# The de Rham cohomology
of soft function algebras

Igor Baskov

###### Abstract.

We study the dg-algebra $\Omega^{\bullet}_{A|\mathbb{R}}$ of algebraic de Rham
forms of a real soft function algebra $A$, i.e., the algebra of global
sections of a soft subsheaf of $C_{X}$, the sheaf of continuous functions on a
space $X$. We obtain a canonical splitting
$\mathrm{H}^{n}(\Omega^{\bullet}_{A|\mathbb{R}})\cong\mathrm{H}^{n}(X,\mathbb{R})\oplus
V$, where $V$ is some vector space. In particular, we consider the cases
$A=C(X)$ for $X$ a compact Hausdorff space and $A=C^{\infty}(X)$ for $X$ a
compact smooth manifold. For the algebra $\mathrm{PPol}_{K}(|K|)$ of piecewise
polynomial functions on a polyhedron $K$ the above splitting reduces to a
canonical isomorphism
$\mathrm{H}^{*}(\Omega^{\bullet}_{\mathrm{PPol}_{K}(|K|)|\mathbb{R}})\cong\mathrm{H}^{*}(|K|,\mathbb{R})$.
We also prove that the algebraic de Rham cohomology
$\mathrm{H}^{n}(\Omega^{\bullet}_{C(X)|\mathbb{R}})$ is nontrivial for each
$n\geq 1$ if $X$ is an infinite compact Hausdorff space.

This research was supported by the Ministry of Science and Higher Education of
the Russian Federation, agreement 075-15-2019-1620 date 08/11/2019 and
075-15-2022-289 date 06/04/2022.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Sheaves
    2. 2.2 Algebraic sets
    3. 2.3 Sheaves of singular cochains
    4. 2.4 The de Rham dg-algebra $\Omega^{\bullet}_{A|k}$ of $A$
  3. 3 The map $\Lambda_{\mathcal{F}}:\mathbb{H}^{*}(X,\underline{k}_{X}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k})$
    1. 3.1 The global sections of ${}^{+}\Omega^{n}_{\mathcal{F}|k}$
    2. 3.2 The construction and naturality of $\Lambda_{\mathcal{F}}:\mathbb{H}^{*}(X,\underline{k}_{X}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k})$
  4. 4 Splitting for the algebra of smooth functions
    1. 4.1 Canonical isomorphism $\Theta$
    2. 4.2 Morphisms $\pi$ and $\underline{\pi}$
    3. 4.3 Main splitting theorem
  5. 5 Simplicial dg-algebra of flat cochains $\Omega^{\bullet}_{\flat}(\Delta^{-})$
    1. 5.1 Flat cochains
    2. 5.2 The attributes of the simplicial dg-algebra $\Omega^{\bullet}_{\flat}(\Delta^{-})$
  6. 6 The map $\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$
    1. 6.1 The pullback of an algebraic de Rham form
    2. 6.2 The map $\xi_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B)$
    3. 6.3 The map $\Phi_{B}:\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})\to\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])$
    4. 6.4 The map $\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$
  7. 7 Identifying the map $\Psi_{C^{\infty}(M)}$
    1. 7.1 The map $\xi^{\mathrm{sm}}_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B)$
    2. 7.2 Comparing $\xi_{B}$ and $\xi^{\mathrm{sm}}_{B}$
    3. 7.3 Morphisms $\varkappa$ and $\underline{\varkappa}$
    4. 7.4 Identifying the map $\Psi_{B}$
    5. 7.5 The calculation of $\Psi_{C^{\infty}(M)}$
  8. 8 The composition $\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])\xrightarrow{\Lambda_{\mathcal{F}}}\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}})\xrightarrow{\Psi_{\mathcal{F}(X)}}\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$
  9. 9 Piecewise polynomial functions
    1. 9.1 Polyhedra and rectilinear maps
    2. 9.2 The maps $\Lambda_{\mathrm{PPol}_{K}}$ and $\Psi_{\mathrm{PPol}_{K}(|K|)|\mathbb{R}}$
  10. 10 When are the maps $\Lambda$ and $\Psi$ isomorphisms?
    1. 10.1 $\mathrm{H}^{0}(\Omega^{\bullet}_{A|k})$ for a function algebra $A$
    2. 10.2 $\Lambda$ and $\Psi$ are not isomorphisms in general

##  1\. Introduction

All algebras are assumed to be commutative and all dg-algebras are assumed to
be graded-commutative. To an algebra $A$ over a field $k$ one associates a dg-
algebra $\Omega^{\bullet}_{A|k}$ with $\Omega^{0}_{A|k}=A$, called the de Rham
dg-algebra, in a standard way (see Subsection 2.4 or [15, $\S 3$]). Our main
focus will be the cases of $\mathbb{R}$-algebras $A=C(X)$ for a compact
Hausdorff space $X$, $A=C^{\infty}(M)$ for a smooth manifold $M$ (possibly
with boundary) and $A=\mathrm{PPol}_{K}(|K|)$, the algebra of piecewise
polynomial functions on a polyhedron $K$. Here all functions are assumed to be
real-valued, but our results will hold for complex-valued functions also. We
study the cohomology groups $\mathrm{H}^{*}(\Omega^{\bullet}_{A|k})$.

There is a canonical morphism of dg-algebras
$\pi:\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}}\to\Omega^{\bullet}(M)$, where
$\Omega^{\bullet}(M)$ is the dg-algebra of smooth differential
$\mathbb{R}$-forms on $M$ (see Subsection 4.2). The morphism $\pi$ is the
identity in degree $0$. It is not an isomorphism. For example, the equality
$df(t)=(\partial f/\partial t)dt$ holds in
$\Omega^{1}_{C^{\infty}(\mathbb{R})|\mathbb{R}}$ if and only if $f(t)$ is an
algebraic function of $t$ ([19, Corollary of Proposition $1$]). Moreover, if
$M$ is of dimension $\geq 1$, then the cardinality of any set of generators
for $\Omega^{1}_{C^{\infty}(M)|\mathbb{R}}$ as a $C^{\infty}(M)$-module is at
least that of $\mathbb{R}$ ([8, Corollary $15$]). The map on cohomology,
induced by $\pi$, is not an isomorphism either, see [19] for the proof that
the closed form $dt/(1+t^{2})$ is not exact in
$\Omega^{\bullet}_{C^{\infty}(\mathbb{R})|\mathbb{R}}$. On the other hand,
consider the algebra $A$ of regular functions on a smooth affine variety $V$
over $\mathbb{C}$. It is a subalgebra of $C^{\infty}(V,\mathbb{C})$ with
inclusion $i:A\hookrightarrow C^{\infty}(V,\mathbb{C})$. Consider the
composition

$\Omega^{\bullet}_{A|\mathbb{C}}\xrightarrow{\Omega_{i}}\Omega^{\bullet}_{C^{\infty}(V)|\mathbb{C}}\xrightarrow{\pi_{\mathbb{C}}}\Omega^{\bullet}(V,\mathbb{C}),$

where $\pi_{\mathbb{C}}$ is the analogue of $\pi$ over $\mathbb{C}$.
Grothendieck’s comparison theorem states that the induced map
$\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{C}})\to\mathrm{H}^{*}(\Omega^{\bullet}(V,\mathbb{C}))$
is an isomorphism ([10, Theorem $1^{\prime}$]).

For a soft sheaf of $k$-algebras $\mathcal{F}$ on a compact Hausdorff space
$X$ we construct (see Section 3) a linear map

$\Lambda_{\mathcal{F}}:\mathbb{H}^{*}(X,\underline{k}_{X}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k}).$

Here the domain is the cohomology of $X$ with coefficients in the constant
sheaf $\underline{k}_{X}$. This map is natural with respect to morphisms of
ringed spaces (see Proposition 12). We are mostly interested in the case
$k=\mathbb{R}$.

We prove (see Theorem 13) that for the sheaf of smooth functions
$C^{\infty}_{M}$ on a smooth manifold $M$ the following diagram is
commutative:

${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M)),}$$\scriptstyle{\Theta}$$\scriptstyle{\Lambda_{C^{\infty}_{M}}}$$\scriptstyle{\mathrm{H}(\pi)}$

where $\Theta$ is the canonical isomorphism (see Subsection 4.1). In
particular, the map

$\mathrm{H}(\pi):\mathrm{H}^{n}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})\to\mathrm{H}^{n}(\Omega^{\bullet}(M))$

is surjective for $n\geq 0$. This is a generalization of a result obtained by
Gómez, namely that $\mathrm{H}(\pi)$ is surjective for $n$ even (see [7,
4.Theorem]).

Next, for an arbitrary space $X$ and a subalgebra $i:A\hookrightarrow C(X)$,
we construct a linear map (see Section 6)

$\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0]).$

Our construction of $\Psi_{A}$ relies on local Lipschitz contractibility of
algebraic sets (Theorem 4) due to Shartser ([22, Theorem $4.18$]). This map is
natural with respect to continuous maps of spaces covered by a homomorphism of
algebras (see Proposition 26); in particular,
$\Psi_{A}=\Psi_{C(X)}\circ\Omega_{i}$. We prove (see Theorem 40) that for a
compact Hausdorff space $X$ and a soft subsheaf $\mathcal{F}$ of $C_{X}$, the
sheaf of continuous functions, the composition
$\Psi_{\mathcal{F}(X)}\circ\Lambda_{\mathcal{F}}$ coincides with the identity
map. Thus, the groups $\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$
canonically split off of $\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)})$.

We also check (see Theorem 36) that for a smooth manifold $M$ the following
diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}(M)).}$$\scriptstyle{\Psi_{C^{\infty}(M)}}$$\scriptstyle{\mathrm{H}(\pi)}$$\scriptstyle{\Theta}$

For the sheaf of piecewise polynomial functions $\mathrm{PPol}_{K}$ on a
polyhedron $K$ (see Section 9) we prove that the morphisms
$\Lambda_{\mathrm{PPol}_{K}}$ and $\Psi_{\mathrm{PPol}_{K}(|K|)}$ are
isomorphisms (see Theorem 51).

In Section 10 we describe the group $\mathrm{H}^{0}(\Omega^{\bullet}_{A|k})$
for a general function algebra $A$ over a field $k$ of characteristic $0$
(Corollary 56) calculated by Gómez in [8]. The related result is [19,
Proposition $5$] (note that the cohomology considered there is, in general,
different from ours). We show that for a soft subsheaf of algebras
$\mathcal{F}\subset C_{X}$ on a compact Hausdorff space $X$ the morphisms
$\Lambda_{\mathcal{F}}$ and $\Psi_{\mathcal{F}(X)}$ are isomorphisms in degree
$0$ (Proposition 57 and Corollary 58). We also prove that for an infinite
compact Hausdorff space $X$ the maps
$\Lambda_{C_{X}}:\mathbb{H}^{n}(X,\underline{\mathbb{R}}_{X}[0])\to\mathrm{H}^{n}(\Omega^{\bullet}_{C(X)|\mathbb{R}})$
and
$\Psi_{C(X)}:\mathrm{H}^{n}(\Omega^{\bullet}_{C(X)|\mathbb{R}})\to\mathbb{H}^{n}(X,\underline{\mathbb{R}}_{X}[0])$
are not isomorphisms in degrees $n>0$. The same is true for the algebra of
smooth functions on a smooth manifold, see Subsection 10.2.

From our results one can deduce the similar results for $\mathbb{C}$-algebras.

In the paper we only consider algebraic structures, however one can consider
topological algebras. Using projective tensor products, one can define the dg-
algebra $\tilde{\Omega}^{\bullet}_{\mathcal{A}|\mathbb{C}}$ for a Fréchet
$\mathbb{C}$-algebra $\mathcal{A}$ (denoted by
$\Omega^{\bullet}_{\mathrm{ab}}\mathcal{A}$ in [9, $\S 8$]). This dg-algebra
is a topological analogue of the de Rham dg-algebra.

Consider the Fréchet algebra $\mathcal{A}=C^{\infty}(M,\mathbb{C})$ for a
compact smooth manifold $M$. The dg-algebra
$\tilde{\Omega}^{\bullet}_{\mathcal{A}|\mathbb{C}}$ is isomorphic to
$\Omega^{\bullet}(M,\mathbb{C})$, see [9, Proposition $8.1$].

Consider the Banach algebra $\mathcal{A}=C(X,\mathbb{C})$ for a compact
Hausdorff space $X$. Then $\tilde{\Omega}^{n}_{\mathcal{A}|\mathbb{C}}=0$ for
$n\geq 1$. To see that, first note that by [14, $\S 8$] or [4, Remarks $47$,
d], the continuous Hochschild cohomology
$\mathrm{H}\mathrm{H}^{n}(\mathcal{A},\mathcal{A}^{*})$ is zero for positive
$n$. Then, by [14, Corollary $1.3$] the continuous Hochschild homology of
$\mathcal{A}$ is zero in positive degrees, in particular,
$\mathrm{H}\mathrm{H}_{1}(\mathcal{A})=0$. By [9, p. 346], we have
$\mathrm{H}\mathrm{H}_{1}(\mathcal{A})\cong\tilde{\Omega}^{1}_{\mathcal{A}|\mathbb{C}}$.
Hence, by the construction of $\tilde{\Omega}^{n}_{\mathcal{A}|\mathbb{C}}$
this space is zero for $n\geq 1$.

### Acknowledgement

I would like to thank Dr. Semёn Podkorytov for his patience, numerous fruitful
discussions and help drafting this paper. I am grateful to the St. Petersburg
Department of Steklov Mathematical Institute of Russian Academy of Sciences
for their financial assistence.

##  2\. Preliminaries

###  2.1. Sheaves

Here we outline the basic definitions and facts of sheaf theory needed in the
paper. For this we follow the books of Godement [6], Wedhorn [24] and Bredon
[2].

We refer to [24, Definition $10.2$] for the definition of the hypercohomology
groups $\mathbb{H}^{*}(X,\mathcal{F}^{\bullet})$ of a complex of sheaves
$\mathcal{F}^{\bullet}$ on a space $X$. By a complex we always mean a non-
negative cochain complex. For a sheaf $\mathcal{F}$ we denote by
$\mathcal{F}[0]$ the complex of sheaves with $\mathcal{F}$ in degree $0$ and
other terms zero.

Let $\mathcal{F}^{\bullet}$ be a complex of sheaves on $X$. Then there is a
canonical homomorphism

$\Upsilon:\mathrm{H}^{*}(\mathcal{F}^{\bullet}(X))\to\mathbb{H}^{*}(X,\mathcal{F}^{\bullet}),$

natural with respect to morphisms of complexes of sheaves. The map $\Upsilon$
is an isomorphism in degree $0$. If the sheaves $\mathcal{F}^{n}$ are acyclic
then $\Upsilon$ is an isomorphism, see [24, Theorem and Definition $10.4$,
Proposition $10.8$].

######  Lemma 1.

Take complexes of sheaves $\mathcal{F}^{\bullet}$ and $\mathcal{G}^{\bullet}$
on topological spaces $X$ and $Y$, respectively. Suppose $f:X\to Y$ is a
continuous map and $\varphi:\mathcal{G}^{\bullet}\to
f_{*}\mathcal{F}^{\bullet}$ is a morphism of complexes of sheaves. Then there
is the induced map on hypercohomology
$f^{*}:\mathbb{H}(Y,\mathcal{G}^{\bullet})\to\mathbb{H}(X,\mathcal{F}^{\bullet})$.
Moreover, the following diagram is commutative:

${\mathbb{H}^{*}(X,\mathcal{F}^{\bullet})}$${\mathrm{H}^{*}(\mathcal{F}^{\bullet}(X))}$${\mathbb{H}^{*}(Y,\mathcal{G}^{\bullet})}$${\mathrm{H}^{*}(\mathcal{G}^{\bullet}(Y)).}$$\scriptstyle{\Upsilon}$$\scriptstyle{f^{*}}$$\scriptstyle{\Upsilon}$$\scriptstyle{\mathrm{H}(\varphi(Y))}$

###### Proof.

See [23]. ∎

For a sheaf $\mathcal{F}$ on $X$ and a closed subset $Z\subset X$ define

$\mathcal{F}(Z):=\underset{\text{open}\,U\supset
Z}{\underset{\longrightarrow}{\operatorname{lim\,}}}\mathcal{F}(U).$

For a sheaf $\mathcal{F}$ we fix the notation for the restriction map
$\mathrm{res}_{W,W^{\prime}}:\mathcal{F}(W)\to\mathcal{F}(W^{\prime})$ for
sets $W^{\prime}\subset W$ open or closed.

A sheaf $\mathcal{F}$ is called soft if for every closed set $Z\subset X$ the
restriction map $\mathrm{res}_{X,Z}$ is surjective. Any soft sheaf on a
compact Hausdorff space is acyclic [24, Proposition $10.17$]. A sheaf
$\mathcal{F}$ is called flabby if for every open set $U\subset X$ the
restriction map $\mathcal{F}(X)\to\mathcal{F}(U)$ is surjective. A flabby
sheaf is acyclic.

For a presheaf $F$ denote by ${}^{+}F$ the sheafification of $F$. Let
$\mathrm{sh}:F\to{}^{+}F$ denote the sheafification map.

###  2.2. Algebraic sets

An algebraic set in $\mathbb{R}^{m}$ is the set of solutions of a system of
polynomial equations in $\mathbb{R}^{m}$.

######  Definition.

Take $B$ a finitely generated $\mathbb{R}$-algebra. Following [18, $3.4$], we
define the real spectrum $\mathrm{spec}_{\mathbb{R}}B$ of $B$ as the set of
algebra homomorphisms $\psi:B\to\mathbb{R}$.

A set of generators $b_{1},\dots,b_{n}$ of $B$ gives rise to an injective map
$i:\mathrm{spec}_{\mathbb{R}}B\hookrightarrow\mathbb{R}^{m}$,
$i(\psi):=(\psi(b_{1}),\dots,\psi(b_{n}))$. We call such a map a distinguished
embedding. The image of a distinguished embedding is an algebraic set. Equip
$\mathrm{spec}_{\mathbb{R}}B$ with the induced topology under some
distinguished embedding. This topology does not depend on the choice of a
distinguished embedding because for any two distinguished embeddings
$i:\mathrm{spec}_{\mathbb{R}}B\hookrightarrow\mathbb{R}^{m}$ and
$j:\mathrm{spec}_{\mathbb{R}}B\hookrightarrow\mathbb{R}^{l}$ there exists a
polynomial map $p:\mathbb{R}^{m}\to\mathbb{R}^{l}$ such that the following
diagram is commutative

${\mathrm{spec}_{\mathbb{R}}B}$${\mathbb{R}^{m}}$${\mathbb{R}^{l}.}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{p}$

A polynomial map is smooth, hence, we can talk of maps into
$\mathrm{spec}_{\mathbb{R}}B$ being smooth or locally Lipschitz. Hence,
$\mathrm{spec}_{\mathbb{R}}B$ becomes a predifferentiable space (see [3, $\S
1$] for the definition).

If $\varphi:B^{\prime}\to B$ is a homomorphism of finitely generated
$\mathbb{R}$-algebras, then define
$\mathrm{spec}_{\mathbb{R}}\varphi:\mathrm{spec}_{\mathbb{R}}B\to\mathrm{spec}_{\mathbb{R}}B^{\prime}$
as $(\mathrm{spec}_{\mathbb{R}}\varphi)(\psi):=\psi\circ\varphi$. For $b\in B$
define the function $\hat{b}\in C(\mathrm{spec}_{\mathbb{R}}B)$ by
$\hat{b}(\psi):=\psi(b)$. We have
$\widehat{\varphi(c)}=\widehat{c}\circ(\mathrm{spec}_{\mathbb{R}}\varphi)$ in
$C(\mathrm{spec}_{\mathbb{R}}B)$ for $c\in B^{\prime}$.

Now, let’s assume that $B$ is a finitely generated $\mathbb{R}$-subalgebra of
$C(X)$ for some topological space $X$. To each point $x\in X$ we associate the
homomorphism $B\to\mathbb{R}$, $b\mapsto b(x)$. We get a continuous map
$\Gamma_{B}:X\to\mathrm{spec}_{\mathbb{R}}B$. If $M$ is a smooth manifold and
$B$ is a finitely generated subalgebra of $C^{\infty}(M)$ then $\Gamma_{B}$ is
smooth.

For $b\in B$ we have

(1) $\hat{b}\circ\Gamma_{B}=b.$

Take $f:X\to Y$ a continuous map of topological spaces. Suppose that $B\subset
C(X)$ and $B^{\prime}\subset C(Y)$ are finitely generated subalgebras and
$\varphi:B^{\prime}\to B$ is a homomorphism such that the following diagram is
commutative:

${C(X)}$${B}$${C(Y)}$${B^{\prime}.}$$\scriptstyle{f^{*}}$$\scriptstyle{\varphi}$

Then the following diagram is commutative:

${X}$${\mathrm{spec}_{\mathbb{R}}B}$${Y}$${\mathrm{spec}_{\mathbb{R}}B^{\prime}.}$$\scriptstyle{\Gamma_{B}}$$\scriptstyle{f}$$\scriptstyle{\mathrm{spec}_{\mathbb{R}}\varphi}$$\scriptstyle{\Gamma_{B^{\prime}}}$

###  2.3. Sheaves of singular cochains

For a topological space $X$ let $\mathrm{S}_{n}(X)$ denote the space of
singular $\mathbb{R}$-chains in $X$ and let $\mathrm{S}^{n}(X)$ denote the
dual space, the space of singular cochains. We will need the following
presheaves $\mathrm{S}^{n}_{X}$ for $n\geq 0$:

$U\mapsto\mathrm{S}^{n}(U).$

The standard differential on cochains gives rise to a complex of presheaves
$\mathrm{S}^{\bullet}_{X}$ on $X$. For a smooth manifold $M$ (possibly with
boundary) let $\mathrm{S}_{\mathrm{sm},n}(M)$ denote the space of smooth
singular $\mathbb{R}$-chains in $M$ and let $\mathrm{S}^{n}_{\mathrm{sm}}(M)$
denote the dual space. We will need the following presheaves
$\mathrm{S}^{n}_{\mathrm{sm},M}$ for $n\geq 0$:

$U\mapsto\mathrm{S}^{n}_{\mathrm{sm}}(U).$

The standard differential on cochains gives rise to a complex of presheaves
$\mathrm{S}^{\bullet}_{\mathrm{sm},M}$ on $M$. More generally, the complex of
presheaves $\mathrm{S}^{\bullet}_{\mathrm{sm},Q}$ is well defined for $Q$ a
predifferentiable space and functorial with respect to smooth maps between
predifferentiable spaces. In particular,
$\mathrm{S}^{\bullet}_{\mathrm{sm},\mathrm{spec}_{\mathbb{R}}B}$ are well
defined.

For a set $V\subset\mathbb{R}^{m}$ let $\mathrm{S}_{\mathrm{Lip},n}(V)$ denote
the space of Lipschitz singular $\mathbb{R}$-chains in $V$ and let
$\mathrm{S}^{n}_{\mathrm{Lip}}(V)$ denote the dual space. We will need the
following presheaves $\mathrm{S}^{n}_{\mathrm{Lip},V}$ for $n\geq 0$:

$U\mapsto\mathrm{S}^{n}_{\mathrm{Lip}}(U).$

The standard differential on cochains gives rise to a complex of presheaves
$\mathrm{S}^{\bullet}_{\mathrm{Lip},V}$ on $V$. The complex of presheaves
$\mathrm{S}^{\bullet}_{\mathrm{Lip},\mathrm{spec}_{\mathbb{R}}B}$ is well
defined and functorial with respect to algebra homomorphisms.

The sheaves ${}^{+}\mathrm{S}^{n}_{X}$ are flabby by [24, proof of Theorem
$11.13$] and, hence, are acyclic. Similarly, one can prove that the sheaves
${}^{+}\mathrm{S}^{n}_{\mathrm{sm},Q}$ and
${}^{+}\mathrm{S}^{n}_{\mathrm{Lip},V}$ are flabby.

For the complex of sheaves ${}^{+}\mathrm{S}^{\bullet}_{X}$ we define the
morphism of complexes of sheaves, called coaugmentation,
$\epsilon:\underline{\mathbb{R}}_{X}[0]\to{}^{+}\mathrm{S}^{\bullet}_{X}$ by
$\epsilon(1):=1$. For the complexes
${}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},Q}$ and
${}^{+}\mathrm{S}^{\bullet}_{\mathrm{Lip},V}$ the coaugmentation is defined
similarly.

######  Lemma 2.

  1. (1)

For a topological space $X$ the morphism of complexes

$\mathrm{sh}:\mathrm{S}^{\bullet}_{X}(X)\to{}^{+}\mathrm{S}^{\bullet}_{X}(X)$

is a quasi-isomorphism.

  2. (2)

For a predifferentiable space $Q$ the morphism of complexes

$\mathrm{sh}:\mathrm{S}^{\bullet}_{\mathrm{sm},Q}(Q)\to{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},Q}(Q)$

is a quasi-isomorphism.

  3. (3)

For a subspace $V\subset\mathbb{R}^{m}$ the morphism of complexes

$\mathrm{sh}:\mathrm{S}^{\bullet}_{\mathrm{Lip},V}(V)\to{}^{+}\mathrm{S}^{\bullet}_{\mathrm{Lip},V}(V)$

is a quasi-isomorphism.

See [2, $7,$ p. $26$] for the proof of the first case. For the other two a
similar proof applies.

######  Lemma 3.

  1. (1)

For a locally contractible $X$ the coaugmentation
$\epsilon:\underline{\mathbb{R}}_{X}[0]\to{}^{+}\mathrm{S}^{\bullet}_{X}$ is a
quasi-isomorphism.

  2. (2)

For a smooth manifold $M$ the coaugmentation
$\epsilon:\underline{\mathbb{R}}_{M}[0]\to{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},M}$
is a quasi-isomorphism.

  3. (3)

For a locally Lipschitz contractible set $V\subset\mathbb{R}^{m}$ the
coaugmentation
$\epsilon:\underline{\mathbb{R}}_{V}[0]\to{}^{+}\mathrm{S}^{\bullet}_{\mathrm{Lip},V}$
is a quasi-isomorphism.

See [2, Example II.$1.2$] for the case of ${}^{+}\mathrm{S}^{\bullet}_{X}$,
the rest is analogous.

An important result needed in this work is the local Lipschitz contractability
of algebraic sets due to Shartser [22, Theorem $4.1.8$].

######  Theorem 4.

Take an algebraic set $V\subset\mathbb{R}^{m}$ and a point $v_{0}\in V$. Then
there exist an open set $U\subset\mathbb{R}^{m}$ with $v_{0}\in U$ and a
Lipschitz map $F:U\times[0,1]\to U$ such that

  1. (1)

$F(u,0)=u$ for $u\in U$;

  2. (2)

$F(u,1)=v_{0}$ for $u\in U$;

  3. (3)

$F(v,t)\in V$ for all $v\in V\cap U$ and $t\in[0,1]$.

######  Corollary 5.

The coaugmentation
$\epsilon:\underline{\mathbb{R}}_{V}[0]\to{}^{+}\mathrm{S}^{\bullet}_{\mathrm{Lip},V}$
is a quasi-isomorphism for $V$ an algebraic set in $\mathbb{R}^{m}$.

###### Proof.

Follows directly from Lemma 3 and Theorem 4. ∎

###  2.4. The de Rham dg-algebra $\Omega^{\bullet}_{A|k}$ of $A$

To a $k$-algebra $A$ one associates the dg-algebra $\Omega^{\bullet}_{A|k}$
([15, Theorem $3.2$]) with $\Omega^{0}_{A|k}=A$. It has the following
universal property: for any dg-algebra $E$ and any algebra homomorphism
$f:A\to E^{0}$ there exists a unique morphism of dg-algebras
$F:\Omega^{\bullet}_{A|k}\to E$ such that $F|_{A}=f$:

${A}$${\Omega^{\bullet}_{A|k}}$${E.}$$\scriptstyle{f}$$\scriptstyle{F}$

The elements of $\Omega^{n}_{A|k}$ are called algebraic $n$-forms. The dg-
algebra $\Omega^{\bullet}_{A|k}$ is covariant in the algebra $A$.

######  Lemma 6.

Suppose $A$ and $B$ are $k$-algebras and $\varphi:A\to B$ is a surjective
homomorphism of algebras. Then the induced morphism
$\Omega_{\varphi}:\Omega^{\bullet}_{A|k}\to\Omega^{\bullet}_{B|k}$ is
surjective and its kernel is the ideal of $\Omega^{\bullet}_{A|k}$ generated
by $\mathrm{Ker}\,\varphi$ and $d(\mathrm{Ker}\,\varphi)$.

###### Proof.

Set $T:=\mathrm{Ker}\,\varphi$, then it is enough to consider the case $B=A/T$
and $\varphi:A\to B$ being the canonical projection. Take the ideal $I$ of
$\Omega^{\bullet}_{A|k}$ generated by $T$ and $dT$. $I$ is a dg-ideal, hence,
$\Omega^{\bullet}_{A|k}/I$ is a dg-algebra and the canonical projection
$\pi:\Omega^{\bullet}_{A|k}\to\Omega^{\bullet}_{A|k}/I$ is a morphism of dg-
algebras. By the universal property there exists a unique morphism of dg-
algebras $\mathcal{L}:\Omega^{\bullet}_{B|k}\to\Omega^{\bullet}_{A|k}/I$ such
that $\mathcal{L}|_{B}$ is the identity map. On the other hand, the kernel of
$\Omega_{\varphi}$ contains $T$ and $dT$. Therefore, the morphism
$\Omega_{\varphi}$ decomposes as the composition of dg-algebra morphisms
$\Omega^{\bullet}_{A|k}\xrightarrow{\pi}\Omega^{\bullet}_{A|k}/I\xrightarrow{\mathcal{M}}\Omega^{\bullet}_{B|k}$.
The composition

$\Omega^{\bullet}_{B|k}\xrightarrow{\mathcal{L}}\Omega^{\bullet}_{A|k}/I\xrightarrow{\mathcal{M}}\Omega^{\bullet}_{B|k}$

is clearly the identity map by the universal property of
$\Omega^{\bullet}_{B|k}$. Consider the diagram

${\Omega^{\bullet}_{A|k}}$${\Omega^{\bullet}_{B|k}}$${\Omega^{\bullet}_{A|k}/I}$${\Omega^{\bullet}_{A|k}/I.}$$\scriptstyle{\pi}$$\scriptstyle{\Omega_{f}}$$\scriptstyle{\pi}$$\scriptstyle{\mathcal{L}}$$\scriptstyle{\mathcal{M}}$$\scriptstyle{=}$

The projection map $\pi$ is surjective and the left triangle is clearly
commutative. The upper triangle is commutative by the universal property of
$\Omega^{\bullet}_{A|k}$. Hence, the right triangle is commutative. Therefore,
the morphisms $\mathcal{L}$ and $\mathcal{M}$ are mutually inverse. As the
morphism $\pi$ is surjective and its kernel is generated by $T$ and $dT$, we
have $\Omega_{f}$ is surjective and its kernel is generated by $T$ and $dT$. ∎

######  Lemma 7.

Suppose a $k$-algebra $A$ is the filtered colimit of $k$-algebras $A_{i}$.
Then
$\Omega^{n}_{A|k}\cong\underset{\longrightarrow}{\operatorname{lim\,}}\Omega^{n}_{A_{i}|k}$.

The proof can be found in [15, Proposition $4.1$].

We can generalize the notion of algebraic forms to the case of sheaves. For a
topological space $X$ and a sheaf of $k$-algebras $\mathcal{F}$ on $X$
consider the following presheaf $\Omega^{n}_{\mathcal{F}|k}$ of
$\mathcal{F}$-modules:

$U\mapsto\Omega^{n}_{\mathcal{F}(U)|k}.$

For every open $U$ we obtain the complex $\Omega^{\bullet}_{\mathcal{F}(U)|k}$
and, hence, we have the complex of presheaves
$\Omega^{\bullet}_{\mathcal{F}|k}$. The associated sheaves form the complex of
sheaves ${}^{+}\Omega^{\bullet}_{\mathcal{F}|k}$.

For sets $W^{\prime}\subset W$ open or closed we fix the notations for the
restriction map
$\mathrm{res}_{W,W^{\prime}}:\Omega^{\bullet}_{\mathcal{F}(W)|k}\to\Omega^{\bullet}_{\mathcal{F}(W^{\prime})|k}$
and
$\mathrm{res}_{W,W^{\prime}}:{}^{+}\Omega^{\bullet}_{\mathcal{F}|k}(W)\to{}^{+}\Omega^{\bullet}_{\mathcal{F}|k}(W^{\prime})$
induced by the restriction map
$\mathrm{res}_{W,W^{\prime}}:\mathcal{F}(W)\to\mathcal{F}(W^{\prime})$.

Consider the morphism of complexes of presheaves
$\mathbb{R}_{X}[0]\to\Omega^{\bullet}_{\mathcal{F}|k}$, $1\mapsto 1$, where
$\mathbb{R}_{X}$ is the constant presheaf. The sheafification functor gives
rise to a morphism of complexes of sheaves, the coaugmentation,
$\epsilon:\underline{\mathbb{R}}_{X}[0]\to{}^{+}\Omega^{\bullet}_{\mathcal{F}|k}$.

######  Lemma 8.

For a soft sheaf of algebras $\mathcal{F}$ on $X$ the sheaves
${}^{+}\Omega^{n}_{\mathcal{F}|k}$ are soft.

###### Proof.

The associated sheaf ${}^{+}\Omega^{n}_{\mathcal{F}|k}$ has the natural
structure of an $\mathcal{F}$-module. So it is a sheaf of modules over a soft
sheaf and, therefore, is soft by [6, Theorem II.$3.7.1$]. ∎

##  3\. The map
$\Lambda_{\mathcal{F}}:\mathbb{H}^{*}(X,\underline{k}_{X}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k})$

###  3.1. The global sections of ${}^{+}\Omega^{n}_{\mathcal{F}|k}$

In this subsection we take $\mathcal{F}$ to be a soft sheaf of $k$-algebras on
a compact Hausdorff space $X$.

######  Lemma 9.

Suppose that $S,F\subset X$ are closed sets such that $S\cap F=\emptyset$.
Then there exists a section $g\in\mathcal{F}(X)$ such that
$\mathrm{res}_{X,S}(g)=0$ and $\mathrm{res}_{X,F}(g)=1$.

The proof can be found in [6, Theorem II.$3.7.2$].

######  Lemma 10.

Suppose that $U\subset X$ is an open set. Take
$\omega\in\Omega^{n}_{\mathcal{F}(X)|k}$ such that $\omega|_{U}=0$ in
$\Omega^{n}_{\mathcal{F}(U)|k}$. Take also a section
$\varphi\in\mathcal{F}(X)$ such that $\mathrm{supp}\,\varphi\subset U$. Then
$\varphi\omega=0$.

###### Proof.

Put $S:=\mathrm{supp}\,\varphi$. The restriction homomorphism
$\mathrm{res}_{X,S}:\mathcal{F}(X)\to\mathcal{F}(S)$ is surjective as
$\mathcal{F}$ is soft. By Lemma 6 the kernel of
$\mathrm{res}_{X,S}:\Omega^{\bullet}_{\mathcal{F}(X)|k}\to\Omega^{\bullet}_{\mathcal{F}(S)|k}$
is the ideal generated by $\mathrm{Ker}\,\mathrm{res}_{X,S}$ and
$d(\mathrm{Ker}\,\mathrm{res}_{X,S})$. By assumption
$\omega\in\mathrm{Ker}\,\mathrm{res}_{X,S}$ and, hence, it is enough to prove
the statement for $\omega$ of the form

$\omega=dt\wedge\lambda+u\eta,$

where $t,u\in\mathcal{F}(X)$,
$\lambda,\eta\in\Omega^{\bullet}_{\mathcal{F}(X)|k}$ and
$\mathrm{res}_{X,S}(t)=\mathrm{res}_{X,S}(u)=0$. We have

$\varphi\omega=(\varphi dt)\wedge\lambda+(\varphi u)\eta.$

We have $\varphi u=0$ as the supports of $\varphi$ and $u$ do not intersect.
It remains to prove that $\varphi dt=0$. By Lemma 9 there exists a section
$g\in\mathcal{F}(X)$ such that $\mathrm{res}_{X,S}(g)=0$ and
$\mathrm{res}_{X,\mathrm{supp}\,t}(g)=1$. We have

$\varphi dt=\varphi d(tg)=(\varphi t)dg+(\varphi g)dt=0$

as $t=tg$, $\varphi t=0$ and $\varphi g=0$. ∎

######  Lemma 11.

Suppose $\mathcal{F}$ is a soft sheaf of algebras on $X$. Then the
sheafification
$\mathrm{sh}:\Omega^{n}_{\mathcal{F}(X)|k}=\Omega^{n}_{\mathcal{F}|k}(X)\to{}^{+}\Omega^{n}_{\mathcal{F}|k}(X)$
is an isomorphism.

###### Proof.

First we prove injectivity. Take an $n$-form
$\omega\in\Omega^{n}_{\mathcal{F}|k}(X)$ such that its image in
${}^{+}\Omega^{n}_{\mathcal{F}|k}(X)$ is $0$. Then there is a finite open
cover $(U_{i})$ of $X$ such that the restrictions $\omega|_{U_{i}}$ are $0$
for all $i$. Choose a partition of unity $(\varphi_{i}\in\mathcal{F}(X))$
subordinate to the cover $(U_{i})$. It can always be done by [6, Theorem
II.$3.6.1$]. By Lemma 10, $\varphi_{i}\omega=0$ for all $i$. Now,
$\omega=\sum_{i}\varphi_{i}\omega=0$.

For surjectivity, take a global section of the sheaf
${}^{+}\Omega^{n}_{\mathcal{F}|k}$. It can be represented by a set of pairs
$(U_{i},\omega_{i})$ where the sets $U_{i}$ form a finite open cover of $X$
and $\omega_{i}\in\Omega^{n}_{\mathcal{F}(U_{i})|k}$ such that the germs of
$\omega_{i}$ agree at every point. We seek a form
$\omega\in\Omega^{n}_{\mathcal{F}(X)|k}$ such that
$\omega_{x}=(\omega_{i})_{x}$ in $\Omega^{n}_{\mathcal{F}_{x}|k}$ for each $i$
and $x\in U_{i}$. Take a partition of unity $(\varphi_{i})$ subordinate to the
cover $(U_{i})$ and denote $F_{i}:=\mathrm{supp}\,\varphi_{i}$. The
restriction map
$\mathrm{res}_{X,S}:\Omega^{n}_{\mathcal{F}(X)|k}\to\Omega^{n}_{\mathcal{F}(S)|k}$
is surjective for any closed $S$ by Lemma 6. Hence, we can extend the forms
$\mathrm{res}_{X,F_{i}}(\omega_{i})$ to some forms
$\bar{\omega}_{i}\in\Omega^{n}_{\mathcal{F}(X)|k}$. Now, put
$\omega:=\sum_{i}\varphi_{i}\bar{\omega}_{i}\in\Omega^{n}_{\mathcal{F}(X)|k}$.
Take $i$ and a point $x\in U_{i}$. Introduce $J:=\\{j\,|\,x\in F_{j}\\}$. We
have

$\omega_{x}=(\sum_{j}\varphi_{j}\bar{\omega}_{j})_{x}=(\sum_{j\in
J}\varphi_{j}\bar{\omega}_{j})_{x}=(\sum_{j\in
J}\varphi_{j}\omega_{j})_{x}=(\sum_{j\in
J}\varphi_{j})_{x}(\omega_{i})_{x}=(\omega_{i})_{x}.$

∎

###  3.2. The construction and naturality of
$\Lambda_{\mathcal{F}}:\mathbb{H}^{*}(X,\underline{k}_{X}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k})$

######  Definition.

For a soft sheaf of algebras $\mathcal{F}$ on a compact Hausdorff space $X$ we
define the map $\Lambda_{\mathcal{F}}$ by the following diagram:

${\mathbb{H}^{*}(X,\underline{k}_{X}[0])}$${\mathbb{H}^{*}(X,{}^{+}\Omega^{\bullet}_{\mathcal{F}|k})}$${\mathrm{H}^{*}({}^{+}\Omega^{\bullet}_{\mathcal{F}|k}(X))}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k}).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Lambda_{\mathcal{F}}}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$

Here the map $\mathrm{H}(\mathrm{sh})$ is an isomorphism by Lemma 11 and the
map $\Upsilon$ is an isomorphism as the sheaves
${}^{+}\Omega^{\bullet}_{\mathcal{F}|k}$ are acyclic by Lemma 8.

A morphism of ringed spaces $(f,\varphi):(X,\mathcal{F})\to(Y,\mathcal{G})$
consists of a continuous map $f:X\to Y$ and a morphism of sheaves
$\varphi:\mathcal{G}\to f_{*}\mathcal{F}$.

######  Proposition 12.

The linear map $\Lambda_{-}$ is natural with respect to morphisms of ringed
spaces in the following sense: for a morphism of ringed spaces
$(f,\varphi):(X,\mathcal{F})\to(Y,\mathcal{G})$ the following diagram is
commutative:

${\mathbb{H}^{*}(X,\underline{k}_{X}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k})}$${\mathbb{H}^{*}(Y,\underline{k}_{Y}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{G}(Y)|k}).}$$\scriptstyle{\Lambda_{\mathcal{F}}}$$\scriptstyle{\Lambda_{\mathcal{G}}}$$\scriptstyle{f^{*}}$$\scriptstyle{\mathrm{H}(\Omega_{\varphi(Y)})}$

###### Proof.

The morphism of sheaves $\varphi:\mathcal{G}\to f_{*}\mathcal{F}$ defines the
morphism of complexes of sheaves
${}^{+}\Omega_{\varphi}:{}^{+}\Omega^{\bullet}_{\mathcal{G}|k}\to
f_{*}{}^{+}\Omega^{\bullet}_{\mathcal{F}|k}$.

Consider the diagram

${\mathbb{H}^{*}(X,\underline{k}_{X}[0])}$${\mathbb{H}^{*}(X,{}^{+}\Omega^{\bullet}_{\mathcal{F}|k})}$${\mathrm{H}^{*}({}^{+}\Omega^{\bullet}_{\mathcal{F}|k}(X))}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|k})}$${\mathbb{H}^{*}(Y,\underline{k}_{Y}[0])}$${\mathbb{H}^{*}(Y,{}^{+}\Omega^{\bullet}_{\mathcal{G}|k})}$${\mathrm{H}^{*}({}^{+}\Omega^{\bullet}_{\mathcal{G}|k}(Y))}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{G}(Y)|k}).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Lambda_{\mathcal{F}}}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\cong}$$\scriptstyle{f^{*}}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Lambda_{\mathcal{G}}}$$\scriptstyle{f^{*}}$$\scriptstyle{\mathrm{H}({}^{+}\Omega_{\varphi}(Y))}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\Omega_{\varphi(Y)})}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\cong}$

The middle square is commutative by Lemma 1. The other squares are commutative
for obvious reasons. ∎

##  4\. Splitting for the algebra of smooth functions

In this section $M$ is a compact smooth manifold. We prove that for a smooth
manifold $M$ the groups $\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])$
canonically split off of
$\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})$.

###  4.1. Canonical isomorphism $\Theta$

Denote by $\Omega^{\bullet}_{M}$ the complex of sheaves of smooth differential
forms on $M$. We often denote by $\Omega^{\bullet}(M)$ the complex
$\Omega^{\bullet}_{M}(M)$. Consider the morphism of complexes, the
coaugmentation,
$\epsilon:\underline{\mathbb{R}}_{M}[0]\to\Omega^{\bullet}_{M}$ defined by
$\epsilon(1):=1$. We define $\Theta$ by the following commutative diagram

${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathbb{H}^{*}(M,\Omega_{M}^{\bullet})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M)).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\Theta}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$

The map $\mathbb{H}(\epsilon)$ in this case is an isomorphism (as $\epsilon$
is a quasi-isomorphism) and $\Upsilon$ is an isomorphism and so $\Theta$ is an
isomorphism.

###  4.2. Morphisms $\pi$ and $\underline{\pi}$

  1. (1)

Take an open set $U\subset M$; the identity map
$C^{\infty}(U)\to\Omega^{0}(U)$ can be uniquely extended to a morphism of dg-
algebras
$\pi:\Omega^{\bullet}_{C^{\infty}(U)|\mathbb{R}}\to\Omega^{\bullet}(U)$ by the
universal property of $\Omega^{\bullet}_{C^{\infty}(U)|\mathbb{R}}$. The
following diagram is commutative:

${C^{\infty}(U)}$${\Omega^{\bullet}_{C^{\infty}(U)|\mathbb{R}}}$${\Omega^{\bullet}(U).}$$\scriptstyle{\pi}$

  2. (2)

This way we obtain a morphism of complexes of presheaves
$\pi:\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}}\to\Omega^{\bullet}_{M}$.

  3. (3)

By the universal property of sheafification we have the morphism of complexes
of sheaves
$\underline{\pi}:{}^{+}\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}}\to\Omega^{\bullet}_{M}$.

The following diagrams are commutative:

(2)
${\underline{\mathbb{R}}_{M}[0]}$${{}^{+}\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}}}$${\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}}}$${\Omega^{\bullet}_{M}}$${\Omega^{\bullet}_{M},}$${{}^{+}\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}}.}$$\scriptstyle{\epsilon}$$\scriptstyle{\epsilon}$$\scriptstyle{\underline{\pi}}$$\scriptstyle{\pi}$$\scriptstyle{\mathrm{sh}}$$\scriptstyle{\underline{\pi}}$

###  4.3. Main splitting theorem

######  Theorem 13.

The following diagram is commutative:

${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M)).}$$\scriptstyle{\Lambda_{C^{\infty}_{M}}}$$\scriptstyle{\Theta}$$\scriptstyle{\mathrm{H}(\pi)}$

###### Proof.

Consider the diagram

${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathbb{H}^{*}(M,{}^{+}\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}})}$${\mathbb{H}^{*}(M,\Omega^{\bullet}_{M})}$${\mathrm{H}^{*}({}^{+}\Omega^{\bullet}_{C^{\infty}_{M}|\mathbb{R}}(M))}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}}).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Theta}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Lambda_{C^{\infty}_{M}}}$$\scriptstyle{\mathbb{H}(\underline{\pi})}$$\scriptstyle{\Upsilon}$$\scriptstyle{\mathrm{H}(\underline{\pi})}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\mathrm{H}(\pi)}$

The left quadrangle is commutative by the definition of
$\Lambda_{C^{\infty}_{M}}$. The upper triangle and the bottom right triangles
are commutative by Diagrams 2. The upper right square is commutative by the
naturality of $\Upsilon$. The upper triangle with $\Theta$ is commutative by
the definition of $\Theta$. As the right arrow $\Upsilon$ is an isomorphism,
the whole diagram is commutative. ∎

##  5\. Simplicial dg-algebra of flat cochains
$\Omega^{\bullet}_{\flat}(\Delta^{-})$

###  5.1. Flat cochains

We present the dg-algebra $\Omega_{\flat}^{\bullet}(\Delta^{n})$ of flat
cochains on the closed $n$-simplex. We follow [25], see also [12]. The dg-
algebras $\Omega_{\flat}^{\bullet}(\Delta^{n})$, $n\geq 0$, will form the
simplicial dg-algebra $\Omega^{\bullet}_{\flat}(\Delta^{-})$.

For a convex set $V\subset\mathbb{R}^{m}$ we denote by $\mathrm{aff}V$ the
affine hull of $V$. We denote by $\mathrm{relint}\,V$ the interior of $V$
relative to $\mathrm{aff}V$.

######  Definition.

For a convex set $V\subset\mathbb{R}^{m}$ denote by
$\mathrm{S}^{\text{aff}}_{k}(V)$ the vector space of affine singular
$k$-chains in $V$ with coefficients in $\mathbb{R}$. The boundary map
$\partial:\mathrm{S}^{\text{aff}}_{k}(V)\to\mathrm{S}^{\text{aff}}_{k-1}(V)$
is defined in the usual manner: we denote by
$\gamma_{i}:\Delta^{k-1}\to\Delta^{k}$ the $i$-th face embedding, then for a
simplex $\sigma:\Delta^{k}\to V$ we denote by
$\partial_{i}\sigma:\Delta^{k-1}\to V$ the composition $\sigma\circ\gamma_{i}$
and by $\partial\sigma$ the singular chain
$\sum_{i=0}^{k}(-1)^{i}\partial_{i}\sigma$. The vector spaces
$\mathrm{S}^{\text{aff}}_{k}(V)$ together with the boundary map form the chain
complex $\mathrm{S}^{\text{aff}}_{\bullet}(V)$.

An affine map $f:V\to V^{\prime}$ induces the morphism of complexes

$\mathrm{S}^{\text{aff}}(f):\mathrm{S}^{\text{aff}}_{\bullet}(V)\to\mathrm{S}^{\text{aff}}_{\bullet}(V^{\prime}).$

Hence, the correspondence $V\mapsto\mathrm{S}^{\text{aff}}_{\bullet}(V)$ is
covariant with respect to affine maps.

Define the mass $|\alpha|$ of an affine $k$-chain
$\alpha=\sum\lambda_{i}\sigma_{i}$, where $\sigma_{i}$ are distinct singular
simplices, as

$|\alpha|:=\sum|\lambda_{i}||\sigma_{i}|_{\mathrm{L}}$

where $|\sigma_{i}|_{\mathrm{L}}$ denotes the Lebesgue $k$-measure of
$\sigma_{i}$. Define the flat seminorm $|\cdot|^{\flat}$ on
$\mathrm{S}^{\text{aff}}_{k}(V)$ as

$|\alpha|^{\flat}:=\inf_{\beta\in\mathrm{S}^{\text{aff}}_{k+1}(V)}\\{|\alpha-\partial\beta|+|\beta|\\}.$

######  Lemma 14.

The map
$\mathrm{S}^{\text{aff}}_{k}(V)\to\mathrm{S}^{\text{aff}}_{k}(V^{\prime})$
induced by an inclusion of convex sets $V\hookrightarrow V^{\prime}$ preserves
the flat seminorm.

###### Proof.

Follows from [25, Lemma V.2b]. ∎

######  Lemma 15.

Let $V$ be a convex set. The map
$\mathrm{S}^{\text{aff}}_{k}(\mathrm{relint}\,V)\to\mathrm{S}^{\text{aff}}_{k}(V)$
induced by the inclusion $\mathrm{relint}\,V\hookrightarrow V$ has image dense
with respect to the flat seminorm.

###### Proof.

Any singular affine simplex in $V$ can be approximated by one in
$\mathrm{relint}\,V$. Cf. [25, VIII.1(h)]. ∎

######  Definition.

If $V$ is a convex set we define $\Omega_{\flat}^{k}(V)$ as the vector space
of linear functionals $\mathrm{S}^{\text{aff}}_{k}(V)\to\mathbb{R}$ bounded
with respect to the seminorm $|\cdot|^{\flat}$. We call the elements of
$\Omega_{\flat}^{k}(V)$ flat cochains on $V$. We define the differential $dX$
of a cochain $X$ by the formula
$\left<dX,\alpha\right>:=\left<X,\partial\alpha\right>$. We obtain the complex
$\Omega_{\flat}^{\bullet}(V)$. This definition is equivalent to the one given
by Whitney in [25, VIII.1(b) and VII.10].

The complex $\mathrm{S}^{\text{aff}}_{\bullet}(V)$ was covariant in $V$ with
respect to affine maps, so the complex $\Omega_{\flat}^{\bullet}(V)$ is
contravariant in $V$ with respect to affine maps.

We refer the reader to [25, IX.$14$] where Whitney defines the graded-
commutative multiplication of flat cochains in open sets. The multiplication
is natural with respect to affine maps [25, X.$11$]. For an open convex set
$V$ the complex $\Omega_{\flat}^{\bullet}(V)$ becomes a dg-algebra, which is
contravariant with respect to affine maps. Hence, the multiplication is well
defined in relatively open convex sets. Next we wish to define the product of
two flat cochains on a closed convex set $V$. For this we need the following
lemma.

######  Lemma 16.

For a closed convex set $V\subset\mathbb{R}^{m}$ the inclusion
$\mathrm{relint}\,V\hookrightarrow V$ induces an isomorphism
$\rho:\Omega^{\bullet}_{\flat}(V)\to\Omega^{\bullet}_{\flat}(\mathrm{relint}\,V)$.

###### Proof.

Follows from Lemmas 14 and 15. ∎

For a closed convex set $V\subset\mathbb{R}^{m}$ we introduce the
multiplication in $\Omega^{\bullet}_{\flat}(V)$ in the way that the
isomorphism of complexes
$\rho:\Omega^{\bullet}_{\flat}(V)\to\Omega^{\bullet}_{\flat}(\mathrm{relint}\,V)$
from Lemma 16 becomes an isomorphism of dg-algebras. This multiplication and
its naturality are implicit in [25, VII.$12$]. In Proposition 19 we show that
the multiplication is natural with respect to affine maps.

Take $V$ a closed convex set. The inclusion $V\hookrightarrow\mathrm{aff}V$
induces the map
$\pi:\Omega^{\bullet}_{\flat}(\mathrm{aff}V)\to\Omega^{\bullet}_{\flat}(V)$.

######  Lemma 17.

The map
$\pi:\Omega^{\bullet}_{\flat}(\mathrm{aff}V)\to\Omega^{\bullet}_{\flat}(V)$ is
surjective.

###### Proof.

Follows from Lemma 14 by the Hahn-Banach theorem. Alternatively, see [25,
VIII.1(h)] and apply Lemma 16. ∎

######  Lemma 18.

The map
$\pi:\Omega^{\bullet}_{\flat}(\mathrm{aff}V)\to\Omega^{\bullet}_{\flat}(V)$ is
a morphism of dg-algebras.

###### Proof.

Consider the following diagram

${\Omega^{\bullet}_{\flat}(\mathrm{aff}V)}$${\Omega^{\bullet}_{\flat}(V)}$${\Omega^{\bullet}_{\flat}(\mathrm{relint}\,V).}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$

The composition $\rho\circ\pi$ is induced by the inclusion
$\mathrm{relint}\,V\hookrightarrow\mathrm{aff}V$ and, hence, is a morphism of
dg-algebras. Since $\rho$ is an isomorphism of dg-algebras, $\pi$ is a
morphism of dg-algebras. ∎

######  Proposition 19.

Consider $f:V\to V^{\prime}$ an affine map of closed convex sets. Then the
induced morphism
$\Omega_{\flat}(f):\Omega^{\bullet}_{\flat}(V^{\prime})\to\Omega^{\bullet}_{\flat}(V)$
preserves multiplication.

###### Proof.

Consider the following commutative diagram:

${\Omega^{\bullet}_{\flat}(\mathrm{aff}V^{\prime})}$${\Omega^{\bullet}_{\flat}(\mathrm{aff}V)}$${\Omega^{\bullet}_{\flat}(V^{\prime})}$${\Omega^{\bullet}_{\flat}(V).}$$\scriptstyle{\pi}$$\scriptstyle{\Omega_{\flat}(\mathrm{aff}f)}$$\scriptstyle{\pi}$$\scriptstyle{\Omega_{\flat}(f)}$

Here $\mathrm{aff}f:\mathrm{aff}V\to\mathrm{aff}V^{\prime}$ is the affine
extension of $f$. The left vertical arrow is surjective by Lemma 17. The
vertical arrows are morphisms of dg-algebras by Lemma 18. The map
$\Omega_{\flat}(\mathrm{aff}f)$ is a morphism of dg-algebras as it is induced
by an affine map of relatively open sets. Therefore, $\Omega_{\flat}(f)$ is a
morphism of dg-algebras. ∎

An order-preserving map $[n]\to[l]$ induces an affine map
$\Delta^{n}\to\Delta^{l}$. Hence, we obtain the simplicial dg-algebra
$\Omega^{\bullet}_{\flat}(\Delta^{-})$ defined as
$[n]\mapsto\Omega^{\bullet}_{\flat}(\Delta^{n})$. In particular, there are
face maps
$\partial_{i}:\Omega^{\bullet}_{\flat}(\Delta^{n})\to\Omega^{\bullet}_{\flat}(\Delta^{k-1})$
induced by the face embeddings $\gamma_{i}:\Delta^{k-1}\to\Delta^{n}$.

###  5.2. The attributes of the simplicial dg-algebra
$\Omega^{\bullet}_{\flat}(\Delta^{-})$

(1) We define the linear maps
$\mathfrak{J}_{n}:\Omega_{\flat}^{n}(\Delta^{n})\to\mathbb{R}$ as

$\mathfrak{J}_{n}(\omega):=\left<\omega,\mathrm{id}_{\Delta^{n}}\right>.$

######  Proposition 20 (Stokes’ formula).

For every $\eta\in\Omega_{\flat}^{n-1}(\Delta^{n})$ the following formula
holds

$\mathfrak{J}_{n}(d\eta)=\sum_{i=0}^{n}(-1)^{i}\mathfrak{J}_{n-1}(\partial_{i}\eta).$

###### Proof.

We have

$\mathfrak{J}_{n}(d\eta)=\left<d\eta,\mathrm{id}_{\Delta^{n}}\right>=\left<\eta,\partial\,\mathrm{id}_{\Delta^{n}}\right>=\sum_{i=0}^{n}(-1)^{i}\left<\eta,\gamma_{i}\right>=$
$=\sum_{i=0}^{n}(-1)^{i}\left<\partial_{i}\eta,\mathrm{id}_{\Delta^{n-1}}\right>=\sum_{i=0}^{n}(-1)^{i}\mathfrak{J}_{n-1}(\partial_{i}\eta).$

∎

(2) The Lipschitz functions on $\Delta^{n}$ form the algebra
$\mathrm{Lip}(\Delta^{n})$. The correspondence
$[n]\mapsto\mathrm{Lip}(\Delta^{n})$ gives rise to the simplicial algebra
$\mathrm{Lip}(\Delta^{-})$. Every Lipschitz function
$f\in\mathrm{Lip}(\Delta^{n})$ defines a flat $0$-cochain $\zeta(f)$ on
$\Delta^{n}$ in the following way. Take an affine simplex
$\sigma:\\{v_{0}\\}=\Delta^{0}\to\Delta^{n}$ and set

$\left<\zeta(f),\sigma\right>:=f(\sigma(v_{0})).$

It is easy to check that $0$-cochain $\zeta(f)$ is flat (see [25, Theorem
VII.4B]. We obtain the linear map
$\zeta:\mathrm{Lip}(\Delta^{n})\to\Omega^{\bullet}_{\flat}(\Delta^{n})$ with
image in $\Omega^{0}_{\flat}(\Delta^{n})$. The composition
$\rho\circ\zeta:\mathrm{Lip}(\Delta^{n})\to\Omega^{\bullet}_{\flat}(\Delta^{n})\xrightarrow{\cong}\Omega^{\bullet}_{\flat}(\mathrm{relint}\,\Delta^{n})$
is a homomorphism of algebras (by the definition of multiplication on open
sets [25, IX.$14$]). Hence, $\zeta$ is a homomorphism of algebras. The map
$\zeta$ clearly preserves the simplicial structure and we obtain the morphism
of simplicial algebras

$\zeta:\mathrm{Lip}(\Delta^{-})\to\Omega^{0}_{\flat}(\Delta^{-}).$

(3) Consider the simplicial dg-algebra $\Omega^{\bullet}(\Delta^{-})$ of
smooth differential forms. Every smooth form $\theta\in\Omega^{k}(\Delta^{n})$
gives rise to a flat cochain $\nabla(\theta)\in\Omega^{k}_{\flat}(\Delta^{n})$
by

$\left<\nabla(\theta),\sigma\right>:=\int_{\Delta^{k}}\sigma^{*}\theta$

for affine $\sigma:\Delta^{k}\to\Delta^{n}$. The flatness can be easily
checked (see [25, V.14 and Theorem V.10A]). This correspondence gives rise to
a simplicial linear map

$\nabla:\Omega^{\bullet}(\Delta^{-})\to\Omega^{\bullet}_{\flat}(\Delta^{-}).$

By Stokes’ formula for smooth forms the differential is preserved. The
composition

$\Omega^{\bullet}(\Delta^{n})\xrightarrow{\nabla}\Omega^{\bullet}_{\flat}(\Delta^{n})\xrightarrow{\rho}\Omega^{\bullet}_{\flat}(\mathrm{relint}\,\Delta^{n})$

preserves multiplication by the definition of multiplication of flat cochains
[25, X.$14$]. Hence, by the definition of multiplication in
$\Omega^{\bullet}_{\flat}(\Delta^{n})$ the map $\nabla$ preserves
multiplication. Therefore, the map

$\nabla:\Omega^{\bullet}(\Delta^{-})\to\Omega^{\bullet}_{\flat}(\Delta^{-})$

is a morphism of simplicial dg-algebras.

(4) We define the linear maps
$\mathfrak{T}_{n}:\Omega^{n}(\Delta^{n})\to\mathbb{R}$ as

$\mathfrak{T}_{n}(\omega):=\int_{\Delta^{n}}\omega.$

The diagram

${\Omega^{n}(\Delta^{n})}$${\mathbb{R}}$${\Omega^{n}_{\flat}(\Delta^{n})}$$\scriptstyle{\mathfrak{T}_{n}}$$\scriptstyle{\nabla}$$\scriptstyle{\mathfrak{J}_{n}}$

is commutative by construction.

(5) The smooth functions form the simplicial algebra $C^{\infty}(\Delta^{-})$.
It embeds via
$i:C^{\infty}(\Delta^{-})\hookrightarrow\mathrm{Lip}(\Delta^{-})$ in the
simplicial algebra $\mathrm{Lip}(\Delta^{-})$. The following diagram of
morphisms of simplicial algebras is commutative:

${C^{\infty}(\Delta^{-})}$${\Omega^{\bullet}(\Delta^{-})}$${\mathrm{Lip}(\Delta^{-})}$${\Omega^{\bullet}_{\flat}(\Delta^{-}).}$$\scriptstyle{i}$$\scriptstyle{\nabla}$$\scriptstyle{\zeta}$

##  6\. The map
$\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$

###  6.1. The pullback of an algebraic de Rham form

Take $B$ a finitely generated $\mathbb{R}$-algebra. We construct a morphism of
dg-algebras
$\mu(\sigma):\Omega^{\bullet}_{B|\mathbb{R}}\to\Omega_{\flat}^{\bullet}(\Delta^{n})$
for every singular Lipschitz simplex
$\sigma:\Delta^{n}\to\mathrm{spec}_{\mathbb{R}}B$. For such $\sigma$ we define
an algebra homomorphism $\theta(\sigma):B\to\mathrm{Lip}(\Delta^{n})$ as
$b\mapsto\hat{b}\circ\sigma$. By the universal property of
$\Omega^{\bullet}_{B|\mathbb{R}}$ there exists a unique morphism of dg-
algebras
$\mu(\sigma):\Omega_{B|\mathbb{R}}^{\bullet}\to\Omega_{\flat}^{\bullet}(\Delta^{n})$
making the following diagram commute:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega_{\flat}^{\bullet}(\Delta^{n})}$${B}$${\mathrm{Lip}(\Delta^{n}).}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\theta(\sigma)}$$\scriptstyle{\zeta}$

######  Lemma 21.

The following diagram is commutative:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}_{\flat}(\Delta^{n})}$${\Omega^{\bullet}_{\flat}(\Delta^{n-1}).}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\mu(\partial_{i}\sigma)}$$\scriptstyle{\partial_{i}}$

###### Proof.

Consider the following diagram:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}_{\flat}(\Delta^{n})}$${\Omega_{\flat}^{\bullet}(\Delta^{n-1})}$${B}$${\mathrm{Lip}(\Delta^{n})}$${\mathrm{Lip}(\Delta^{n-1}).}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\mu(\partial_{i}\sigma)}$$\scriptstyle{\partial_{i}}$$\scriptstyle{\theta(\sigma)}$$\scriptstyle{\theta(\partial_{i}\sigma)}$$\scriptstyle{\zeta}$$\scriptstyle{\partial_{i}}$$\scriptstyle{\zeta}$

The right square obviously commutes. By the definition of $\mu$ the left
square and the outer contour commute. The bottom triangle commutes by the
definition of $\theta$. Hence, by the universal property of
$\Omega^{\bullet}_{B|\mathbb{R}}$ the upper triangle also commutes. ∎

The map $\mu(\sigma)$ is natural in algebra, namely:

######  Lemma 22.

Suppose $\varphi:B^{\prime}\to B$ is a homomorphism of finitely generated
$\mathbb{R}$-algebras. Consider a commutative diagram

${\Delta^{n}}$${\mathrm{spec}_{\mathbb{R}}B}$${\mathrm{spec}_{\mathbb{R}}B^{\prime}}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma^{\prime}}$$\scriptstyle{\mathrm{spec}_{\mathbb{R}}\varphi}$

with $\sigma$ and $\sigma^{\prime}$ Lipschitz. Then the following diagram
commutes:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega_{\flat}^{\bullet}(\Delta^{n})}$${\Omega^{\bullet}_{B^{\prime}|\mathbb{R}}.}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\Omega_{\varphi}}$$\scriptstyle{\mu(\sigma^{\prime})}$

###### Proof.

Consider the following diagram:

${\Omega^{\bullet}_{B^{\prime}|\mathbb{R}}}$${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega_{\flat}^{\bullet}(\Delta^{n})}$${B^{\prime}}$${B}$${\mathrm{Lip}(\Delta^{n}).}$$\scriptstyle{\Omega_{\varphi}}$$\scriptstyle{\mu(\sigma^{\prime})}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\varphi}$$\scriptstyle{\theta(\sigma^{\prime})}$$\scriptstyle{\theta(\sigma)}$$\scriptstyle{\zeta}$

The left square obviously commutes. By the definition of $\mu$ the right
square and the outer contour commute. The bottom triangle commutes: for $c\in
B^{\prime}$

$\theta(\sigma)(\varphi(c))=\widehat{\varphi(c)}\circ\sigma=\hat{c}\circ(\mathrm{spec}_{\mathbb{R}}\varphi)\circ\sigma=\hat{c}\circ\sigma^{\prime}=\theta(\sigma^{\prime})(c).$

Hence, by the universal property of $\Omega^{\bullet}_{B^{\prime}|\mathbb{R}}$
the upper triangle also commutes. ∎

###  6.2. The map
$\xi_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B)$

Let $B$ be a finitely generated $\mathbb{R}$-algebra. To an algebraic $n$-form
$\omega\in\Omega^{n}_{B|\mathbb{R}}$ we associate a Lipschitz singular cochain
$\xi_{B}(\omega)$. On a Lipschitz singular simplex
$\sigma:\Delta^{n}\to\mathrm{spec}_{\mathbb{R}}B$ we define $\xi_{B}(\omega)$
as

$\left<\xi_{B}(\omega),\sigma\right>:=\mathfrak{J}_{n}(\mu(\sigma)(\omega))\in\mathbb{R}.$

######  Proposition 23.

The above gives a morphism of complexes

$\xi_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B).$

###### Proof.

The only part that needs to be checked is that the map preserves $d$. Take a
Lipschitz singular simplex $\sigma:\Delta^{n}\to\mathrm{spec}_{\mathbb{R}}B$
and $\eta\in\Omega^{n-1}_{B|\mathbb{R}}$, then

$\left<\xi_{B}(d\eta),\sigma\right>=\mathfrak{J}_{n}(\mu(\sigma)(d\eta))\overset{(2)}{=}\mathfrak{J}_{n}(d(\mu(\sigma)(\eta)))\overset{(3)}{=}\sum_{i=0}^{n}(-1)^{i}\mathfrak{J}_{n-1}(\partial_{i}(\mu(\sigma)(\eta)))\overset{(4)}{=}$
$\overset{(4)}{=}\sum_{i=0}^{n}(-1)^{i}\mathfrak{J}_{n-1}(\mu(\partial_{i}\sigma)(\eta))=\sum_{i=0}^{n}(-1)^{i}\left<\xi_{B}(\eta),\partial_{i}\sigma\right>=\left<\xi_{B}(\omega),\partial\sigma\right>=\left<d\xi_{B}(\omega),\sigma\right>.$

The second equality follows from the fact that $\mu(\sigma)$ is a morphism of
complexes. The third equality follows from Proposition 20. The fourth equality
follows from Lemma 21. ∎

Lemma 22 allows us to prove the naturality of $\xi$ in algebras:

######  Lemma 24.

Suppose $\varphi:B^{\prime}\to B$ is a homomorphism of finitely generated
$\mathbb{R}$-algebras. Then the following diagram is commutative

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B)}$${\Omega^{\bullet}_{B^{\prime}|\mathbb{R}}}$${\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B^{\prime}).}$$\scriptstyle{\xi_{B}}$$\scriptstyle{\Omega_{\varphi}}$$\scriptstyle{\xi_{B^{\prime}}}$$\scriptstyle{\mathrm{S}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}\varphi)}$

###  6.3. The map
$\Phi_{B}:\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})\to\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])$

We define the homomorphism $\Phi_{B}$ as the vertical map making the following
diagram commutative:

${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])}$${\mathbb{H}^{*}({\mathrm{spec}_{\mathbb{R}}B},{}^{+}\mathrm{S}^{\bullet}_{\mathrm{Lip},{\mathrm{spec}_{\mathbb{R}}B}})}$${\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})}$${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{Lip}}({\mathrm{spec}_{\mathbb{R}}B})).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\xi_{B})}$$\scriptstyle{\Phi_{B}}$$\scriptstyle{\cong}$$\scriptstyle{\Upsilon\circ\mathrm{H}(\mathrm{sh})}$

Here $\mathbb{H}(\epsilon)$ is an isomorphism by Corollary 5.

######  Lemma 25.

Let $\varphi:B^{\prime}\to B$ be a homomorphism of finitely generated
$\mathbb{R}$-algebras. Then the following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{B^{\prime}|\mathbb{R}})}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B^{\prime},\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B^{\prime}}[0]).}$$\scriptstyle{\Phi_{B}}$$\scriptstyle{\mathrm{H}(\Omega_{\varphi})}$$\scriptstyle{\Phi_{B^{\prime}}}$$\scriptstyle{(\mathrm{spec}_{\mathbb{R}}\varphi)^{*}}$

###### Proof.

It follows directly from Lemma 24 and naturality of
$\Upsilon\circ\mathrm{H}(\mathrm{sh})$. ∎

######  Remark.

There are other ways to define a map, analogous to $\Phi_{B}$. One way is to
consider semi-algebraic cochains, instead of Lipschitz ones, see [11]. Another
way would be to use the filtered de Rham complex, see [20, Proposition
$7.24$]. Our construction of $\Phi_{B}$ allows us to relate it to the
classical de Rham complex $\Omega^{\bullet}(M)$ in the case $B\subset
C^{\infty}(M)$, see Lemma 35.

###  6.4. The map
$\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$

For a topological space $X$ and a subalgebra $A\subset C(X)$ write $A$ as the
filtered colimit of its finitely generated subalgebras:
$A=\underset{\longrightarrow}{\operatorname{lim\,}}B$. The functors $\Omega$
and $\mathrm{H}$ preserve filtered colimits. For an inclusion of finitely
generated subalgebras $i:B^{\prime}\hookrightarrow B$ of $C(X)$ consider the
diagram

${\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{B^{\prime}|\mathbb{R}})}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B^{\prime},\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B^{\prime}}[0]).}$$\scriptstyle{\Phi_{B}}$$\scriptstyle{\Gamma^{*}_{B}}$$\scriptstyle{\mathrm{H}(\Omega_{i})}$$\scriptstyle{\Phi_{B^{\prime}}}$$\scriptstyle{(\mathrm{spec}_{\mathbb{R}}i)^{*}}$$\scriptstyle{\Gamma^{*}_{B^{\prime}}}$

The left square is commutative by Lemma 25 and the right triangle is
commutative by the naturality of $\Gamma_{B}$ (see Subsection 2.2). We pass to
the colimit of $\Gamma^{*}_{B}\circ\Phi_{B}$ over all finitely generated
subalgebras $B\subset A$ and obtain the map

$\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0]).$

######  Proposition 26.

Take $f:X\to Y$ a continuous map of topological spaces. Suppose that $A\subset
C(X)$ and $A^{\prime}\subset C(Y)$ are subalgebras and $\varphi:A^{\prime}\to
A$ is a homomorphism such that the following diagram is commutative:

${C(X)}$${A}$${C(Y)}$${A^{\prime}.}$$\scriptstyle{f^{*}}$$\scriptstyle{\varphi}$

Then the following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{A^{\prime}|\mathbb{R}})}$${\mathbb{H}^{*}(Y,\underline{\mathbb{R}}_{Y}[0]).}$$\scriptstyle{\Psi_{A}}$$\scriptstyle{\mathrm{H}(\Omega_{\varphi})}$$\scriptstyle{\Psi_{A^{\prime}}}$$\scriptstyle{f^{*}}$

###### Proof.

It follows directly from Lemma 25 and naturality of $\Gamma$. ∎

##  7\. Identifying the map $\Psi_{C^{\infty}(M)}$

###  7.1. The map
$\xi^{\mathrm{sm}}_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B)$

This subsection mirrors the Subsections 6.1 and 6.2, so the proofs will be
omitted.

Take $B$ a finitely generated $\mathbb{R}$-algebra. We construct a morphism of
dg-algebras
$\mu^{\mathrm{sm}}(\sigma):\Omega^{\bullet}_{B|\mathbb{R}}\to\Omega^{\bullet}(\Delta^{n})$
for every smooth singular simplex
$\sigma:\Delta^{n}\to\mathrm{spec}_{\mathbb{R}}B$. For such $\sigma$ we define
an algebra homomorphism $\theta^{\mathrm{sm}}(\sigma):B\to
C^{\infty}(\Delta^{n})$ as $b\mapsto\hat{b}\circ\sigma$. By the universal
property of $\Omega^{\bullet}_{B|\mathbb{R}}$ there exists a unique morphism
of dg-algebras
$\mu^{\mathrm{sm}}(\sigma):\Omega_{B|\mathbb{R}}^{\bullet}\to\Omega^{\bullet}(\Delta^{n})$
making the following diagram commute:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}(\Delta^{n})}$${B}$${C^{\infty}(\Delta^{n}).}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\theta^{\mathrm{sm}}(\sigma)}$

######  Lemma 27 (cf. Lemma 21).

The following diagram is commutative:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}(\Delta^{n})}$${\Omega^{\bullet}(\Delta^{n-1}).}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\mu^{\mathrm{sm}}(\partial_{i}\sigma)}$$\scriptstyle{\partial_{i}}$

The map $\mu^{\mathrm{sm}}(\sigma)$ is natural in algebra, namely:

######  Lemma 28 (cf. Lemma 22).

Suppose $\varphi:B^{\prime}\to B$ is a homomorphism of finitely generated
$\mathbb{R}$-algebras. Consider a commutative diagram

${\Delta^{n}}$${\mathrm{spec}_{\mathbb{R}}B}$${\mathrm{spec}_{\mathbb{R}}B^{\prime}}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma^{\prime}}$$\scriptstyle{\mathrm{spec}_{\mathbb{R}}\varphi}$

with $\sigma$ and $\sigma^{\prime}$ smooth. Then the following diagram
commutes:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega_{\flat}^{\bullet}(\Delta^{n})}$${\Omega^{\bullet}_{B^{\prime}|\mathbb{R}}.}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\Omega_{\varphi}}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma^{\prime})}$

Next we construct the map
$\xi^{\mathrm{sm}}_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B)$
for a finitely generated $\mathbb{R}$-algebra $B$. For an algebraic $n$-form
$\omega\in\Omega^{n}_{B|\mathbb{R}}$ and a smooth simplex
$\sigma:\Delta^{n}\to\mathrm{spec}_{\mathbb{R}}B$ we set

$\left<\xi^{\mathrm{sm}}_{B}(\omega),\sigma\right>:=\mathfrak{T}_{n}(\mu^{\mathrm{sm}}(\sigma)(\omega))\in\mathbb{R}$

(the map $\mathfrak{T}_{n}$ was defined in Paragraph 5.2(4)).

######  Proposition 29 (cf. Proposition 23).

The above gives a morphism of complexes

$\xi^{\mathrm{sm}}_{B}:\Omega^{\bullet}_{B|\mathbb{R}}\to\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B).$

Lemma 28 allows us to prove the naturality of $\xi^{\mathrm{sm}}$ in algebras:

######  Lemma 30 (cf. Lemma 24).

Suppose $\varphi:B^{\prime}\to B$ is a homomorphism of finitely generated
$\mathbb{R}$-algebras. Then the following diagram is commutative:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B)}$${\Omega^{\bullet}_{B^{\prime}|\mathbb{R}}}$${\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B^{\prime}).}$$\scriptstyle{\xi^{\mathrm{sm}}_{B}}$$\scriptstyle{\Omega_{\varphi}}$$\scriptstyle{\xi^{\mathrm{sm}}_{B^{\prime}}}$$\scriptstyle{\mathrm{S}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}\varphi)}$

###  7.2. Comparing $\xi_{B}$ and $\xi^{\mathrm{sm}}_{B}$

Take $B$ a finitely generated $\mathbb{R}$-algebra.

######  Lemma 31.

Take $\sigma:\Delta^{n}\to\mathrm{spec}_{\mathbb{R}}B$ a smooth simplex. Then
the following diagram is commutative:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}(\Delta^{n})}$${\Omega_{\flat}^{\bullet}(\Delta^{n}).}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\nabla}$

###### Proof.

Consider the following diagram:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}(\Delta^{n})}$${\Omega_{\flat}^{\bullet}(\Delta^{n})}$${B}$${C^{\infty}(\Delta^{n})}$${\mathrm{Lip}(\Delta^{n}).}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\nabla}$$\scriptstyle{\theta^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\theta(\sigma)}$$\scriptstyle{\zeta}$

The left square and the outer contour commute by the definitions of $\mu$ and
$\mu^{\mathrm{sm}}$. The right square commutes by Paragraph 5.2(5). The bottom
triangle commutes by the definitions of $\theta$ and $\theta^{\mathrm{sm}}$.
Hence, by the universal property of $\Omega^{\bullet}_{B|\mathbb{R}}$ the
upper triangle also commutes. ∎

######  Lemma 32.

The following diagram is commutative:

(3)
${\Omega^{\bullet}_{B|\mathbb{R}}}$${\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B)}$${\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B).}$$\scriptstyle{\xi_{B}}$$\scriptstyle{\xi^{\mathrm{sm}}_{B}}$$\scriptstyle{\mathrm{res}}$

Here $\mathrm{res}$ is the restriction of Lipschitz cochains to smooth chains.

###### Proof.

Consider the diagram:

(4)
${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}(\Delta^{n})}$${\mathbb{R}}$${\Omega^{\bullet}_{\flat}(\Delta^{n}).}$$\scriptstyle{\mu^{\mathrm{sm}}(\sigma)}$$\scriptstyle{\mu(\sigma)}$$\scriptstyle{\nabla}$$\scriptstyle{\mathfrak{T}_{n}}$$\scriptstyle{\mathfrak{J}_{n}}$

The left triangle is commutative by Lemma 31 The right triangle is commutative
by Paragraph 5.2(4). Hence, the statement follows by the definitions of $\xi$
and $\xi^{\mathrm{sm}}$. ∎

###  7.3. Morphisms $\varkappa$ and $\underline{\varkappa}$

Let $M$ be a smooth manifold. Take an open set $U\subset M$, a smooth simplex
$\sigma:\Delta^{n}\to U$ and a smooth $n$-form $\omega\in\Omega^{n}(U)$. We
define
$\left<\varkappa(\omega),\sigma\right>:=\mathfrak{T}_{n}(\sigma^{*}(\omega))$.
By Stokes’ formula we obtain the morphism of complexes

$\varkappa:\Omega^{\bullet}(U)\to\mathrm{S}^{\bullet}_{\mathrm{sm}}(U).$

This way we obtain a morphism of complexes of presheaves
$\varkappa:\Omega^{\bullet}_{M}\to\mathrm{S}^{\bullet}_{\mathrm{sm},M}$.
Taking the composition with the sheafification map we get the morphism of
complexes of sheaves

$\underline{\varkappa}:\Omega^{\bullet}_{M}\to{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},M}.$

The following diagram is commutative:

(5)
${\underline{\mathbb{R}}_{M}[0]}$${\Omega^{\bullet}_{M}}$${{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},M}.}$$\scriptstyle{\epsilon}$$\scriptstyle{\epsilon}$$\scriptstyle{\underline{\varkappa}}$

The following diagram is also commutative:

(6)
${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{sm}}(M))}$${\mathrm{H}^{*}({}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm}}(M))}$${\mathbb{H}^{*}(M,{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},M})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathbb{H}^{*}(M,\Omega^{\bullet}_{M})}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\Upsilon}$$\scriptstyle{\mathrm{H}(\varkappa)}$$\scriptstyle{\mathrm{H}(\underline{\varkappa}(M))}$$\scriptstyle{\Upsilon}$$\scriptstyle{\mathbb{H}(\underline{\varkappa})}$

###  7.4. Identifying the map $\Psi_{B}$

In this subsection we take $M$ a smooth manifold.

######  Lemma 33.

For the inclusion of a finitely generated subalgebra $i:B\hookrightarrow
C^{\infty}(M)$ and a smooth singular simplex $\sigma:\Delta^{n}\to M$ the
following diagram commutes:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}}}$${\Omega^{\bullet}(M)}$${\Omega^{\bullet}(\Delta^{n}).}$$\scriptstyle{\mu^{\mathrm{sm}}(\Gamma_{B}\circ\sigma)}$$\scriptstyle{\Omega_{i}}$$\scriptstyle{\pi}$$\scriptstyle{\sigma^{*}}$

###### Proof.

Consider the diagram:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}}}$${\Omega^{\bullet}(M)}$${\Omega^{\bullet}(\Delta^{n})}$${B}$${C^{\infty}(M)}$${C^{\infty}(M)}$${C^{\infty}(\Delta^{n}).}$$\scriptstyle{\mu^{\mathrm{sm}}(\Gamma_{B}\circ\sigma)}$$\scriptstyle{\Omega_{i}}$$\scriptstyle{\pi}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\theta^{\mathrm{sm}}(\Gamma_{B}\circ\sigma)}$$\scriptstyle{i}$$\scriptstyle{\sigma^{*}}$

The left and right squares clearly commute. The middle square commutes by the
definition of $\pi$. The outer contour commutes by the definition of
$\mu^{\mathrm{sm}}$. For $b\in B$ we have

$\theta^{\mathrm{sm}}(\Gamma_{B}\circ\sigma)(b)=\hat{b}\circ\Gamma_{B}\circ\sigma=b\circ\sigma=\sigma^{*}(b)$

by the definition of $\theta^{\mathrm{sm}}$ and Equation 1. Hence, the bottom
quadrangle commutes. By the universal property of
$\Omega^{\bullet}_{B|\mathbb{R}}$ the upper quadrangle commutes. ∎

######  Lemma 34.

For the inclusion of a finitely generated subalgebra $i:B\hookrightarrow
C^{\infty}(M)$ the following diagram is commutative:

${\Omega^{\bullet}_{B|\mathbb{R}}}$${\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B)}$${\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}}}$${\Omega^{\bullet}(M)}$${\mathrm{S}^{\bullet}_{\mathrm{sm}}(M)}$$\scriptstyle{\xi^{\mathrm{sm}}_{B}}$$\scriptstyle{\Omega_{i}}$$\scriptstyle{\mathrm{S}_{\mathrm{sm}}(\Gamma_{B})}$$\scriptstyle{\pi}$$\scriptstyle{\varkappa}$

###### Proof.

For a form $\omega\in\Omega^{\bullet}_{B|\mathbb{R}}$ we have

$\left<\mathrm{S}_{\mathrm{sm}}(\Gamma_{B})(\xi^{\mathrm{sm}}_{B}(\omega)),\sigma\right>=\left<\xi^{\mathrm{sm}}_{B}(\omega),\Gamma_{B}\circ\sigma\right>=\mathfrak{T}_{n}(\mu^{\mathrm{sm}}(\Gamma_{B}\circ\sigma)(\omega))\overset{(3)}{=}$
$\overset{(3)}{=}\mathfrak{T}_{n}(\sigma^{*}(\pi(\Omega_{i}(\omega))))=\left<\varkappa(\pi(\Omega_{i}(\omega))),\sigma\right>.$

The equality (3) follows from Lemma 33. ∎

######  Lemma 35.

For the inclusion of a finitely generated subalgebra $i:B\hookrightarrow
C^{\infty}(M)$ the following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Phi_{B}}$$\scriptstyle{\mathrm{H}(\Omega_{i})}$$\scriptstyle{\Gamma_{B}^{*}}$$\scriptstyle{\mathrm{H}(\pi)}$$\scriptstyle{\Theta}$

###### Proof.

Consider the diagram:

(7)
${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B))}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,{}^{+}\mathrm{S}^{\bullet}_{\mathrm{Lip},\mathrm{spec}_{\mathbb{R}}B})}$${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B))}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},\mathrm{spec}_{\mathbb{R}}B})}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])}$${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{sm}}(M))}$${\mathbb{H}^{*}(M,{}^{+}\mathrm{S}^{\bullet}_{\mathrm{sm},M})}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathbb{H}^{*}(M,\Omega^{\bullet}_{M})}$$\scriptstyle{\Upsilon\circ\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\mathrm{res})}$$\scriptstyle{\mathbb{H}(\mathrm{res})}$$\scriptstyle{\mathrm{H}(\mathrm{S}_{\mathrm{sm}}(\Gamma_{B}))}$$\scriptstyle{\Upsilon\circ\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\cong}$$\scriptstyle{\Gamma_{B}^{*}}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Gamma_{B}^{*}}$$\scriptstyle{\Upsilon\circ\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\cong}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\Upsilon}$$\scriptstyle{\mathrm{H}(\varkappa)}$$\scriptstyle{\mathbb{H}(\underline{\varkappa})}$

The diagram without the bottom row commutes for obvious reasons. The bottom
left quadrangle commutes by Diagram 6. The bottom right triangle commutes by
Diagram 5. We do not know if the morphism $\mathbb{H}(\epsilon)$ in the middle
row is an isomorphism.

Next, we consider the following diagram:

(8)
${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{sm}}(M))}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathbb{H}^{*}(M,\Omega^{\bullet}_{M}).}$$\scriptstyle{\mathbb{H}(\epsilon)^{-1}\circ\Upsilon\circ\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\varkappa)}$$\scriptstyle{\Upsilon}$$\scriptstyle{\Theta^{-1}}$

By the bottom stage of the previous diagram the outer contour commutes. The
bottom triangle commutes by the definition of $\Theta$, Subsection 4.1. The
vertical map $\mathbb{H}(\epsilon)$ is an isomorphism. Therefore, the upper
triangle is commutative.

Consider the following diagram:

${\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})}$${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{Lip}}(\mathrm{spec}_{\mathbb{R}}B))}$${\mathbb{H}^{*}(\mathrm{spec}_{\mathbb{R}}B,\underline{\mathbb{R}}_{\mathrm{spec}_{\mathbb{R}}B}[0])}$${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{sm}}(\mathrm{spec}_{\mathbb{R}}B))}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\mathrm{S}^{\bullet}_{\mathrm{sm}}(M))}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$.$\scriptstyle{\mathrm{H}(\Omega_{i})}$$\scriptstyle{\Phi_{B}}$$\scriptstyle{\mathrm{H}(\xi_{B})}$$\scriptstyle{\mathrm{H}(\xi_{B}^{\mathrm{sm}})}$$\scriptstyle{\mathrm{H}(\mathrm{res})}$$\scriptstyle{\mathbb{H}(\epsilon)^{-1}\circ\Upsilon\circ\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\Gamma_{B}^{*}}$$\scriptstyle{\mathrm{H}(\mathrm{S}_{\mathrm{sm}}(\Gamma_{B}))}$$\scriptstyle{\mathrm{H}(\pi)}$$\scriptstyle{\mathbb{H}(\epsilon)^{-1}\circ\Upsilon\circ\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\mathrm{H}(\varkappa)}$$\scriptstyle{\Theta^{-1}}$

The upper triangle is commutative by the definition of $\Phi_{B}$. The left
pentagon is commutative by Lemma 34. The left triangle is commutative by
Diagram 3. The right pentagon is commutative by Diagram 7. The bottom right
triangle commutes by Diagram 8 and $\Theta$ is an isomorphism. Thus, the whole
diagram is commutative. Therefore, the claim follows. ∎

###  7.5. The calculation of $\Psi_{C^{\infty}(M)}$

In the previous section we have constructed a morphism
$\Psi_{A}:\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$
for every topological space $X$ and subalgebra $A\subset C(X)$. In case $X=M$
a smooth manifold and $A=C^{\infty}(M)$ we would like to calculate this map
explicitly.

By passing in Lemma 35 to colimit over all finitely generated subalgebras
$B\subset C^{\infty}(M)$ we get

######  Theorem 36.

The following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Psi_{C^{\infty}(M)}}$$\scriptstyle{\mathrm{H}(\pi)}$$\scriptstyle{\Theta}$

##  8\. The composition
$\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])\xrightarrow{\Lambda_{\mathcal{F}}}\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}})\xrightarrow{\Psi_{\mathcal{F}(X)}}\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$

######  Lemma 37.

Take $M$ a compact smooth manifold. Then the following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Psi_{C^{\infty}(M)}}$$\scriptstyle{\Lambda_{C^{\infty}_{M}}}$$\scriptstyle{\mathrm{id}}$

###### Proof.

Consider the following diagram:

${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}(M))}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Psi_{C^{\infty}(M)}}$$\scriptstyle{\mathrm{H}(\pi)}$$\scriptstyle{\Lambda_{C^{\infty}_{M}}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\Theta}$

The right triangle commutes by Theorem 36. The outer contour is commutative by
Theorem 13 . Since $\Theta$ is an isomorphism, the left triangle also
commutes. ∎

######  Corollary 38.

For a compact smooth manifold $M$ the following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{C(M)|\mathbb{R}})}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Psi_{C(M)}}$$\scriptstyle{\Lambda_{C_{M}}}$$\scriptstyle{\mathrm{id}}$

###### Proof.

The inclusion morphism of sheaves $C^{\infty}_{M}\hookrightarrow C_{M}$ allows
us to consider the diagram

${\mathrm{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C(M)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Lambda_{C_{M}}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\Lambda_{C^{\infty}_{M}}}$$\scriptstyle{\Psi_{C(M)}}$$\scriptstyle{\Psi_{C^{\infty}(M)}}$

The outer contour is commutative by Lemma 37. The right triangle is
commutative by Proposition 26. The upper triangle is commutative by
Proposition 12. Hence, the left triangle is commutative. ∎

######  Corollary 39.

For a compact Hausdorff space $X$ the following diagram is commutative:

${\mathrm{H}^{*}(\Omega^{\bullet}_{C(X)|\mathbb{R}})}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0]).}$$\scriptstyle{\Psi_{C(X)}}$$\scriptstyle{\Lambda_{C_{X}}}$$\scriptstyle{\mathrm{id}}$

###### Proof.

Choose a cohomology class
$\lambda\in\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])$. We show that
$(\Psi_{C(X)}\circ\Lambda_{C_{X}})(\lambda)=\lambda$. First, by [6, II.$5.10$]
there is a polyhedron $N$ (the geometric realization of some nerve) and a
continuous map $f:X\to N$, such that $\lambda=f^{*}(\delta)$ for some
$\delta\in\mathbb{H}^{*}(N,\underline{\mathbb{R}}_{N}[0])$. Second, there
exists a compact smooth manifold (with boundary) $M$ such that $N\subset M$
and $N$ is a deformation retract of $M$ (see [13, Theorem $1$]). There exists
$\gamma\in\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0])$ such that
$\gamma|_{N}=\delta$. Consider the composition
$g:X\xrightarrow{f}N\hookrightarrow M$. We have $\lambda=g^{*}(\gamma)$.
Consider the diagram

${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C(X)|\mathbb{R}})}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{X}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{C(M)|\mathbb{R}})}$${\mathbb{H}^{*}(M,\underline{\mathbb{R}}_{M}[0]).}$$\scriptstyle{\Lambda_{C_{X}}}$$\scriptstyle{\Psi_{C(X)}}$$\scriptstyle{\Lambda_{C_{M}}}$$\scriptstyle{g^{*}}$$\scriptstyle{\Psi_{C(M)}}$$\scriptstyle{\mathrm{H}(\Omega_{g^{*}})}$$\scriptstyle{g^{*}}$

This diagram is commutative by Propositions 12 and 26. By Corollary 38 the
equality $(\Psi_{C(X)}\circ\Lambda_{C_{X}})(\lambda)=\lambda$ follows. ∎

######  Theorem 40.

For a compact Hausdorff space $X$ and a soft subsheaf of algebras
$\mathcal{F}\hookrightarrow C_{X}$ the following diagram is commutative

${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}})}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathbb{H}^{*}(X,\underline{\mathbb{R}}_{X}[0]).}$$\scriptstyle{\Psi_{\mathcal{F}(X)}}$$\scriptstyle{\Lambda_{\mathcal{F}}}$$\scriptstyle{\mathrm{id}}$

###### Proof.

It immediately follows from naturality of $\Lambda$ and $\Psi$ (Propositions
12 and 26) and Corollary 39. ∎

##  9\. Piecewise polynomial functions

###  9.1. Polyhedra and rectilinear maps

A polyhedron $K$ is a finite set of affine simplices in $\mathbb{R}^{m}$ such
that

  1. (1)

for any $a\in K$ and any face $b\subset a$ we have $b\in K$;

  2. (2)

if $a,b\in K$ then $a\cap b$ is either a common face of $a$ and $b$ or empty.

A subset $P$ of $K$ that is also a polyhedron is called a subpolyhedron of
$K$. Define the space $|K|\subset\mathbb{R}^{m}$ as the union of all simplices
of $K$.

We say that a function $f:|K|\to|K^{\prime}|$ is a rectilinear map if for any
$a\in K$ there exists $b\in K^{\prime}$ such that $f(a)\subset b$ and $f$ maps
$a$ to $b$ affinely. We call two rectilinear maps $f,g:|K|\to|K^{\prime}|$
adjacent if for every simplex $a\in K$ the set $f(a)\cup g(a)$ is contained in
a simplex of $K^{\prime}$.

We say that a polyhedron $P$ is a minor of a polyhedron $K$ if $|P|\subset|K|$
and $|P|\hookrightarrow|K|$ is a rectilinear map (in other words, for any
$a\in P$ there is $b\in K$ such that $a\subset b$). We call a minor $P$ of $K$
a subdivision if $|P|=|K|$.

We call a polyhedron $S$ a star with the center $x\in\mathbb{R}^{m}$ if
$\\{x\\}$ is a vertex of $S$ and each maximal by inclusion simplex of $S$ has
$\\{x\\}$ as a vertex. Take a polyhedron $K$ with $x\in|K|$, then a minor $S$
of $K$ is called a star neighborhood of $x$ if $S$ is a star with center $x$
and $x\in\operatorname{int}_{|K|}|S|$.

For a polyhedron $K$ consider the algebra $\operatorname{Pol}(K)$ of functions
$\varphi:|K|\to\mathbb{R}$ such that for each simplex $a\in K$ $\varphi|_{a}$
is a polynomial. This algebra was considered, for example, in [1]. The algebra
$\operatorname{Pol}(K)$ is contravariant with respect to rectilinear maps, in
particular, if $i:|P|\hookrightarrow|K|$ is an inclusion of a minor we have
the restriction homomorphism
$\operatorname{Pol}(i):\operatorname{Pol}(K)\to\operatorname{Pol}(P)$.

######  Lemma 41.

Suppose $K,P_{1},\dots,P_{l}$ are polyhedra such that $P_{i}$ are minors of
$K$. Then there exist subdivisions $K^{\prime}$ of $K$ and $P_{i}^{\prime}$ of
$P_{i}$ such that $P_{i}^{\prime}$ is a subpolyhedron of $K^{\prime}$ for each
$i$.

###### Proof.

See [21, Addendum $2.12$]. ∎

######  Lemma 42.

Take $K$ a polyhedron, $U\subset|K|$ a set open in $|K|$ and $D\subset U$ a
set closed in $|K|$. Then there exists a minor $P$ of $K$ such that
$|P|\subset U$ and $D\subset\operatorname{int}_{|K|}|P|$.

######  Definition.

Let us define the sheaf of piecewise polynomial functions on a polyhedron.

  1. (1)

For a polyhedron $K$ and a subset $U\subset|K|$ open in $|K|$ we call a
function $s:U\to\mathbb{R}$ piecewise polynomial if for each point $x\in U$
there exists a minor $K_{x}$ of $K$, such that $|K_{x}|\subset U$ with
$x\in\operatorname{int}_{|K|}|K_{x}|$ and
$s|_{|K_{x}|}\in\operatorname{Pol}(K_{x})$.

  2. (2)

The set $\mathrm{PPol}(U)$ of piecewise polynomial functions on $U$ forms an
algebra (use Lemma 41).

  3. (3)

Take two sets $V\subset U$ open in $|K|$. It is not hard to see that the
restriction of a piecewise polynomial function $s\in\mathrm{PPol}(U)$ to $V$
is piecewise polynomial. Hence, the correspondence $U\mapsto\mathrm{PPol}(U)$
defines a sheaf on $|K|$ which we denote by $\mathrm{PPol}_{K}$.

######  Proposition 43.

For a polyhedron $K$ the sheaf $\mathrm{PPol}_{K}$ is soft.

###### Proof.

Take a subset $D\subset|K|$ closed in $|K|$ and an element of
$\mathrm{PPol}_{K}(D)$, which is represented by a set $U\subset|K|$ open in
$|K|$ with $D\subset U$ and a section $s\in\mathrm{PPol}_{K}(U)$. By Lemma 42
there exists a minor $P$ of $K$ such that
$D\subset\operatorname{int}_{|K|}|P|$ and $|P|\subset U$. Apply Lemma 42 again
to obtain a minor $P_{2}$ of $K$ such that
$|P|\subset\operatorname{int}_{|K|}|P_{2}|$ and $|P_{2}|\subset U$. By Lemma
41 there exist subdivisions $K^{\prime}$ of $K$, $P^{\prime}_{2}$ of $P_{2}$
and $P^{\prime}$ of $P$ such that $P^{\prime}$ and $P^{\prime}_{2}$ are
subpolyhedra of $K^{\prime}$. Take an element
$t\in\operatorname{Pol}(K^{\prime})$ such that $t|_{|P^{\prime}|}\equiv 1$ and
$t|_{|K^{\prime}|-|P^{\prime}_{2}|}\equiv 0$. The function $ts$ is a global
section of $\mathrm{PPol}_{K}$ and $ts$ and $s$ coincide in
$\mathrm{PPol}_{K}(D)$. ∎

###  9.2. The maps $\Lambda_{\mathrm{PPol}_{K}}$ and
$\Psi_{\mathrm{PPol}_{K}(|K|)|\mathbb{R}}$

Here we prove that for the sheaf of piecewise polynomial functions on $K$ the
maps $\Lambda_{\mathrm{PPol}_{K}}$ and
$\Psi_{\mathrm{PPol}_{K}(|K|)|\mathbb{R}}$ are in fact isomorphisms.

The following notion can be found in [5]. Two morphisms of
$\mathbb{R}$-algebras $\varphi_{0},\varphi_{1}:A\to B$ are called simply
homotopic if there exists a homomorphism $H:A\to B\otimes\mathbb{R}[t]$ such
that the following diagram is commutative for $\lambda=0,1$:

(9)
${A}$${B\otimes\mathbb{R}[t]}$${B.}$$\scriptstyle{\varphi_{\lambda}}$$\scriptstyle{H}$$\scriptstyle{t\mapsto\lambda}$

The following is a well known definition of homotopic morphisms of dg-algebras
and can be found in [16, II.$1$].

######  Definition.

Two morphisms of dg-algebras $\varphi_{0},\varphi_{1}:E\to E^{\prime}$ are
called homotopic if there exists a morphism $H:E\to
E^{\prime}\otimes\Omega^{\bullet}_{\mathbb{R}[t]|\mathbb{R}}$ such that the
following diagram is commutative for $\lambda=0,1$:

${E}$${E^{\prime}\otimes\Omega^{\bullet}_{\mathbb{R}[t]|\mathbb{R}}}$${E^{\prime}.}$$\scriptstyle{\varphi_{\lambda}}$$\scriptstyle{H}$$\scriptstyle{t\mapsto\lambda}$

Here $t\mapsto\lambda$ is the dg-algebra morphism that is the identity on
$E^{\prime}$ and sends $t$ to $\lambda$.

######  Lemma 44.

Homotopic morphisms of dg-algebras induce equal maps on the cohomology groups.

###### Proof.

See [16, Lemma II$.1$]. ∎

######  Proposition 45.

Suppose the morphisms $\varphi_{0},\varphi_{1}:A\to B$ of algebras are simply
homotopic. Then the induced morphisms
$\mathrm{H}(\Omega_{\varphi_{\lambda}}):\mathrm{H}^{*}(\Omega^{\bullet}_{A|\mathbb{R}})\to\mathrm{H}^{*}(\Omega^{\bullet}_{B|\mathbb{R}})$
for $\lambda=0,1$ are equal.

###### Proof.

We prove that the morphisms of dg-algebras
$\Omega_{\varphi_{\lambda}}:\Omega^{\bullet}_{A|\mathbb{R}}\to\Omega^{\bullet}_{B|\mathbb{R}}$
are homotopic for $\lambda=0,1$. By Lemma 44 it will imply that the maps on
the cohomology are equal. As $\varphi_{\lambda}$ are simply homotopic for
$\lambda=0,1$, we have a homomorphism $H:A\to B\otimes\mathbb{R}[t]$ such that
Diagram 9 commutes. The obvious morphisms
$\Omega^{\bullet}_{B|\mathbb{R}}\to\Omega^{\bullet}_{B\otimes\mathbb{R}[t]|\mathbb{R}}$
and
$\Omega^{\bullet}_{\mathbb{R}[t]|\mathbb{R}}\to\Omega^{\bullet}_{B\otimes\mathbb{R}[t]|\mathbb{R}}$
form the canonical morphism
$u:\Omega^{\bullet}_{B|\mathbb{R}}\otimes\Omega^{\bullet}_{\mathbb{R}[t]|\mathbb{R}}\to\Omega^{\bullet}_{B\otimes\mathbb{R}[t]|\mathbb{R}}$,
which is an isomorphism by [15, Corollary $4.2$].

Consider the following commutative diagram:

${\Omega^{\bullet}_{A|\mathbb{R}}}$${\Omega^{\bullet}_{B|\mathbb{R}}\otimes\Omega^{\bullet}_{\mathbb{R}[t]|\mathbb{R}}}$${\Omega^{\bullet}_{B\otimes\mathbb{R}[t]|\mathbb{R}}}$${\Omega^{\bullet}_{B|\mathbb{R}}.}$$\scriptstyle{u^{-1}\circ\Omega_{H}}$$\scriptstyle{\Omega_{H}}$$\scriptstyle{\Omega_{\varphi_{\lambda}}}$$\scriptstyle{\cong}$$\scriptstyle{u}$$\scriptstyle{t\mapsto\lambda}$$\scriptstyle{t\mapsto\lambda}$

We obtain that $\Omega_{\varphi_{\lambda}}$ are homotopic for $\lambda=0,1$. ∎

######  Lemma 46.

Suppose $f_{0},f_{1}:|K|\to|K^{\prime}|$ are two adjacent rectilinear maps of
polyhedra. Then the induced homomorphisms
$f_{0}^{*},f_{1}^{*}:\operatorname{Pol}(K^{\prime})\to\operatorname{Pol}(K)$
are simply homotopic.

###### Proof.

Consider the algebra $T$ of functions $s$ on $|K|\times[0,1]$ such that for
each $a\in K$ the restriction $s|_{a\times[0,1]}$ is a polynomial function.
The map $\operatorname{Pol}(K)\otimes\mathbb{R}[t]\to T$ sending $\beta\otimes
p(t)$ to the function $(x,t)\mapsto\beta(x)p(t)$ is an isomorphism. Construct
the homomorphism $H:\operatorname{Pol}(K^{\prime})\to T$ as
$H(\alpha)(x,t):=\alpha(f_{0}(x)(1-t)+f_{1}(x)t)$. As $f_{0}$ and $f_{1}$ are
adjacent, $H(\alpha)\in T$. Also,
$H(\alpha)|_{t=\lambda}=\alpha(f_{\lambda}(x))=f_{\lambda}^{*}(\alpha)$ for
$\lambda=0,1$. The needed homotopy map is the lift of the homomorphism $H$ to
$\operatorname{Pol}(K)\otimes\mathbb{R}[t]$. ∎

For a dg-algebra $E$ we consider the morphism of complexes, the
coaugmentation, $\epsilon:\mathbb{R}[0]\to E$ defined by $\epsilon(1)=1$.

######  Corollary 47 (Poincaré lemma).

Suppose $S$ is a star with center $x$. Then the coaugmentation
$\epsilon:\mathbb{R}[0]\to\Omega^{\bullet}_{\operatorname{Pol}(S)|\mathbb{R}}$
is a quasi-isomorphism.

###### Proof.

We denote by $Q$ the one-point polyhedron $\\{\\{x\\}\\}$. Consider the
rectilinear maps $\mathrm{col}:|S|\to|Q|$ and $i:|Q|\hookrightarrow|S|$. By
the definition of a star the composition $i\circ\mathrm{col}$ is adjacent to
the identity $\mathrm{id}:|S|\to|S|$. Consider the following diagram:

${\mathrm{H}^{*}(\mathbb{R}[0])}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\operatorname{Pol}(S)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\operatorname{Pol}(Q)|\mathbb{R}})}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\operatorname{Pol}(S)|\mathbb{R}})}$$\scriptstyle{\mathrm{H}(\epsilon)}$$\scriptstyle{\mathrm{H}(\epsilon)}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\epsilon)}$$\scriptstyle{\mathrm{H}(\Omega_{\operatorname{Pol}(i)})}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\mathrm{H}(\Omega_{\operatorname{Pol}(\mathrm{col})})}$

By Proposition 45 and Lemma 46 the bottom triangle is commutative. The rest of
the diagram commutes for obvious reasons. One can easily see that the morphism
of complexes
$\epsilon:\mathbb{R}[0]\to\Omega^{\bullet}_{\operatorname{Pol}(Q)|\mathbb{R}}$
is an isomorphism. Hence,
$\mathrm{H}(\epsilon):\mathrm{H}^{*}(\mathbb{R}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\operatorname{Pol}(K)|\mathbb{R}})$
is an isomorphism. ∎

######  Lemma 48.

Take $K$ a polyhedron and $x\in|K|$. Suppose $S_{0},S_{1}$ are star
neighborhoods of $x$ in $K$, then there exists a star neighborhood
$S^{\prime}$ of $x$ in $K$ which is a minor of $S_{i}$ for $i=0,1$.

###### Proof.

Follows from Lemma 41. ∎

Consider a polyhedron $K$ and a point $x\in|K|$. Let $\Sigma(x)$ be the set of
all star neighborhoods $S$ of $x$ in $K$. Define a partial order on
$\Sigma(x)$ as $S\leq S^{\prime}$ if $S^{\prime}$ is a minor of $S$. By Lemma
48 the set $\Sigma(x)$ is directed. Each star neighborhood $S$ of $x$ in $K$
gives rise to a homomorphism
$\operatorname{Pol}(S)\to(\mathrm{PPol}_{K})_{x}$. By taking the colimit over
the directed set $\Sigma(x)$ we obtain the homomorphism

$\chi:\underset{S\in\Sigma(x)}{\underset{\longrightarrow}{\operatorname{lim\,}}}\operatorname{Pol}(S)\to(\mathrm{PPol}_{K})_{x}.$

######  Lemma 49.

The map $\chi$ is an isomorphism.

###### Proof.

For injectivity, take a star neighborhood $S$ of $x$ in $K$ and
$t\in\operatorname{Pol}(S)$ such that $\chi(t)=0$. There exists a set $U$ open
in $|K|$ with $x\in U$ such that $t|_{U}=0$ as a function. Take a star
neighborhood $S^{\prime}$ of $x$ such that $|S^{\prime}|\subset U$ and
$S^{\prime}$ is a minor of $S$. We observe that
$t|_{|S^{\prime}|}=(t|_{U})|_{|S^{\prime}|}=0$ and, hence, $t$ is zero in
$\underset{\longrightarrow}{\operatorname{lim\,}}_{S\in\Sigma(x)}\mathrm{PPol}_{K}(S)$.

For surjectivity, take a germ $g\in(\mathrm{PPol}_{K})_{x}$, a set
$U\subset|K|$ open in $|K|$ and a section $\tilde{g}\in\mathrm{PPol}_{K}(U)$
such that $\tilde{g}_{x}=g$. As $\tilde{g}\in\mathrm{PPol}_{K}(U)$ there is a
minor $P$ of $K$ such that $|P|\subset U$, $x\in\operatorname{int}_{|K|}|P|$
and $\tilde{g}|_{|P|}\in\operatorname{Pol}(P)$. Take a star neighborhood $S$
of $x$ in $P$. Then $\chi$ maps $\tilde{g}|_{|S|}$ to $g$. ∎

######  Proposition 50.

For a polyhedron $K$ the coaugmentation

$\epsilon:\underline{\mathbb{R}}_{|K|}[0]\to{}^{+}\Omega^{\bullet}_{\mathrm{PPol}_{K}|\mathbb{R}}$

is a quasi-isomorphism.

###### Proof.

We prove that $\epsilon$ is a quasi-isomorphism on stalks. We have

$(\underline{\mathbb{R}}_{|K|}[0])_{x}\to({}^{+}\Omega^{\bullet}_{\mathrm{PPol}_{K}|\mathbb{R}})_{x}=(\Omega^{\bullet}_{\mathrm{PPol}_{K}|\mathbb{R}})_{x}=\Omega^{\bullet}_{(\mathrm{PPol}_{K})_{x}|\mathbb{R}}\cong\underset{S\in\Sigma(x)}{\underset{\longrightarrow}{\operatorname{lim\,}}}\Omega^{\bullet}_{\operatorname{Pol}(S)|\mathbb{R}}.$

The last isomorphism follows from Lemma 49. By Corollary 47 the morphism
$(\underline{\mathbb{R}}_{|K|}[0])_{x}\to\underset{\longrightarrow}{\operatorname{lim\,}}_{S\in\Sigma(x)}\Omega^{\bullet}_{\operatorname{Pol}(S)|\mathbb{R}}$
is a quasi-isomorphism. ∎

######  Theorem 51.

The maps

$\Lambda_{\mathrm{PPol}_{K}}:\mathbb{H}^{*}(|K|,\underline{\mathbb{R}}_{|K|}[0])\to\mathrm{H}^{*}(\Omega^{\bullet}_{\mathrm{PPol}_{K}(|K|)|\mathbb{R}})$

and

$\Psi_{\mathrm{PPol}_{K}(|K|)}:\mathrm{H}^{*}(\Omega^{\bullet}_{\mathrm{PPol}_{K}(|K|)|\mathbb{R}})\to\mathbb{H}^{*}(|K|,\underline{\mathbb{R}}_{|K|}[0])$

are isomorphisms.

###### Proof.

By Theorem 40 it is enough to prove that $\Lambda_{\mathrm{PPol}_{K}}$ is an
isomorphism. Recall that $\Lambda_{\mathrm{PPol}_{K}}$ is defined as the
diagonal map in the following diagram

${\mathbb{H}^{*}({|K|},\underline{\mathbb{R}}_{|K|}[0])}$${\mathbb{H}^{*}({|K|},{}^{+}\Omega^{\bullet}_{\mathrm{PPol}_{K}|\mathbb{R}})}$${\mathrm{H}^{*}({}^{+}\Omega^{\bullet}_{\mathrm{PPol}_{K}|\mathbb{R}}({|K|}))}$${\mathrm{H}^{*}(\Omega^{\bullet}_{\mathrm{PPol}_{K}({|K|})|\mathbb{R}}).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Lambda_{\mathrm{PPol}_{K}}}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$

By Proposition 50 the map $\mathbb{H}(\epsilon)$ is an isomorphism, hence,
$\Lambda_{\mathrm{PPol}_{K}}$ is an isomorphism. ∎

##  10\. When are the maps $\Lambda$ and $\Psi$ isomorphisms?

###  10.1. $\mathrm{H}^{0}(\Omega^{\bullet}_{A|k})$ for a function algebra $A$

Let $k$ be a field.

######  Definition.

We call the elements $a_{1},\dots,a_{n}$ of a $k$-algebra $A$ algebraically
independent over $k$ if there is no non-zero polynomial $q$ over $k$ such that
$q(a_{1},\dots,a_{n})=0$. We call an element $a$ algebraic over $k$ if there
is a non-zero polynomial $p$ over $k$ such that $p(a)=0$.

######  Lemma 52.

Let $A$ be a $k$-algebra. Suppose the elements $a_{1},\dots,a_{n}$ of $A$ are
algebraically independent over $k$. Then there exists a prime ideal
$\mathfrak{p}\subset A$ such that the images of $a_{1},\dots,a_{n}$ in
$A/\mathfrak{p}$ are algebraically independent over $k$.

###### Proof.

See [8, Proposition $6$]. ∎

######  Lemma 53.

Suppose $K/k$ is a field extension for $k$ of characteristic $0$. Consider the
differential $d:K\to\Omega^{1}_{K|k}$. Then for $x\in K$ we have $dx=0$ iff
$x$ is algebraic over $k$.

###### Proof.

See [17, $\S 26$]. ∎

######  Lemma 54.

Consider a $k$-algebra $A$ for $k$ of characteristic $0$ and take $a\in A$
such that $da=0$ in $\Omega^{1}_{A|k}$. Then $a$ is algebraic.

This is [8, Proposition $7$] where the assumption on characteristic being zero
is implicit.

###### Proof.

Assume $a$ is transcendental. By Lemma 52 there exists a prime
$\mathfrak{p}\subset A$ such that the image $\bar{a}$ of $a$ in
$A/\mathfrak{p}$ is transcendental. Hence, $\bar{a}$ is transcendental in the
field of fractions $\mathrm{Frac}(A/\mathfrak{p})$. By Lemma 53 $d\bar{a}\neq
0$ in $\Omega^{1}_{\mathrm{Frac}(A/\mathfrak{p})|k}$, which is a
contradiction.

∎

We call a subalgebra $A$ of $\mathrm{Maps}(X,k)$ a function algebra on a set
$X$.

######  Theorem 55.

Take $A$ a function algebra on $X$ and $a\in A$. Then $da=0$ in
$\Omega^{1}_{A|k}$ if and only if $a$ takes a finite number of values.

This theorem is proved in [8, proof of Theorem $8$]. We give a proof, which is
different in the ‘‘if’’ direction. A related result is [19, Proposition 3].

###### Proof.

Assume first $da=0$. By Lemma 54 there exists a non-zero polynomial $p$ such
that $p(a)=0$. Hence, $a$ takes a finite number of values.

Conversely, assume $a$ attains distinct values $r_{1},\dots,r_{m}\in k$ and
consider the polynomial $q(t)=(t-r_{1})\dots(t-r_{m})$. Clearly $q(a)=0$ and
$\gcd(q,q^{\prime})=1$. Hence, there exist polynomials $h_{0},h_{1}\in k[t]$
such that $h_{0}q+h_{1}q^{\prime}=1$. Substitute $a$ into this equality and
obtain $h_{1}(a)q^{\prime}(a)=1$; hence, $q^{\prime}(a)$ is invertible. Take
the equality $q(a)=0$ and apply $d$ to both sides, we get $q^{\prime}(a)da=0$
and subsequently $da=0$. ∎

######  Corollary 56.

Take $A$ a function algebra on a space $X$. Then
$\mathrm{H}^{0}(\Omega^{\bullet}_{A|k})$ is the vector space of functions
$a\in A$ that take a finite number of values.

A related result is [19, Proposition $5$].

######  Proposition 57.

For a soft subsheaf of algebras $\mathcal{F}\hookrightarrow C_{X}$ on a
compact Hausdorff space $X$ the map
$\Lambda_{\mathcal{F}}:\mathbb{H}^{0}(X,\underline{\mathbb{R}}[0])\to\mathrm{H}^{0}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}})$
is an isomorphism.

###### Proof.

By Corollary 56 the space
$\mathrm{H}^{0}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}})$ consists of
functions $f\in\mathcal{F}(X)$ that take a finite number of values. As
$\mathcal{F}$ is a subsheaf of $C_{X}$ and $X$ is compact, the functions in
$\mathcal{F}(X)$ that take a finite number of values are exactly the locally
constant functions from $\mathcal{F}(X)$. The morphism of sheaves of algebras
$\underline{\mathbb{R}}_{X}\to\mathcal{F}$ induces a homomorphism
$\underline{\mathbb{R}}_{X}(X)\to\mathcal{F}(X)=\Omega^{0}_{\mathcal{F}(X)|\mathbb{R}}$.
By above, this homomorphism can be extended to the morphism of complexes
$\tilde{\epsilon}:\underline{\mathbb{R}}_{X}(X)[0]\to\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}}$
and $\mathrm{H}(\tilde{\epsilon})$ is an isomorphism in degree $0$.

The coaugmentation
$\epsilon:\underline{\mathbb{R}}_{X}[0]\to{}^{+}\Omega^{\bullet}_{\mathcal{F}|\mathbb{R}}$
induces a morphism of complexes
$\epsilon(X):\underline{\mathbb{R}}_{X}(X)[0]\to{}^{+}\Omega^{\bullet}_{\mathcal{F}|\mathbb{R}}(X)$.
The following diagram commutes:

${\underline{\mathbb{R}}_{X}(X)[0]}$${{}^{+}\Omega^{\bullet}_{\mathcal{F}|\mathbb{R}}(X)}$${\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}}.}$$\scriptstyle{\epsilon(X)}$$\scriptstyle{\tilde{\epsilon}}$$\scriptstyle{\mathrm{sh}}$

Consider the diagram:

${\mathbb{H}^{0}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathrm{H}^{0}(\underline{\mathbb{R}}_{X}(X)[0])}$${\mathbb{H}^{0}(X,{}^{+}\Omega^{\bullet}_{\mathcal{F}|\mathbb{R}})}$${\mathrm{H}^{0}({}^{+}\Omega^{\bullet}_{\mathcal{F}|\mathbb{R}}(X))}$${\mathrm{H}^{0}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}}).}$$\scriptstyle{\mathbb{H}(\epsilon)}$$\scriptstyle{\Lambda_{\mathcal{F}}}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\epsilon(X))}$$\scriptstyle{\mathrm{H}(\tilde{\epsilon})}$$\scriptstyle{\cong}$$\scriptstyle{\Upsilon}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{H}(\mathrm{sh})}$$\scriptstyle{\cong}$

The outer contour commutes by the definition of $\Lambda_{\mathcal{F}}$. The
right triangle commutes by the above diagram. The left square obviously
commutes. Hence, the whole diagram is commutative. The upper map $\Upsilon$ is
an isomorphism by a property of $\Upsilon$ (see Subsection 2.1). Therefore,
$\Lambda_{\mathcal{F}}$ is an isomorphism. ∎

It turns out that the group
$\mathrm{H}^{0}(\Omega^{\bullet}_{C(X)|\mathbb{R}})$ behaves just as we
expect.

######  Corollary 58.

For a soft subsheaf of algebras $\mathcal{F}$ of $C_{X}$ on a compact
Hausdorff space $X$ the map

$\Psi_{\mathcal{F}(X)}:\mathrm{H}^{0}(\Omega^{\bullet}_{\mathcal{F}(X)|\mathbb{R}})\to\mathbb{H}^{0}(X,\underline{\mathbb{R}}[0])$

is an isomorphism.

###### Proof.

It follows directly from Proposition 57 and Theorem 40. ∎

###  10.2. $\Lambda$ and $\Psi$ are not isomorphisms in general

Take $k$ a field.

######  Lemma 59.

Suppose $k$ is of characteristic $0$. Suppose $F/k$ is a field extension and
$L/F$ is a finite field extension. Then the map

$\mathrm{H}(\Omega_{i}):\mathrm{H}^{*}(\Omega^{\bullet}_{F|k})\to\mathrm{H}^{*}(\Omega^{\bullet}_{L|k})$

induced by the inclusion $i:F\hookrightarrow L$ is injective.

###### Proof.

There is a morphism of complexes
$\Sigma:\Omega^{\bullet}_{L|k}\to\Omega^{\bullet}_{F|k}$ with the following
properties (see [15, $\S 16$]):

  1. (1)

if we consider $\Omega^{\bullet}_{L|k}$ as a $\Omega^{\bullet}_{F|k}$-module,
then $\Sigma$ is $\Omega^{\bullet}_{F|k}$-linear;

  2. (2)

the restriction of $\Sigma$ to the elements of degree $0$ coincides with the
trace $\sigma:L\to F$.

Therefore, the composition

$\Omega^{\bullet}_{F|k}\xrightarrow{\Omega_{i}}\Omega^{\bullet}_{L|k}\xrightarrow{\Sigma}\Omega^{\bullet}_{F|k}$

is the multiplication by $\sigma(1)=[L:F]$. As $k$ is of characteristic zero
the map $\mathrm{H}(\Omega_{i})$ is an injection. ∎

######  Lemma 60.

Let $K/k$ be a field extension, where $K$ is an infinite field, and
$K(\Gamma)/K$, where $\Gamma=\\{\gamma_{1},\dots,\gamma_{l}\\}$, be a purely
transcendental extension. Then the map

$\mathrm{H}(\Omega_{i}):\mathrm{H}^{*}(\Omega^{\bullet}_{K|k})\to\mathrm{H}^{*}(\Omega^{\bullet}_{K(\Gamma)|k}),$

induced by the inclusion $i:K\hookrightarrow K(\Gamma)$, is injective.

###### Proof.

The field $K(\Gamma)$ is the colimit over finite sets $S\subset
K[\Gamma]-\\{0\\}$ of the localizations of $K[\Gamma]$:

$K(\Gamma)\cong\underset{S\subset
K[\Gamma]-\\{0\\}}{\underset{\longrightarrow}{\operatorname{lim\,}}}K[\Gamma][S^{-1}].$

Therefore, it is enough to prove that the map

$\mathrm{H}(\Omega_{i_{S}}):\mathrm{H}^{*}(\Omega^{\bullet}_{K|k})\to\mathrm{H}^{*}(\Omega^{\bullet}_{K[\Gamma][S^{-1}]|k})$

induced by the inclusion $i_{S}:K\hookrightarrow K[\Gamma][S^{-1}]$ is
injective for each $S$.

There exists a point $(\bar{\gamma}_{1},\dots,\bar{\gamma}_{l})\in K^{l}$
which is not a zero of any element of $S$. Consider the homomorphism
$K[\Gamma]\to K$, $\gamma_{j}\mapsto\bar{\gamma}_{j}$. By the universal
property of localization it can be extended to a homomorphism
$\tau:K[\Gamma][S^{-1}]\to K$ such that $\tau|_{K}$ is the identity.
Therefore, the composition

$\mathrm{H}^{*}(\Omega^{\bullet}_{K|k})\xrightarrow{\mathrm{H}(\Omega_{i_{S}})}\mathrm{H}^{*}(\Omega^{\bullet}_{K[\Gamma][S^{-1}]|k})\xrightarrow{\mathrm{H}(\Omega_{\tau})}\mathrm{H}^{*}(\Omega^{\bullet}_{K|k})$

is the identity map, which suffices. ∎

######  Lemma 61.

Suppose the characteristic of $k$ is zero. The equation

(10) $\sum_{i=1}^{n}x_{i}(\partial/\partial x_{i})F_{i}=c$

where $c\in k$ and $c\neq 0$, has no solutions in rational functions $F_{i}\in
k(x_{1},\dots,x_{n})$.

###### Proof.

We proceed by induction on $n$. For $n=0$ the claim is obvious. Take
$l\in\mathbb{N}\cup\\{0\\}$ such that, for each $i$, $x_{1}$ is not a factor
in the denominators of $x_{1}^{l}F_{i}$. Multiply the equation (10) by
$x_{1}^{l}$ and apply $\left(\partial/\partial x_{1}\right)^{l}$ to both sides
of the equation. We get

$\sum_{i=1}^{n}\left(\partial/\partial
x_{1}\right)^{l}(x_{1}^{l}x_{i}(\partial/\partial x_{i})F_{i})=l!c.$

Notice that the operators $x(\partial/\partial x)$ and
$\left(\partial/\partial x\right)^{l}x^{l}$ commute. That can be easily
checked by induction. Hence, the operators $x_{i}(\partial/\partial x_{i})$
and $\left(\partial/\partial x_{j}\right)^{l}x^{l}_{j}$ commute for any $i$
and $j$. We get

$\sum_{i=1}^{n}x_{i}(\partial/\partial x_{i})\left(\partial/\partial
x_{1}\right)^{l}(x_{1}^{l}F_{i})=l!c.$

Put $G_{i}:=\left(\partial/\partial x_{1}\right)^{l}(x_{1}^{l}F_{i})$. We
obtain

$\sum_{i=1}^{n}x_{i}(\partial/\partial x_{i})G_{i}=l!c.$

But none of $G_{i}$ has $x_{1}$ as a factor in the denominator so we can
substitute $x_{1}:=0$ in the above equation and obtain

$\sum_{i=2}^{n}x_{i}(\partial/\partial x_{i})(G_{i}|_{x_{1}=0})=l!c.$

By the induction hypothesis this equation has no solutions in rational
functions. ∎

######  Lemma 62.

Suppose the characteristic of $k$ is zero. The closed form

$\omega=\dfrac{dx_{1}}{x_{1}}\wedge\dots\wedge\dfrac{dx_{n}}{x_{n}}\in\Omega^{n}_{k(x_{1},\dots,x_{n})|k}$

is not exact.

###### Proof.

Suppose there exists $\eta\in\Omega^{n-1}_{k(x_{1},\dots,x_{n})|k}$ such that
$d\eta=\omega$. Write $\eta$ as

$\eta=\sum_{i=1}^{n}F_{i}dx_{1}\wedge\dots\wedge\widehat{dx_{i}}\wedge\dots\wedge
dx_{n}$

where $F_{i}\in k(x_{1},\dots,x_{n})$. Then

$d\eta=\left[\sum_{i=1}^{n}(-1)^{i+1}(\partial/\partial
x_{i})F_{i}\right]dx_{1}\wedge\dots\wedge dx_{n}.$

The equality $d\eta=\omega$ then takes the form

$\left[\sum_{i=1}^{n}(-1)^{i+1}(\partial/\partial
x_{i})F_{i}\right]dx_{1}\wedge\dots\wedge dx_{n}=\dfrac{1}{x_{1}\dots
x_{n}}dx_{1}\wedge\dots\wedge dx_{n}.$

The vector $dx_{1}\wedge\dots\wedge dx_{n}$ of the vector space
$\Omega^{n}_{k(x_{1},\dots,x_{n})|k}$ over the field $k(x_{1},\dots,x_{n})$ is
not zero [17, p. $201$]. Hence,

$\sum_{i=1}^{n}(-1)^{i+1}(\partial/\partial x_{i})F_{i}=\dfrac{1}{x_{1}\dots
x_{n}}.$

Put $G_{i}:=(-1)^{i+1}x_{1}\dots\hat{x}_{i}\dots x_{n}F_{i}$. Then the above
equation takes the form

$\sum_{i=1}^{n}x_{i}(\partial/\partial x_{i})G_{i}=1.$

By Lemma 61 this equation has no solutions in rational functions. ∎

######  Theorem 63.

Suppose $k$ is of characteristic $0$. Take $A$ a $k$-algebra and a set of
invertible elements $a_{1},\dots,a_{n}$ in $A$ that are algebraically
independent over $k$. Then the closed form

$\omega=\dfrac{da_{1}}{a_{1}}\wedge\dots\wedge\dfrac{da_{n}}{a_{n}}\in\Omega^{n}_{A|k}$

is not exact.

###### Proof.

By Lemma 7 it is enough to consider $A$ being finitely generated. By Lemma 52
there is a prime ideal $\mathfrak{p}\subset A$ such that the elements
$a_{1},\dots,a_{n}$ are still algebraically independent in $A/\mathfrak{p}$.
Consider the field of fractions $\mathrm{Frac}(A/\mathfrak{p})$ which is
finitely generated over $k$ as a field. The elements $a_{1},\dots,a_{n}$ are
algebraically independent in $\mathrm{Frac}(A/\mathfrak{p})$.

Take a finite transcendental basis
$\hat{\Gamma}=\\{a_{1},\dots,a_{n},\gamma_{1},\dots,\gamma_{l}\\}$ of
$\mathrm{Frac}(A/\mathfrak{p})$ over $k$. The field
$\mathrm{Frac}(A/\mathfrak{p})$ is a finite extension of $k(\hat{\Gamma})$. By
Lemma 59 and Lemma 60 the composition $k(a_{1},\dots,a_{n})\to
k(\hat{\Gamma})\to\mathrm{Frac}(A/\mathfrak{p})$ induces an injection on the
de Rham cohomology.

The image of $\omega$ in $\Omega^{n}_{\mathrm{Frac}(A/\mathfrak{p})|k}$
coincides with the image of the closed form

$\omega_{0}=\dfrac{da_{1}}{a_{1}}\wedge\dots\wedge\dfrac{da_{n}}{a_{n}}\in\Omega^{n}_{k(a_{1},\dots,a_{n})|k},$

which is not exact by Lemma 62. ∎

######  Theorem 64.

Let $X$ be a topological space and $f\in C(X)$ take an infinite number of
distinct values. Take a subalgebra $A\subset C(X)$ such that $e^{\lambda f}\in
A$ for all $\lambda\in\mathbb{R}$. Then for each $n\geq 1$
$\mathrm{H}^{n}(\Omega^{\bullet}_{A|\mathbb{R}})\neq 0$, moreover, the map

$\Psi_{A}:\mathrm{H}^{n}(\Omega^{\bullet}_{A|R})\to\mathbb{H}^{n}(X,\underline{\mathbb{R}}_{X}[0])$

is not injective.

###### Proof.

Choose a set of linearly independent over $\mathbb{Q}$ numbers
$\lambda_{1},\dots,\lambda_{n}\in\mathbb{R}$ and consider the functions
$a_{i}:=e^{\lambda_{i}f(x)}\in A$. These functions are algebraically
independent and invertible, hence, the closed form

$\omega=\dfrac{da_{1}}{a_{1}}\wedge\dots\wedge\dfrac{da_{n}}{a_{n}}\in\Omega^{n}_{A|\mathbb{R}}$

is not exact by Theorem 63.

For the second part, we prove that $\Psi_{A}([\omega])=0$. Denote by
$i:A\hookrightarrow C(X)$ the inclusion. Consider the algebra $E\subset
C(\mathbb{R})$ generated as an algebra by the functions $e^{\lambda t}\in
C(\mathbb{R})$. We have the commutative diagram

${C(X)}$${A}$${C(\mathbb{R})}$${E,}$$\scriptstyle{f^{*}}$$\scriptstyle{\varphi}$

where the homomorphism $\varphi$ is induced by the homomorphism $f^{*}$. Then
the following diagram is commutative by Proposition 26:

${\mathrm{H}^{n}(\Omega^{\bullet}_{A|\mathbb{R}})}$${\mathbb{H}^{n}(X,\underline{\mathbb{R}}_{X}[0])}$${\mathrm{H}^{n}(\Omega^{\bullet}_{E|\mathbb{R}})}$${\mathbb{H}^{n}(\mathbb{R},\underline{\mathbb{R}}_{\mathbb{R}}[0])\hbox
to0.0pt{=0.\hss}}$$\scriptstyle{\Psi_{A}}$$\scriptstyle{\mathrm{H}(\Omega_{\varphi})}$$\scriptstyle{\Psi_{E}}$$\scriptstyle{f^{*}}$

The equality
$\mathbb{H}^{n}(\mathbb{R},\underline{\mathbb{R}}_{\mathbb{R}}[0])=0$ is well
known.

Consider the functions $\tilde{a}_{i}:=e^{\lambda_{i}t}\in E$ and the closed
form

$\tilde{\omega}=\dfrac{d\tilde{a}_{1}}{\tilde{a}_{1}}\wedge\dots\wedge\dfrac{d\tilde{a}_{n}}{\tilde{a}_{n}}\in\Omega^{n}_{E|\mathbb{R}}.$

We have $\omega=\Omega_{\varphi}(\tilde{\omega})$. By the commutativity of the
above diagram, $\Psi_{A}([\omega])=0$. Since $[\omega]\neq 0$, the map
$\Psi_{A}$ is not injective. ∎

######  Corollary 65.

Suppose $X$ is a compact Hausdorff space and $\mathcal{F}$ is a soft subsheaf
of $C_{X}$ such that $\mathcal{F}(X)$ satisfies the conditions imposed on the
algebra $A$ in Theorem 64. Then
$\Lambda_{\mathcal{F}}:\mathbb{H}^{n}(X,\underline{\mathbb{R}}_{X}[0])\to\mathrm{H}^{n}(\Omega^{\bullet}_{\mathcal{F}|\mathbb{R}})$
is not surjective.

###### Proof.

By Theorem 40 the composition
$\Psi_{\mathcal{F}(X)}\circ\Lambda_{\mathcal{F}}=\mathrm{id}$. By Theorem 64
$\Psi_{\mathcal{F}(X)}$ is not injective and, hence, $\Lambda_{\mathcal{F}}$
is not surjective. ∎

As the first example one can consider a smooth manifold $M$ of positive
dimension and $A=C^{\infty}(M)$. Then
$\mathrm{H}^{n}(\Omega^{\bullet}_{C^{\infty}(M)|\mathbb{R}})\neq 0$ for $n\geq
1$. Also, one can consider an infinite compact Hausdorff space $X$ and
$A=C(X)$. Then $\mathrm{H}^{n}(\Omega^{\bullet}_{C(X)|\mathbb{R}})\neq 0$ for
$n\geq 1$.

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St. Petersburg Department of Steklov Mathematical Institute
of Russian Academy of Sciences

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