
Lg.$

Therefore $h\circ\tilde{k}=\tilde{k^{\prime}}\circ Lg$ and we are done. ∎

######  Corollary 8.12.

Let $A_{1}\xrightarrow{h}A_{2}$ be a map of presheaves over a space $M$,
$B_{1}\xrightarrow{g}B_{2}$ a map of presheaves over a space $N$, $f:M\to N$ a
continuous map and $f_{i}:B_{i}\to f_{*}A_{i}$, $i=1,2$ maps of presheaves
making the diagram

$\textstyle{B_{1}}$$\textstyle{f_{*}A_{1}}$$\textstyle{B_{2}}$$\textstyle{f_{*}A_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{g}$$\scriptstyle{f_{*}h}$$\scriptstyle{f_{2}}$

commute. Let $\eta_{A_{i}}:A_{i}\to\mathsf{sh}(A_{i})$,
$\eta_{B_{i}}:B_{i}\to\mathsf{sh}(B_{i})$ denote the sheafifications and let
$\tilde{f}_{i}:\mathsf{sh}(B_{i})\to f_{*}(\mathsf{sh}(A_{i}))$ be the
canonical maps with

$\tilde{f}_{i}\circ\eta_{B_{i}}=f_{*}(\eta_{A_{i}})\circ f_{i},\quad i=1,2.$

Then the diagram

$\textstyle{\mathsf{sh}(B_{1})}$$\textstyle{f_{*}\mathsf{sh}(A_{1})}$$\textstyle{\mathsf{sh}(B_{2})}$$\textstyle{f_{*}\mathsf{sh}(A_{2})}$$\scriptstyle{\tilde{f}_{1}}$$\scriptstyle{\mathsf{sh}(g)}$$\scriptstyle{f_{*}\mathsf{sh}(h)}$$\scriptstyle{\tilde{f}_{2}}$

commutes.

###### Proof.

We know that the sheafification functor $\mathsf{sh}$ is left adjoint to the
inclusion $i:\mathsf{Sh}\hookrightarrow\mathsf{Psh}$. Apply Lemma 8.11 to the
commutative diagram

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15.8924pt\raise 28.4526pt\hbox{\hbox{\kern
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f_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 15.8924pt\raise
28.4526pt\hbox{\hbox{\kern 0.0pt\raise
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42.67891pt\raise 7.11316pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
42.67891pt\raise-6.2263pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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f_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
15.8924pt\raise-14.2263pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
∎

We are now in position to prove that $C^{\infty}$-algebraic differential forms
pull back along maps of $C^{\infty}$-ringed spaces, that pullback commutes
with exterior multiplication and exterior derivatives.

######  Theorem 8.13.

A map $(f,f_{\\#}):(M,\mathscr{A})\to(N,\mathscr{B})$ of $C^{\infty}$-ringed
spaces induces a map

$\bar{f}:\bm{\Upomega}^{\bullet}_{\mathscr{B}}\to
f_{*}(\bm{\Upomega}^{\bullet}_{\mathscr{A}})$

of sheaves of CDGAs.

###### Proof.

By Lemma 8.7 the map $(f,f_{\\#})$ induces a map
$\tilde{f}:\Lambda^{\bullet}\Omega^{1}_{\mathscr{B}}\to
f_{*}(\Lambda^{\bullet}\Omega^{1}_{\mathscr{A}})$ of presheaves of CDGAs. In
particular for each $k\geq 0$ we have maps
$\tilde{f}:\Lambda^{k}\Omega^{1}_{\mathscr{B}}\to
f_{*}(\Lambda^{k}\Omega^{1}_{\mathscr{A}})$ so that the following two diagrams

$\textstyle{\Lambda^{k}\Omega^{1}_{\mathscr{B}}\times\Lambda^{\ell}\Omega^{1}_{\mathscr{B}}}$$\textstyle{f_{*}\Lambda^{k}\Omega^{1}_{\mathscr{A}}\times
f_{*}\Lambda^{\ell}\Omega^{1}_{\mathscr{A}}}$$\textstyle{\Lambda^{k+\ell}\Omega^{1}_{\mathscr{B}}}$$\textstyle{f_{*}\Lambda^{k+\ell}\Omega^{1}_{\mathscr{A}}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\wedge}$$\scriptstyle{\wedge}$$\scriptstyle{\tilde{f}}$
$\textstyle{\Lambda^{k}\Omega^{1}_{\mathscr{B}}}$$\textstyle{f_{*}\Lambda^{k}\Omega^{1}_{\mathscr{A}}}$$\textstyle{\Lambda^{k+1}\Omega^{1}_{\mathscr{B}}}$$\textstyle{f_{*}\Lambda^{k+1}\Omega^{1}_{\mathscr{A}}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{\tilde{f}}$

commute. Recall the notation:
$\bm{\Upomega}^{k}_{\mathscr{A}}:=\mathsf{sh}(\Lambda^{k}\Omega^{1}_{\mathscr{A}})$
etc. Apply Corollary 8.12 to the two diagrams above and set

$\bar{f}:=\tilde{\tilde{f}}$

(the first $\tilde{\,}$ on $f$ comes from Lemma 8.7 and the second from
Corollary 8.12). We get two commuting diagrams:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
65.44005pt\hbox{{\hbox{\kern-59.50253pt\raise 28.4526pt\hbox{\hbox{\kern
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44.42455pt\raise 28.4526pt\hbox{\hbox{\kern
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f_{*}\bm{\Upomega}^{\ell}_{\mathscr{A}}}$}}}}}{\hbox{\kern-54.18031pt\raise-14.2263pt\hbox{\hbox{\kern
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55.24399pt\raise-14.2263pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 44.42455pt\raise
28.4526pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>7.11316pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
55.24399pt\raise-14.2263pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\quad\textrm{and}\qquad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern 64.41678pt\hbox{{\hbox{\kern-50.83585pt\raise
28.4526pt\hbox{\hbox{\kern
3.0pt\raise-3.43222pt\hbox{$\textstyle{\bm{\Upomega}^{k}_{\mathscr{B}}}$}}}}}{\hbox{\kern
58.58844pt\raise 28.4526pt\hbox{\hbox{\kern
3.0pt\raise-3.41666pt\hbox{$\textstyle{f_{*}\bm{\Upomega}^{k}_{\mathscr{A}}}$}}}}}{\hbox{\kern-54.41364pt\raise-14.2263pt\hbox{\hbox{\kern
3.0pt\raise-3.43222pt\hbox{$\textstyle{\bm{\Upomega}^{k+1}_{\mathscr{B}}}$}}}}}{\hbox{\kern
55.01067pt\raise-14.2263pt\hbox{\hbox{\kern
3.0pt\raise-3.41666pt\hbox{$\textstyle{f_{*}\bm{\Upomega}^{k+1}_{\mathscr{A}}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 58.58844pt\raise
28.4526pt\hbox{\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
71.13152pt\raise-5.8652pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
55.01067pt\raise-14.2263pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}},$

for all $k,\ell\geq 0$. This proves the theorem. ∎

### Integration over chains and Stokes’ theorem

Recall that for any manifold with corners $Q$ the presheaf
$(\Lambda^{\bullet}\Omega^{1}_{C^{\infty}_{Q}}),d,\wedge)$ of
$C^{\infty}$-algebraic de Rham forms agrees with the sheaf
$(\bm{\Upomega}^{\bullet}_{dR,Q},d,\wedge)$ of ordinary differential forms
(Lemma 8.6). Consequently sheafification of
$\Lambda^{\bullet}\Omega^{1}_{C^{\infty}_{Q}}$ does nothing and the possibly
mysterious sheaf
$\bm{\Upomega}^{\bullet}_{C^{\infty}_{Q}}=\mathsf{sh}(\Lambda^{\bullet}\Omega^{1}_{C^{\infty}_{Q}})$
turns out to be the sheaf $(\bm{\Upomega}^{\bullet}_{dR,Q},d,\wedge)$ of
ordinary de Rham differential forms.

######  Definition 8.14.

The standard $k$-simplex $\Delta^{k}$ is the space

$\Delta^{k}=\\{x\in\mathbb{R}^{k+1}\mid x_{i}\geq 0,x_{1}+\cdots+x_{k}=1\\}.$

The $k$-simplex is a manifold with corners, hence a local $C^{\infty}$-ringed
space $(\Delta^{k},C^{\infty}_{\Delta^{k}})$

The $i$th face of the $k$th simplex is the map
$d_{i}:\Delta^{k-1}\to\Delta^{k}$ given by

$d_{i}(x_{1},\ldots,x_{k-1})=(x_{1},\ldots,x_{i-1},0,x_{i},\ldots,x_{k-1}).$

######  Definition 8.15.

A $k$-simplex of a local $C^{\infty}$-ringed space $(M,\mathscr{A})$ is a map
$\sigma:(\Delta^{k},C^{\infty}_{\Delta^{k}})\to(M,\mathscr{A})$ of
$C^{\infty}$-ringed spaces.

We define the vector space $C_{k}(M,\mathscr{A};\mathbb{R})$ of real
$k$-chains of a local $C^{\infty}$-ringed space $(M,\mathscr{A})$ to be the
real vector space generated by the set of $k$-simplexes of $(M,\mathscr{A})$.
We will refer to elements of $C_{k}(M,\mathscr{A};\mathbb{R})$ a $k$-chains
and write them as finite formal sums $\sum n_{\sigma}\sigma$.

######  Definition 8.16.

The boundary $\partial\sigma$ of a $k$-simplex
$\sigma:(\Delta^{k},C^{\infty}_{\Delta^{k}})\to(M,\mathscr{A})$ is the chain
$\partial\sigma\in C_{k-1}(M,\mathscr{A};\mathbb{R})$ defined by

$\partial\sigma=\sum(-1)^{i}\sigma\circ d_{i}$

where $d_{i}:\Delta_{k-1}\to\Delta_{k}$, $i=1,\ldots,k$ are the standard face
maps: $d_{i}(x_{1},\ldots,x_{k-1})=(x_{1},\ldots,x_{i-1},0,x_{i},\ldots
x_{k-1})$.

######  Definition 8.17.

The boundary  $\partial\Delta^{k}$ of the standard $k$-simplex is the chain
$\partial(\operatorname{id})\in
C_{k-1}(\Delta^{k},C^{\infty}_{\Delta_{k}};\mathbb{R})$ where
$\operatorname{id}:(\Delta^{k},C^{\infty}_{\Delta_{k}})\to(\Delta^{k},C^{\infty}_{\Delta_{k}})$
is the identity $k$-simplex.

######  Definition 8.18.

Let $(M,\mathscr{A})$ be a local $C^{\infty}$-ringed space,
$\sigma:(\Delta^{k},C^{\infty}_{\Delta^{k}})\to(M,\mathscr{A})$ a $k$-simplex
and
$\omega\in\bm{\Upomega}^{k}_{\mathscr{A}}(M)=\mathsf{sh}(\Lambda^{k}\Omega^{1}_{\mathscr{A}})(M)$
a global $C^{\infty}$-algebraic differential form. Then, since by Lemma 8.6,
$\mathsf{sh}(\Lambda^{k}\Omega^{1}_{C^{\infty}_{\Delta^{k}}})=\bm{\Upomega}^{k}_{dR}(\Delta^{k})$,
the pullback $\sigma^{*}\omega$ is an ordinary $k$-form on the manifold with
corners $\Delta^{k}$. We define the integral of $\omega$ over $\sigma$ to be

$\int_{\sigma}\omega:=\int_{\Delta^{k}}\sigma^{*}\omega.$

We define the integral of $\omega$ over a $k$-chain $\sum n_{\sigma}\sigma\in
C_{k}(M,\mathscr{A};\mathbb{R})$ to be

$\int_{\sum n_{\sigma}\sigma}\omega:=\sum
n_{\sigma}\int_{\Delta^{k}}\sigma^{*}\omega.$

Recall Stokes’s theorem for simplices (see, for example, [Co, Theorem 8.2.9])
:

######  Theorem 8.19.

For any $k>0$ and for any differential $k-1$-form
$\gamma\in\bm{\Upomega}_{dR}(\Delta^{k})$ on the $k$-simplex $\Delta^{k}$

$\int_{\Delta_{k}}d\gamma=\int_{\partial\Delta_{k}}\gamma.$

As an immediate consequence we get

######  Theorem 8.20.

Let $(M,\mathscr{A})$ be a local $C^{\infty}$-ringed space,
$\gamma\in\bm{\Upomega}^{k-1}_{\mathscr{A}}(M)$ a global
$C^{\infty}$-algebraic de Rham $k-1$ form and
$\sigma:(\Delta^{k},C^{\infty}_{\Delta^{k}})\to(M,\mathscr{A})$ a $k$-simplex.
Then

$\int_{\sigma}d\gamma=\int_{\partial\sigma}\gamma.$

###### Proof.

The proof is a computation:

$\int_{\sigma}d\gamma=\int_{\Delta^{k}}\sigma^{*}(d\gamma)=\int_{\Delta^{k}}d(\sigma^{*}\gamma)=\int_{\partial\Delta^{k}}\sigma^{*}\gamma=\sum(-1)^{i}\int_{\Delta^{k-1}}d_{i}^{*}(\sigma^{*}\gamma)=\int_{\partial\sigma}\gamma.$

∎

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##  Appendix A Kähler differentials on concrete finitely presented
differential spaces

In the appendix we describe a class of local $C^{\infty}$-ringed spaces
$(M,\mathscr{A})$ for which the presheaf of Kähler differentials is
automatically a sheaf. These are concrete differential space
$(M,C^{\infty}(M))$ ((Definition 3.24) ) so that the ring $C^{\infty}(M)$ is
finitely presented — see Definition A.1 below.

######  Definition A.1.

A $C^{\infty}$-ring $\mathscr{A}$ is finitely presented if there is $n>0$ and
a finitely generated ideal $I\subset C^{\infty}(\mathbb{R}^{n})$ so that
$\mathscr{A}\simeq C^{\infty}(\mathbb{R}^{n})/I$.

######  Example A.2.

Let $M=\\{xy=0\\}\subset\mathbb{R}^{2}$. Then
$C^{\infty}(M)=C^{\infty}(\mathbb{R}^{2})|_{M}$ is finitely presented since
the ideal $I_{M}=\\{f\in\mathbb{R}^{2}\mid f|_{M}=0\\}$ is generated by the
function $xy$; see Example 6.22.

In order to have more examples we need to recall a lemma.

######  Lemma A.3.

Suppose $0$ is a regular value of a function a smooth function
$f=(f_{1},\ldots,f_{k}):M\to\mathbb{R}^{k}$ on a manifold $M$. Then the ideal
$I$ of functions that vanish on the embedded submanifold $f^{-1}(0)$ is
generated by the components $f_{1},\ldots,f_{k}$ of $f$.

###### Proof.

See, for example, [MR, p. 24 Lemma 2.1]. ∎

######  Lemma A.4.

Let $U\subset\mathbb{R}^{n}$ be an open set. Then the $C^{\infty}$-ring
$C^{\infty}(U)$ is finitely presented. In particular $C^{\infty}(U)\simeq
C^{\infty}(\mathbb{R}^{n+1})/\langle g\rangle$ for an appropriate choice of
$g\in C^{\infty}(\mathbb{R}^{n+1})$.

###### Proof.

By Lemma 3.4 there is a smooth function $f:\mathbb{R}^{n}\to[0,1]$ so that

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

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

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

The set $F(U)$ is the zero level set of the function
$g:\mathbb{R}^{n+1}\to\mathbb{R}$,

$g(x,y)=yf(x)-1.$

Hence by Lemma A.3 the ideal $I$ of functions in $\mathbb{R}^{n+1}$ that
vanish on $F(U)$ is generated by $g$: $I=\langle g\rangle$. Since
$C^{\infty}(U)\simeq C^{\infty}(F(U))=C^{\infty}(\mathbb{R}^{n+1})/\langle
g\rangle$ we are done. ∎

######  Lemma A.5.

Let $U\subset\mathbb{R}^{n}$ be open, $M\subset U$ closed in $U$ and suppose
there is a smooth map $r:U\to U$ so that $r(U)\subset M$ and
$r|_{M}=\operatorname{id}_{M}$, i.e., suppose $M$ is a smooth retract of $U$.
Then the $C^{\infty}$-ring $C^{\infty}(M)$ is finitely presented.

######  Remark A.6.

The lemma and its consequence, Example A.7 is an observation of Lawvere. See
[D, p. 687]. Note also that any retract of a manifold is a submanifold — see
[Mi, Theorem 1.15].

###### Proof of Lemma A.5.

By Lemma A.4 we have an exact sequence

$0\to J\to C^{\infty}(\mathbb{R}^{n+1})\xrightarrow{q}C^{\infty}(U)\to 0$

with the ideal $J$ generated by one element, call it $g$. Let
$a_{1}=q(x_{1}),\ldots,a_{n+1}=q(x_{n+1})$ denote the generators of
$C^{\infty}(U)$. Let $\imath:M\to U$ denote the inclusion. Then
$r\circ\imath=\operatorname{id}_{M}$. Since $M$ is closed in $U$,
$C^{\infty}(M)=C^{\infty}(U)/I_{M}$ where as before

$I_{M}=\ker(\imath^{*}:C^{\infty}(U)\to C^{\infty}(M))=\\{f\in
C^{\infty}(U)\mid f|_{M}=0\\}.$

We now argue that the set $\\{a_{i}-r^{*}\imath^{*}a_{i}\\}_{i=1}^{n+1}$
generates $I_{M}$. Consider $g\in I_{M}$. Then there is $f\in
C^{\infty}(\mathbb{R}^{n+1})$ so that $g=f(a_{1},\ldots,a_{n+1})$. By
Hadamard’s lemma there are functions $h_{1},\ldots,h_{n+1}\in
C^{\infty}(\mathbb{R}^{2(n+1)})$ so that

$f(z)-f(w)=\sum(z_{i}-w_{i})h_{i}(z,w)$

for all $z,w\in\mathbb{R}^{n+1}$. Since $g\in I_{M}$, $r^{*}\imath^{*}g=0$. On
the other hand

$r^{*}\imath^{*}g=r^{*}\imath^{*}(f(a_{1},\ldots,a_{n+1})=f(a_{1}\circ(\imath\circ
r),\ldots,a_{n+1}\circ(\imath\circ
r))=f(r^{*}\imath^{*}a_{1},\ldots,r^{*}\imath^{*}a_{n+1}).$

Consequently

$g=g-r^{*}\imath^{*}g=f(a_{1},\ldots,a_{n+1})-f(r^{*}\imath^{*}a_{1},\ldots,r^{*}\imath^{*}a_{n+1})=\sum(a_{i}-r^{*}\imath^{*}a_{i})h_{i}(a,r^{*}\imath^{*}a).$

Therefore the set $\\{a_{i}-r^{*}\imath^{*}a_{i}\\}_{i=1}^{n+1}$ generates the
ideal $I_{M}$ as claimed.

Finally
$C^{\infty}(M)=C^{\infty}(U)/I_{M}=(C^{\infty}(\mathbb{R}^{n+1}/\langle
g\rangle)/I_{M}$. Therefore $C^{\infty}(M)=C^{\infty}(\mathbb{R}^{n+1})/K$
where $K$ is generated by $n+2$ elements. ∎

######  Example A.7.

Let $M$ be a (second countable Hausdorff) manifold. Then the $C^{\infty}$-ring
$C^{\infty}(M)$ of smooth functions on $M$ is finitely presented.

###### Proof.

By the Whitney embedding theorem we may assume that $M$ is embedded in
$\mathbb{R}^{n}$ for some $n$. Let $U\subset\mathbb{R}^{n}$ be a tubular
neighborhood of $M$. Then $M$ is a retract of $U$. Hence by Lemma A.5 the
$C^{\infty}$-ring $C^{\infty}(M)$ is finitely presented. ∎

The next proposition shows that if a $C^{\infty}$-ring $\mathscr{A}$ is
finitely presented and $\mathscr{A}=C^{\infty}(\mathbb{R}^{n})/I$ then the
ideal $I$ has to be finitely generated, i.e., being finitely presented does
not depend on the choice of a presentation. The analogous result for
commutative rings and modules over commutative rings is well-known. It is less
well-known for $C^{\infty}$-rings. Our proof follows [J, Proposition 2.17].

######  Proposition A.8.

Let $\mathscr{A}$ be a finitely presented $C^{\infty}$-ring and
$p:C^{\infty}(\mathbb{R}^{n})\to\mathscr{A}$ a surjective map of
$C^{\infty}$-rings. Then $I=\ker p$ is finitely generated.

###### Proof.

Since $\mathscr{A}$ is finitely presented there is $m>0$ and a surjection
$\pi:C^{\infty}(\mathbb{R}^{m})\to\mathscr{A}$ with $J=\ker\pi$ finitely
generated. Since $p$ is surjective and $C^{\infty}(\mathbb{R}^{n})$ is free
there is a map $\varphi:C^{\infty}(\mathbb{R}^{m})\to
C^{\infty}(\mathbb{R}^{n})$ with $p\circ\varphi=\pi$. Similarly there is
$\psi:C^{\infty}(\mathbb{R}^{n})\to C^{\infty}(\mathbb{R}^{m})$ with
$\pi\circ\psi=p$.

Let $z_{1},\ldots,z_{\ell}$ denote the generators of $J$. We argue that $I$ is
generated by
$\\{\varphi(z_{1}),\ldots,\varphi(z_{\ell}),x_{1}-\varphi(\psi(x)),\ldots,x_{n}-\varphi(\psi(x_{n}))\\}$
where $x_{1},\ldots,x_{n}:\mathbb{R}^{n}\to\mathbb{R}$ are the canonical
generators of For any $a\in C^{\infty}(\mathbb{R}^{m})$

$a=\varphi(\psi(a))+(a-\varphi(\psi(a))).$

Now suppose $a\in I$. Then

$0=p(a)=\pi(\psi(a)).$

Hence $\psi(a)\in J$, $\varphi(\psi(a))\in\varphi(J)$ and so
$\varphi(\psi(a))$ can be written as a linear combination of
$\varphi(z_{1}),\ldots,\varphi(z_{\ell})$. It remains to show that
$a-\varphi(\psi(a))$ can be written as a linear combination of
$x_{1}-\varphi(\psi(x)),\ldots,x_{n}-\varphi(\psi(x_{n}))$.

Since $C^{\infty}(\mathbb{R}^{n})$ is free on $x_{1},\ldots,x_{n}$,

$a_{C^{\infty}(\mathbb{R}^{n})}(x_{1},\ldots,x_{n})=a$

for any $a\in C^{\infty}(\mathbb{R}^{n})$. Since $\varphi\circ\psi$ is a map
of $C^{\infty}$-rings

$(\varphi\circ\psi)(a)=(\varphi\circ\psi)(a_{C^{\infty}(\mathbb{R}^{n})}(x_{1},\ldots,x_{n}))=a_{C^{\infty}(\mathbb{R}^{n})}(\varphi\circ\psi)(x_{1}),\ldots,\varphi\circ\psi)(x_{n})).$

Next by Hadamard’s lemma for any $a\in C^{\infty}(\mathbb{R}^{n})$ there exist
$h_{1},\ldots,h_{n}\in C^{\infty}(\mathbb{R}^{2n})$ with

$a(x)-a(y)=\sum(x_{i}-y_{i})h_{i}(x,y)$

for all points $x,y\in\mathbb{R}^{n}$. Therefore

$\begin{split}((\operatorname{id}-\varphi\circ\psi)a)\,(x_{1},\ldots,x_{n})&=a(x_{1},\ldots,x_{n})-a((\varphi\circ\psi)(x_{1}),\ldots,(\varphi\circ\psi)(x_{n}))\\\
&=\sum(x_{i}-\varphi(\psi(x_{i}))h_{i}(x_{1},\ldots,x_{n},(\varphi\circ\psi)(x_{1}),\ldots,(\varphi\circ\psi)(x_{n}))\end{split}$

and we are done. ∎

######  Lemma A.9.

Let $(M,C^{\infty}_{M})$ be concrete differential space (Definition 3.24) with
the $C^{\infty}$-ring $C^{\infty}_{M}(M)$ finitely presented. Let $U\subset M$
be an open set. Then the $C^{\infty}$-ring $C^{\infty}_{M}(U)$ is finitely
presented.

###### Proof.

It is no loss of generality to assume that $M$ is a closed subset of
$\mathbb{R}^{n}$ for some $n$. Then the ideal $I_{M}=\\{f\in
C^{\infty}(\mathbb{R}^{n})\mid f|_{M}=0\\}$ is finitely generated. Since
$U\subset M$ is open there is an open subset $W$ of $\mathbb{R}^{n}$ with
$U=M\cap W$. Then $U$ is closed in $W$ and $C^{\infty}(U)=C^{\infty}(W)/I_{U}$
where $I_{U}=\\{f\in C^{\infty}(W)\mid f|_{U}=0\\}$. We want to show that the
ideal $I_{U}$ is finitely generated.

By the Localization Theorem, [MO], [NGSS, p. 28], for any $f\in C^{\infty}(W)$
there exist $h,g\in C^{\infty}(\mathbb{R}^{n})$ so that

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

and $h|_{W}$ is invertible in $C^{\infty}(U)$. If $f\in I_{U}$ then
$g|_{U}=0$. By Whitney’s theorem there is a smooth function $k\in
C^{\infty}(\mathbb{R}^{n})$ so that $\\{k=0\\}=M\smallsetminus W$. Then
$k|_{W}\not=0$ and $(kh|_{W})f=(kg)|_{W}$. Consequently $kg\in I_{M}$ and
$f=((kh)|_{W})^{-1}(kg)|_{W}$. It follows that if $z_{1},\ldots,z_{k}$
generate $I_{M}$ then their restrictions $z_{1}|_{W},\ldots,z_{k}|_{W}$
generate $I_{U}$. ∎

######  Definition A.10.

A concrete differential space $(M,C^{\infty}_{M})$ is finitely presented if
the $C^{\infty}$-ring $C^{\infty}_{M}(M)$ of global smooth functions is
finitely presented.

######  Remark A.11.

If $(M,C^{\infty}_{M})$ is a finitely presented concrete differential space
then for any open subset $U$ of $M$ the $C^{\infty}$-ring $C^{\infty}_{M}(U)$
is finitely presented as well.

We are now in position to state the main result of the appendix.

######  Theorem A.12.

Let $(M,C^{\infty}_{M})$ be a finitely presented concrete differential space.
Then the presheaf $\Omega^{1}_{C^{\infty}_{M}}$ of Kähler differentials on $M$
is a sheaf.

Note that Theorem A.12 has an easy corollary:

######  Corollary A.13.

The sheaf of ordinary 1-forms $\bm{\Upomega}_{M}^{1}$ on a manifold $M$ is a
sheaf of $C^{\infty}$-Kähler differentials $\Omega^{1}_{C^{\infty}_{M}}$ and
the universal derivation
$d_{C^{\infty}_{M}}:C^{\infty}_{M}\to\Omega^{1}_{C^{\infty}_{M}}$ is the
ordinary exterior derivative $d:C^{\infty}_{M}\to\bm{\Upomega}^{1}_{M}$ .

###### Proof.

By Example A.7 above the ring of smooth functions $C^{\infty}(M)$ on a
manifold $M$ is finitely presented. Consequently the presheaf of
$C^{\infty}$-Kähler differentials $\Omega^{1}_{C^{\infty}_{M}}$ on $M$ is a
sheaf. Locally a manifold is isomorphic to an open subset of $\mathbb{R}^{n}$
and for open subsets the sheafs of 1-forms and of $C^{\infty}$ Kähler
differentials coincide (see Example 6.12). Hence the sheaves $\Omega^{1}$ of
ordinary 1-forms and $\Omega^{1}_{C^{\infty}_{M}}$ of Kähler differentials
agree locally and therefore agree. ∎

We next prove a series of lemmas that we need for our proof of Theorem A.12.

######  Lemma A.14.

Let $N$ be a manifold, $C,D\subset N$ two closed subsets with $C\cap
D=\varnothing$. There is a smooth function $\tau:N\to\mathbb{R}$ with
$0\leq\tau\leq 1$ so that $\tau|_{C}\equiv 0$ and $\tau|_{D}\equiv 1$.

###### Proof.

Choose $\varphi_{1},\varphi_{2}\in C^{\infty}(N)$ with
$C=\\{\varphi_{1}=0\\}$, $D=\\{\varphi_{2}=0\\}$. We may assume that
$0\leq\varphi_{i}\leq 1$, $i=1,2$. Now let
$\tau=\frac{\varphi_{1}}{\varphi_{1}+\varphi_{2}}$. ∎

######  Lemma A.15.

Let $(M,C^{\infty}(M))$ be a concrete differential space,
$\\{U_{1},\ldots,U_{k}\\}$ a finite open cover of $M$, $a\in\Omega_{M}^{1}(M)$
with $a|_{U_{i}}=0$. Then $a=0$

###### Proof.

It is no loss of generality to assume that $M$ is a closed subset of
$\mathbb{R}^{n}$. Choose open sets $\tilde{U}_{i}\subset\mathbb{R}^{n}$ with
$\tilde{U}_{i}\cap M=U_{i}$. Let $\tilde{U}_{0}=\mathbb{R}^{n}\smallsetminus
M$. Choose $\tilde{a}\in\Omega^{1}(\mathbb{R}^{n})$ with $\tilde{a}|_{M}=a$.
We want to show that

$\tilde{a}\in J:=\ker\\{\Omega^{1}(\mathbb{R}^{n})\to\Omega_{M}^{1}(M)\\}.$

By Lemma 6.20,
$J=C^{\infty}(\mathbb{R}^{n})\,dI+I\,dC^{\infty}(\mathbb{R}^{n})$ where as
before $I=\ker\\{C^{\infty}(\mathbb{R}^{n})\to C^{\infty}(M)\\}=\\{f\in
C^{\infty}(\mathbb{R}^{n})\mid f|_{M}=0\\}$. For any index $i$

$\tilde{a}|_{U_{i}}=(\tilde{a}|_{M})|_{U_{i}}=a|_{U_{i}}=0.$

Since $\tilde{a}|_{U_{i}}=(\tilde{a}|_{\tilde{U}_{i}})|_{U_{i}}$,

$(\tilde{a}|_{\tilde{U}_{i}})\in
J_{i}:=\ker\\{\Omega^{1}(\tilde{U}_{i})\to\Omega_{M}^{1}(U_{i})\\}.$

By Lemma 6.21 $J_{i}=C^{\infty}(\tilde{U}_{i})\,dI_{i}$ where $I_{i}=\\{f\in
C^{\infty}(\tilde{U}_{i})\mid f|_{U_{i}}=0\\}$.

Let $\rho_{0},\rho_{1},\ldots,\rho_{k}\in C^{\infty}(\mathbb{R}^{n})$ be a
partition of 1 subordinate to the cover
$\\{\tilde{U}_{0},\ldots,\tilde{U}_{k}\\}$. Since $\rho_{0}\in I$,
$\rho_{0}\tilde{a}\in J$. Since
$\tilde{a}=\rho_{0}\tilde{a}+\sum_{i=1}^{k}\rho_{i}\tilde{a}$ it’s enough to
show that
$\sum_{i=1}^{k}\rho_{i}\tilde{a}=\sum_{i=1}^{k}\rho_{i}(\tilde{a}|_{\tilde{U}_{i}})$
is in $J$.

Fix an index $i$ and consider $\rho_{i}\cdot\tilde{a}|_{\tilde{U}_{i}}$. Since
$\tilde{a}|_{\tilde{U}_{i}}\in J_{i}$ there is $m>0$, $h_{1},\ldots,h_{m}\in
C^{\infty}(\tilde{U}_{i})$, $g_{1},\ldots,g_{m}\in I_{i}$ so that

$\tilde{a}|_{\tilde{U}_{i}}=\sum h_{j}\,dg_{j}.$

By Lemma A.14 we can choose $\tau\in C^{\infty}(\mathbb{R}^{n})$,
$0\leq\tau\leq 1$, with $\tau|_{\operatorname{supp}\rho_{i}}=1$ and
$\operatorname{supp}\tau\subset\tilde{U}_{i}$. Then

$\rho_{i}h_{j}\,dg_{j}=\rho_{i}h_{j}\,d(\tau g_{j})$

for each $j$. Note that $\tau g_{j}\in I$ since $\tau
g_{j}|_{\tilde{U_{i}}\cap M}=0$ and $\tau
g_{j}|_{M\smallsetminus\tilde{U}_{j}}=0$ as well. Hence

$\rho_{i}\tilde{a}=\sum h_{j}\rho_{i}\,dg_{j}=\sum h_{j}\rho_{i}\,d(\tau
g_{j})\in C^{\infty}(\mathbb{R}^{n})\,dI=J$

for all $i$. Therefore $\sum_{i=1}^{k}\rho_{i}\tilde{a}\in J$ and we are done.
∎

######  Lemma A.16.

Let $(M,C^{\infty}_{M})$ be a finitely presented concrete differential space.
Then for any Kähler differential $a\in\Omega^{1}_{C^{\infty}_{M}(M)}$ and any
open cover $\\{U_{\alpha}\\}_{\alpha\in A}$ of $M$

$a|_{U_{\alpha}}=0\quad\textrm{for all }\alpha\in A\qquad\Rightarrow\qquad
a=0.$

###### Proof.

It is no loss of generality to assume that $M$ is a closed subset of
$\mathbb{R}^{n}$ for some $n$. Then by Proposition A.8 the ideal the ideal
$I=\\{f\in C^{\infty}(\mathbb{R}^{n})|f|_{M}=0\\}$ is finitely generated.

The rest of the proof is similar to our proof of Lemma A.15. As before choose
open sets $\tilde{U}_{\alpha}\subset\mathbb{R}^{n}$ with
$\tilde{U}_{\alpha}\cap M=U_{\alpha}$. Let
$\tilde{U}_{0}=\mathbb{R}^{n}\smallsetminus M$. Choose
$\tilde{a}\in\Omega^{1}(\mathbb{R}^{n})$ with $\tilde{a}|_{M}=a$. We want to
show that

$\tilde{a}\in
J:=\ker\\{\Omega^{1}(\mathbb{R}^{n})\to\Omega_{M}^{1}(M)\\}=I\,dC^{\infty}(\mathbb{R}^{n})+C^{\infty}(\mathbb{R}^{n})\,dI.$

Note that for any $f\in I$ and $\gamma\in\Omega^{1}(\mathbb{R}^{n})$ the
product $f\gamma$ is in $J$ because

$(f\gamma)|_{M}=f|_{M}\gamma|_{M}=0\gamma|_{M}.$

Choose a finite set of generators $z_{1},\ldots,z_{k}$ of the ideal $I$.
Choose a partition of unity $\\{\rho_{0}\\}\cup\\{\rho_{\alpha}\\}_{\alpha\in
A}$ subordinate to the cover
$\\{\tilde{U}_{0}\\}\cup\\{\tilde{U}_{\alpha}\\}_{\alpha\in A}$ of
$\mathbb{R}^{n}$. We know that
$\tilde{a}=\rho_{0}\tilde{a}+\sum_{\alpha}\rho_{\alpha}\tilde{a}$ and that
$\rho_{0}\tilde{a}\in J$. We argue that there are
$\gamma_{1},\ldots,\gamma_{k}\in\Omega^{1}(\mathbb{R}^{n})$ and
$\ell_{1},\ldots,\ell_{k}\in C^{\infty}(\mathbb{R}^{n})$ so that

(A.1)
$\sum_{\alpha}\rho_{\alpha}\tilde{a}=\sum_{j}z_{j}\gamma_{j}+\sum_{j}\ell_{j}\,dz_{j}\in
I\Omega^{1}(\mathbb{R}^{n})+C^{\infty}(\mathbb{R}^{n})\,dI\subseteq
IdC^{\infty}(\mathbb{R}^{n})+C^{\infty}(\mathbb{R}^{n})\,dI=J.$

which is enough in order to prove that $\tilde{a}\in J$ and hence
$a=\tilde{a}|_{M}=0$.

Fix an index $\alpha$ and consider
$\rho_{\alpha}\tilde{a}=\rho_{\alpha}(\tilde{a}|_{\tilde{U}_{\alpha}})$. Since

$\tilde{a}|_{\tilde{U}_{\alpha}}\in
J_{\alpha}:=\ker\\{\Omega^{1}(\tilde{U}_{\alpha})\to\Omega^{1}_{M}(U_{\alpha})\\}$

there is $m>0$, $h_{1},\ldots,h_{m}\in C^{\infty}(\tilde{U}_{\alpha})$,
$g_{1},\ldots,g_{m}\in I_{\alpha}=\ker\\{C^{\infty}(\tilde{U}_{\alpha})\to
C^{\infty}(U_{\alpha})\\}$ so that

$\tilde{a}|_{\tilde{U}_{\alpha}}=\sum h_{i}\,dg_{i}$

(here we used Lemma 6.21). Choose $\tau\in C^{\infty}(\tilde{U}_{\alpha})$
with $0\leq\tau\leq 1$, $\operatorname{supp}\tau\subset\tilde{U}_{\alpha}$ and
$\tau|_{\operatorname{supp}\rho_{\alpha}}=1$. Then, as in the proof of Lemma
A.15

$\rho_{\alpha}\tilde{a}=\sum_{i}h_{i}\rho_{\alpha}\,d(\tau g_{i}),$

and $\tau g_{i}\in I$ for all $i$. Consequently

$\tau g_{i}=\sum_{j}c_{ij}z_{j}$

for some $c_{ij}\in C^{\infty}(\mathbb{R}^{n})$. Hence

$\rho_{\alpha}\tilde{a}=\sum_{i,j}\rho_{\alpha}h_{i}z_{j}\,dc_{ij}+\sum_{i,j}\rho_{\alpha}h_{i}c_{ij}\,dz_{j}.$

To reduce the clutter set $b_{j}^{\alpha}:=\sum h_{i}\,dc_{ij}$; these are
1-forms on $\mathbb{R}^{n}$. Similarly set $e^{\alpha}_{j}=\sum h_{i}c_{ij}$;
these are smooth functions on $\mathbb{R}^{n}$. With this notation

$\rho_{\alpha}\tilde{a}=\sum_{j}\rho_{\alpha}z_{j}\,b^{\alpha}_{j}+\sum_{j}\rho_{\alpha}e^{\alpha}_{j}\,dz_{j}.$

Consequently

$\sum_{\alpha}\rho_{\alpha}\tilde{a}=\sum_{j}z_{j}\left(\sum_{\alpha}\rho_{\alpha}b^{\alpha}_{j}\right)+\sum_{j}\left(\sum_{\alpha}\rho_{\alpha}e^{\alpha}_{j}\right)\,dz_{j},$

which is (A.1) with $\gamma_{j}=\sum_{\alpha}\rho_{\alpha}b^{\alpha}_{j}$ and
$\ell_{j}=\sum_{\alpha}\rho_{\alpha}e^{\alpha}_{j}$. So we are done.

∎

######  Lemma A.17.

Let $(M,C^{\infty}(M))$ be a finitely presented concrete differential space,
$\\{U_{\alpha}\\}_{\alpha\in A}$ an open cover of $M$,
$U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}$,
$\\{a_{\alpha}\in\Omega^{1}_{M}(U_{\alpha})\\}_{\alpha\in A}$ a family of
1-forms with $a_{\alpha}|_{U_{\alpha\beta}}=a_{\beta}|_{U_{\alpha\beta}}$ for
all $\alpha,\beta\in A$. Then there is a unique Kähler differential
$a\in\Omega^{1}_{C^{\infty}_{M}(M)}$ with $a|_{U_{\alpha}}=a_{\alpha}$.

###### Proof.

The uniqueness follows directly from Lemma A.16.

We now prove existence of the form $a$. It is no loss of generality to assume
that $M$ is a closed subset of some $\mathbb{R}^{n}$. For each $\alpha\in A$
choose an open set $\tilde{U}_{\alpha}$ in $\mathbb{R}^{n}$ so that
$\tilde{U}_{\alpha}\cap M=U_{\alpha}$. Let
$\tilde{U}_{0}=\mathbb{R}^{n}\smallsetminus M$ (we are tacitly assuming that
$0\not\in A$). Then
$\\{\tilde{U}_{0}\\}\cup\\{\tilde{U}_{\alpha}\\}_{\alpha\in A}$ is an open
cover of $\mathbb{R}^{n}$. Let
$\\{\rho_{0}\\}\cup\\{\rho_{\alpha}\\}_{\alpha\in A}$ be a partition of 1
subordinate to the cover (so that
$\operatorname{supp}\rho_{0}\subseteq\mathbb{R}^{n}\smallsetminus M$ and
$\operatorname{supp}\rho_{\alpha}\subseteq\tilde{U}_{\alpha}$ for all
$\alpha$). For each $a_{\alpha}\in\Omega^{1}_{M}(U_{\alpha})$ choose
$\tilde{a}_{\alpha}\in\Omega^{1}_{C^{\infty}(\tilde{U}_{\alpha})}$. We have
seen that the module $\Omega^{1}_{C^{\infty}(\tilde{U}_{\alpha})}$ of
$C^{\infty}$-Kähler differentials is the module
$\Omega^{1}(\tilde{U}_{\alpha})$ of ordinary 1-forms on the open set
$\tilde{U}_{\alpha}\subseteq\mathbb{R}^{n}$. Let

$\tilde{a}:=\sum_{\alpha\in
A}\rho_{\alpha}\tilde{a}_{\alpha}\qquad\textrm{and}\qquad a:=\tilde{a}|_{M}.$

We need to show that

$a|_{U_{\alpha}}=a_{\alpha}.$

Since the diagram

$\textstyle{\Omega^{1}(\mathbb{R}^{n})}$$\textstyle{\Omega^{1}(\tilde{U}_{\alpha})}$$\textstyle{\Omega^{1}_{M}(M)}$$\textstyle{\Omega^{1}_{M}(U_{\alpha})}$

commutes, it’s enough to show that

$\tilde{a}|_{U_{\alpha}}=a_{\alpha}.$

Claim

$(\rho_{\beta}\tilde{a}_{\beta})|_{U_{\alpha}}=(\rho_{\beta}\tilde{a}_{\alpha})|_{U_{\alpha}}.$

###### Proof of claim.

As before we set $U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}$. Note that

$U_{\alpha}=(U_{\alpha}\cap
U_{\beta})\cup(U_{\alpha}\smallsetminus\tilde{U}_{\beta})=U_{\alpha\beta}\cup(U_{\alpha}\smallsetminus\operatorname{supp}\rho_{\beta}).$

Now
$(\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha})|_{U_{\alpha\beta}}=\rho_{\beta}|_{U_{\alpha\beta}}\cdot(a_{\beta}|_{U_{\alpha\beta}}-a_{\alpha}|_{U_{\alpha\beta}})=0$.
On the other hand
$\rho_{\beta}|_{U_{\alpha}\smallsetminus\operatorname{supp}\rho_{\beta}}=0$ so
$(\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha})|_{U_{\alpha}\smallsetminus\operatorname{supp}\rho_{\beta}}=0$
as well. Since
$\\{U_{\alpha\beta},U_{\alpha}\smallsetminus\operatorname{supp}\rho_{\beta}\\}$
is a finite cover the Claim follows by Lemma A.15 ∎

It follows fron the claim above that for any finite collection $J\subset A$

$\left.\left(\sum_{\beta\in
J}\rho_{\beta}\tilde{a}_{\beta}\right)\right|_{U_{\alpha}}=\left.\left(\sum_{\beta\in
J}\rho_{\beta}\tilde{a}_{\alpha}\right)\right|_{U_{\alpha}}=(\sum_{\beta\in
J}\rho_{\beta})|_{U_{\alpha}}a_{\alpha}.$

Note that for any open $V\subset U_{\alpha}$

$\left(\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha}\right)|_{V}=\left.\left(\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha}\right)|_{U_{\alpha}}\right|_{V}=0|_{V}=0.$

Note next that for any $x\in\tilde{U}_{\alpha}$ there is an open neighborhood
$V_{x}$ of $x$ so that $J_{x}:=\\{\beta\in
A\mid\operatorname{supp}\rho_{\beta}\cap V_{x}\not=\varnothing\\}$ is finite.
And then

$\begin{split}\left.\left(\sum_{\beta\in
A}\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha}\right)\right|_{U_{\alpha}\cap
V_{x}}=\left.\left(\sum_{\beta\in
J_{x}}\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha}\right)\right|_{U_{\alpha}\cap
V_{x}}=0.\end{split}$

We thus have an open cover $\\{V_{x}\\}_{x\in\tilde{U_{\alpha}}}$ of
$\tilde{U}_{\alpha}$ and a 1-form $\gamma=\left(\sum_{\beta\in
A}\rho_{\beta}\tilde{a}_{\beta}-\rho_{\beta}\tilde{a}_{\alpha}\right)$ so that
$\gamma|_{U_{\alpha}\cap V_{x}}=0$. Since $C^{\infty}(M)$ is finitely
presented and $U_{\alpha}\subset M$ is open the $C^{\infty}$-ring
$C^{\infty}(U_{\alpha})$ is finitely presented by Lemma A.9. It follows from
Lemma A.15 that $\gamma$ has to be zero. Therefore

$\left(\sum_{\beta\in
A}\rho_{\beta}\tilde{a}_{\beta}\right)|_{U_{\alpha}}=\left(\sum_{\beta\in
A}\rho_{\beta}\tilde{a}_{\alpha}\right)|_{U_{\alpha}}=((\sum_{\beta\in
A}\rho_{\beta})\tilde{a}_{\alpha})|_{U_{\alpha}}=\tilde{a}_{\alpha}|_{U_{\alpha}}=a_{\alpha}$

and we are done. ∎

###### Proof of Theorem A.12.

Let $U\subset M$ be an open set, $\\{U_{\alpha}\\}_{\alpha\in A}$ an open
cover of $U$, $U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}$,
$\\{a_{\alpha}\in\Omega^{1}_{M}(U_{\alpha})\\}_{\alpha\in A}$ a family of
1-forms with $a_{\alpha}|_{U_{\alpha\beta}}=a_{\beta}|_{U_{\alpha\beta}}$ for
all $\alpha,\beta\in A$. We want to show that there is a unique
$a\in\Omega^{1}_{M}(U)$ with $a|_{U_{\alpha}}=a_{\alpha}$.

The open subset $U$ of $M$ with its induced smooth structure $C^{\infty}(U)$
is a concrete differential space by Lemma 3.26. By Lemma A.9 $C^{\infty}(U)$
is finitely presented. By Lemma A.17 there is a unique $a\in\Omega^{1}_{M}(U)$
with $a|_{U_{\alpha}}=a_{\alpha}$. Hence $\Omega^{1}_{M}$ is a sheaf as
claimed. ∎

######  Remark A.18.

Joyce observes on p. 15 of [J] that if $X\subset\mathbb{R}^{n}$ is a closed
subset whose interior (in $\mathbb{R}^{n}$) is dense in $X$, then the ideal
$I_{X}:=\\{f\in C^{\infty}(\mathbb{R}^{n})\mid f|_{X}=0\\}$ cannot be finitely
generated. The observation relies on the results of Moerdijk and Reyes [MR] on
the so called flat ideals. Consequently the $C^{\infty}$-ring
$C^{\infty}(X):=C^{\infty}(\mathbb{R}^{n})/I_{X}$ is not finitely presented
and the results of the appendix do not apply. In particular the ring of smooth
functions on the standard simplex $\Delta^{k}$ is not finitely presented since

$\Delta^{k}=\\{x\in\mathbb{R}^{k}\mid x_{i}\geq 0,\sum x_{i}\leq 1\\}.$

None the less the presheaf of Kähler differentials
$\Omega^{1}_{C^{\infty}_{\Delta^{k}}}$ is a sheaf [FS, Proposition 4.7.5] .