The local ring of C-infinity functions

Stemwede, 2022-07-09

$$ \newcommand{\ra}{\rightarrow} \newcommand{\kC}{\mathcal{C}} \newcommand{\IR}{\mathbb{R}} \newcommand{\AINF}{\mathbb{A}} \newcommand{\CINF}{\kC^\infty} \newcommand{\half}{\frac{1}{2}} \newcommand{\IFF}{\Leftrightarrow} \newcommand{\inf}{\infty} \newcommand{\Ind}{\mathbb{1}} \newcommand{\IR}{\mathbb{R}} \newcommand{\IA}{\mathbb{A}} \newcommand{\IB}{\mathbb{B}} \newcommand{\IC}{\mathbb{C}} \newcommand{\ID}{\mathbb{D}} \newcommand{\IF}{\mathbb{F}} \newcommand{\IH}{\mathbb{H}} \newcommand{\II}{\mathbb{I}} \newcommand{\IL}{\mathbb{L}} \newcommand{\IN}{\mathbb{N}} \newcommand{\IP}{\mathbb{P}} \newcommand{\IQ}{\mathbb{Q}} \newcommand{\IR}{\mathbb{R}} \newcommand{\IS}{\mathbb{S}} \newcommand{\IV}{\mathbb{V}} \newcommand{\IZ}{\mathbb{Z}} \newcommand{\floor}[1]{\lfloor #1 \rfloor} \newcommand{\ceil}[1]{\lceil #1 \rceil} \newcommand{\set}[2]{\{\, #1 \;\vert\; #2 \,\}} \newcommand{\Set}[2]{\left\{, #1 ;\vert; #2 ,\right\}} \newcommand{\C}{,#} \newcommand{\CSet}[2]{#{, #1 ;\vert; #2 ,}} \newcommand{\qtext}[1]{\quad\text{#1}\quad} \newcommand{\stext}[1]{\;\text{#1}\;} \newcommand{\dbrackets}[1]{ [\![ #1 ]\!]} \newcommand{\KR}{\matcal{R}} \newcommand{\KA}{\matcal{A}} \newcommand{\KB}{\matcal{B}} \newcommand{\KC}{\matcal{C}} \newcommand{\KD}{\matcal{D}} \newcommand{\KF}{\matcal{F}} \newcommand{\KH}{\matcal{H}} \newcommand{\KI}{\matcal{I}} \newcommand{\KL}{\matcal{L}} \newcommand{\KN}{\matcal{N}} \newcommand{\KP}{\matcal{P}} \newcommand{\KQ}{\matcal{Q}} \newcommand{\KR}{\matcal{R}} \newcommand{\KS}{\matcal{S}} \newcommand{\KV}{\matcal{V}} \newcommand{\KZ}{\matcal{Z}} \newcommand{\gc}{\mathfrak{C}} \newcommand{\gd}{\mathfrak{D}} \newcommand{\gM}{\mathfrak{M}} \newcommand{\gm}{\mathfrak{m}} \newcommand{\gf}{\mathfrak{f}} \newcommand{\gu}{\mathfrak{U}} \newcommand{\fa}{\mathfrak{a}} \newcommand{\fg}{\mathfrak{g}} \newcommand{\fn}{\mathfrak{n}} \newcommand{\fk}{\mathfrak{k}} \newcommand{\fm}{\mathfrak{m}} \newcommand{\fp}{\mathfrak{p}} \newcommand{\curly}[1]{\mathcal{#1}} \newcommand{\op}[1]{\mathrm{#1}} \newcommand{\Cat}[1]{\mathfrak{#1}} \newcommand{\cat}[1]{\mathbf{#1}} \newcommand{\vphi}{\varphi} \newcommand{\sphi}{\phi} \newcommand{\eps}{\varepsilon} \newcommand{\tensor}{\otimes} \newcommand{\tensors}{\tensor\dots\tensor} \newcommand{\Tensor}{\bigotimes} \newcommand{\ra}{\rightarrow} \newcommand{\lra}{\longrightarrow} \newcommand{\la}{\leftarrow} \newcommand{\lla}{\longleftarrow} \newcommand{\isom}{\cong} \newcommand{\epi}{\twoheadrightarrow} \newcommand{\mono}{\hookrightarrow} \newcommand{\del}{\partial} \newcommand{\union}{\cup} \newcommand{\dotcup}{\ensuremath{\mathaccent\cdot\cup}} \newcommand{\dunion}{\dotcup} \newcommand{<}{\langle} \renewcommand{>}{\rangle} \newcommand{\inpart}[1]{\in\text{\part}(#1)} \newcommand{\Vsum}{\bigoplus} \newcommand{\vsum}{\oplus} \renewcommand{\S}{\mathfrak{S}} \newcommand{\id}{\mathrm{id}} \newcommand{\rk}{\mathrm{rk}} \newcommand{\Diff}{\mathrm{Diff}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\Pic}{\mathrm{Pic}} \newcommand{\Spec}{\mathrm{Spec}} \newcommand{\End}{\mathrm{End}} \newcommand{\Ext}{\mathrm{Ext}} \DeclareMathOperator{\Supp}{\mathrm{Supp}} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Alt}{\Lambda} \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\td}{td} \DeclareMathOperator{\pr}{pr} $$

Setting

Definition. The local ring of $C^\infty$ functions around $0 \in \IR^n$ is defined as

$$ \AINF^n = \lim_\ra { \CINF(U,\IR) ,|, 0 \in U \subset \IR^n }. $$

The inductive limit is taken along the restriction maps $\rho_V: \CINF(U,\IR) \ra \CINF(V,\IR)$ for $U \subset V$.

More explicitly, an element $a \in \AINF$ is hence represented by a pair $(a,U), a \in \CINF(U,\IR)$. Two pairs $(a,U), (b,V)$ represent the same element in $\AINF^n$ if there is a $W \subset U \cap V, 0 \in W$ so that $\rho_W(a) = \rho_W(b)$.

Naturality. Any ($C^\infty$-)differentiable map $f: \IR^n \ra \IR^m$ with $f(0) = 0$ gives rise to a map $f^*: \AINF^m \ra \AINF^n$ (“pullback”), which sends $(b,V)$ to $(b \circ f, f^{-1}(V))$.

Evaluation. For $a \in \AINF^n$, we denote by $e[a] := ev_0[a] := a(0) \in \IR$, the evaluation of $a$ at $0 \in \IR^n$. This is operatin can be identified with the pullback along the injection $0: \IR^0 \ra \IR^n$, $e = 0^*$.

Constants. For $\lambda \in \IR$ the constant function $\IR^n \ra \IR, x \mapsto \lambda$ by the same letter $\lambda \in \AINF^n$. If we want to stress the function character of $\lambda$ we may use the symbol $\underline{\lambda}$. The resulting linear map $c: \IR \ra \AINF^n$ can be identified with $c = \pi^*$, where $\pi:\IR^n \ra \IR^0$ is the projection to a point.

Elementary Operations. Let $i \in {1,\dots,n}$. We have the following operations on $\AINF^n$.

Coordinate fuctions $x^i \in \AINF^n$ with $x^i(x_1,\dots,x_n) = x_i$ on any $0 \in U \subset \IR^n$,
Partial derivatives $ \del_i = \del_i^x = \frac{\del}{\del x^i}$ those are linear operators $\AINF^n \ra \AINF^n $.
Axis injections $x_i: \IR \ra \IR^n$, giving rise to maps $x_i^*: \AINF^n \ra \AINF^1$ via pullback.

For a multi-index $I \in \IN^n, I = (i_1, \dots, i_n)$, we define:

The monomial $x^I = (x^1)^{i_1} \cdot \dots \cdot (x^n)^{i_n} \in \AINF^n$
The differential operator $\del^I = \del_1^{i_1} \cdot \dots \cdot \del_n^{i_n}$

Elementary Relations.

Product rule. $\del_i[ a \cdot b ] = \del_i[a] \cdot b + a \cdot \del_i[b]$.
Iterated Product Rule. $$ \del^I[a \cdot b] = \sum_{J,K \in \IN^n, J+K=I} \frac{I!}{J! K!} \del^J[a] \cdot \del^K[b]. $$
Chain Rule. $\del_j f^*[b] = \del_j[ b \circ f ] = \sum_k \del_k[b] \circ f \cdot \del_i[f^k] = \sum_k f^{*}\del_k[b] \cdot \del_i[f^k]$, where $f^k$ are the components $f^k = x^k \circ f$.

Taylor Series

Definition. The taylor series of $a \in \AINF^n$ is the formal power series $$ T[a] = \sum_{I \in \IN^n} e \del^I [a] \cdot \frac{x^I}{I!} = \sum_{I \in \IN^n} \frac{\del^I a}{\del x^I}(0) \frac{x^I}{I!} \in \IR \dbrackets{ x^1, \dots, x^n }, $$ where $I! = i_1! \cdot \dots \cdot i_n!$. The degree-k taylor polyonmial is the degree-k truncation of the taylor series: $$ T_k[a] = \sum_{I, |I| \leq k} e \del^I [a] \cdot \frac{x^I}{I!} \in \IR[x^1, \dots, x^n]. $$

Properties.

For a polynomial $p \in \IR[x^1, \dots, x^n]$, we have $T_k[p] = p$ if $k \geq deg(p)$.
For $a,b \in \AINF^n$, we have $$ T(a \cdot b) = T(a) \cdot T(b), \quad\text{and}\quad T_k(a) \cdot T_k(b). $$

Proof The first property follows from the observation that $e \del^I[x^J] = I!$ if $I = J$ and zero otherwise. The second and third property follow from the iterated product rule.

Theorem (Borel) For every power series $p \in \IR\dbrackets{ x^1, \dots, x^n }$ there exists a $C^\infty$ function $a \in \AINF^n$ with $T(a) = p$. In other words, the map $T : \AINF^n \ra \IR\dbrackets{ x^1, \dots, x^n }$ is surjective, and hence $ \IR\dbrackets{ x^1, \dots, x^n } \isom \AINF^n / \ker(T)$.

For a proof see Ieke Moerdijk, Gonzalo E. Reyes: Models for Smooth Infinitesimal Analysis, p13. or ncatlab.

Commutative Algebra

$\IR$-Algebra Structure. The ring $\AINF^n$ inherits the structure of a unital $\IR$ algebra form $\CINF(U,\IR)$. In other words elements $a,b \in \AINF^n$ can be added, multiplied and multiplied by scalars $\lambda \in \IR$, so that the usual associativity and distributivity relations hold. The unit element $1 \in \AINF^n$ is the constant function $1$.

The structure maps $\rho_0: \CINF(U,\IR) \ra \AINF^n$ are morphisms of $\IR$-algebras.
The pullback maps $f^*$ are morphisms of $\IR$-algebras.
The “constants map” $c$ is a morphism of $\IR$-algebras.
The evaluation map $e$ is a morphism of $\IR$-algebras.

Denote the kernel of the evaluation map by $\fm = \ker(e) = \{ a \in \AINF^n | a(0) = 0 \} \subset \AINF^n$. It is a maximal ideal in $\AINF^n$, since the co-domain of $e$ is a field $\IR$.

Hadamard Lemma. The maxiamal $\fm$ ideal is generated by the coordinate fuctions $x^i$, i.e. every element $a \in \fm$ can be written in the form $a = \sum_{i=1}^n x^i \cdot a_i$ with $a_i \in \AINF^n$.

Proof. Let $a \in \fm$, and set $a_i(x) = \int_0^1 (\del_i a)(t \cdot x) dt$. Now $\frac{\del}{\del t} a(t \cdot x) = \sum_{i=1}^n (\del_i a)(tx) \cdot x^i(x)$, hence

$$ \sum_{i=1}^n x^i \cdot a_i = \int_0^1 \sum_{i=1}^n (\del_i a)(tx) \cdot x^i(x) dt = \int_0^1 \frac{\del}{\del t} a(t \cdot x) dt = a(x) - a(0) = a(x). qed. $$

Corollary.

If $a = \sum_{i=1}^n x^i \cdot a_i$ is a Hadamard representation of $a \in \fm$, then $a_i(0) = e \del_i[a]$
The powers $\fm^k$ are generated by the monomials $x^I, I \in \IN^n, |I| = k$.
The intersection $\fm^\inf := \cap_{k \geq 0} \fm^k$ is equal to $\ker(T) = \set{a}{ \del^I[a] = 0 \stext{for all} I}$.
The map $\IR[x^1, \dots, x^n] \ra \AINF^n$, mapping a polyomial to the associated $C^\infty$-function-germ is injective, and induces an isomorphism: $$ \IR[x^1, \dots, x^n] / (x^1, \dots, x^n)^k \lra \AINF^n / \fm^k. $$ In particular, the completions are isomorphic: $\IR\dbrackets{ x^1, \dots, x^n} \isom \hat \AINF^n$.

Lemma.

The evaluation map is the only $\IR$-alebgra morphism $\AINF^n \ra \IR$.

Lemma For an open set $0 \in U \subset \IR^n$, denote the restriction map by $\rho: C^\infty(U, \IR) \ra \AINF^n$, let $\fm_0 \subset C^\infty(U, \IR)$ be the kernel of the evaluation map at $0$. Then,

$\rho$ is surjective and induces and isomorphism $ C^\infty(U, \IR) / \ker(\rho) \isom \AINF^n$.
$\rho$ induces an isomophism $ C^\infty(U, \IR)_{\fm_0} \isom \AINF^n$.

Proof. To prove the first claim choose a representative $(V, a) \in \AINF^n$, with $V \subset U$. Choose $\eps > 0$ small enough so that the ball $B_{2\eps}$ lies in $V$. Choose a bump function $\delta$ with $\delta = 1$ on $B_{\eps}$ and $\delta = 0$ outside of $B_{2\eps}$. Then $a \cdot \delta $ is defined on $V$ and equal to zero outside of $B_{2\eps}$, and hence extends to U.

To prove the second claim we first note that $b(0) \neq 0$ implies that $b(x) \neq 0$ for $x$ in an open subset of $0$, hence $b$ is invertible in $\AINF^n$. This shows that $\rho$ induces a map from the localization. The map is surjective by the claim we just proved. To show the map is injective, assume that the formal quotient $(a,b)$ maps to $a/b = 0 \in \AINF^n$. This means that $a = 0$ in an open neightbourhood $V$ of $0$. Let $\delta$ be a bump function with $\delta(1) = 1$ and $\delta(x) = 0$ outside of $V$. Then $\delta \notin \fm_0$ but $\delta a = 0 \in C^\infty(U, \IR)$, showing that $(a,b) = 0 \in C^\infty(U, \IR)_{\fm_0}$.

Invariant Theory

Group Action. Let $\mathrm{Diff}_0^n$ be the group of local diffeomorphisms of $\IR^n$. Elements $\varphi \in \mathrm{Diff}_0^n$ are represented by diffeomorophisms $\varphi: U \ra V$, with $0 \in U, 0 \in V$. Two representatives $\varphi: U \ra V, \varphi': U' \ra V'$ are equivalent if they agree on $U \cap U'$.

Evenry element $\varphi \in \mathrm{Diff}_0^n$ gives rise to an liear isomorphism $ \varphi^* $ of $\AINF^n$ via the pullback operation. In this way we optain a group action $\mathrm{Diff}_0^n \ra GL(\AINF^n), \varphi \mapsto \varphi^*$.

Question: For any any natural “structure” $F$ on $\AINF^n$, classify elements of $F$ up to diffeomorphism.

Examples:

$F = id$: Classify local functions up to diffeomorphism. In the case $n=1$ we claim that $\AINF^1 / \mathrm{Diff}_0^n = c(\IR) \union \{ x^1 \}$. For $n > 2$
$F = T$ tangent space: Classify tangent vectors up to diffeopmorphis. We have $T(\AINF^n) / \mathrm{Diff}_0^n = \{ 0, \del_1 \}$, reflecting the fact that two non-zero tangent vector can be transformed into each other via a (linear) diffeomorphism.