Zoran Skoda formal geometry

Formal geometry and Borel’s theorem

UNDER CONSTRUCTION!!

Basics

Let $M$, $N$ be real smooth manifolds, $U,V$ open subsets of $N$, $a\in U\cap V\subset M$ and $f\colon U\to N$, $g\colon V\to N$ smooth maps. We say that $f$ is tangent of order $k\in\mathbb{N}$ to $g$ and we right $f\sim_k g$ if $f(a) = g(a)$ and for some (therefore every) pair of smooth charts $\phi\colon W\to\mathbb{R}^m$ around $a$, such that $\phi(a)=0$, $\psi\colon W'\to\mathbb{R}^n$, $\psi(f(a)) = 0$, we have

$$\|\psi(f(x)) - \psi(g(x))\| \leq h(x)\|\phi(x)-\phi(a)\|^k, \,\,\,\,\,\,\forall x\in W\cap f^{-1}(W')\cap g^{-1}(W'),$$

where $h(x)$ is a function defined and continuous on $W\cap f^{-1}(W')\cap g^{-1}(W')$ and such that $h(x)\to 0$ whenever $x\to a$. Equivalently, the Taylor series of all componenets of $f$ and $g$ agree up to order $k$. Tangency of order $k$ at point $x$ is a relation of equivalence on the set of smooth functions locally defined around $x$. The corresponding class of equivalence $f$ is the $k$-jet_ $j^k(f)$. $k$-jets of smooth maps $M\to N$ at point $x\in M$ form a set of $k$-jets $J^k(x)=J^k(M,N,x)$, with canonical structure of a real vector space. Note the obvious projections $J^{k+1}(x)\to J^k(x)$; these define an inverse system of vector spaces. Its limit is the space $J^\infty(x) = J^\infty(M,N,x)\colon = \lim_k J^k(M,N,x)$ of $\infty$-jets at $x$ of smooth maps from $M$ to $N$. A formal function is a an $\infty$-jet at $x$ of smooth functions from $M$ to $\mathbb{R}$.

(Emile Borel’s theorem on formal power series.) Let $\mathcal{C}^\infty_x$ be the vector space of germs of smooth real functions around a point $x\in\mathbb{R}^n$. Then a linear map $\mathrm{can}\colon\mathcal{C}^\infty_x\to J^\infty(\mathbb{R}^n,\mathbb{R},x)$ of taking the Taylor series at $x$ is a surjection.

Taylor series of a smooth function does not need to converge and if it does it may converge to a different function. For example, for the Cauchy function (equal $e^{-1/x^2}$ for $x\neq 0$ and equal $1$ for $x=0$) all Taylor coefficients at $0$ vanish. Of course, the germs of real analytic functions around $x$ could be identified with the corresponding formal functions: the restriction of $\mathrm{can}$ to the subspace of all germs of analytic functions is injective.

Differential operators

Let $R$ be a commutative unital $k$-algebra. For every $r\in R$ denote by $l(r)$ the $k$-linear map $l(r)(r') = r\cdot r'$ of muliplication by $r$ from the left. The function $r\mapsto l(r)$ defines a morphism of rings $l\colon R\to\operatorname{End}_k(R)$. For $f\in End_k(R)$ let us define $[f,r] := f\circ l(r) - l(r)\circ f\in\operatorname{End}_k(R)$. We say that (not necessarily cocomplete!) filtration $F_0\subset F_1\subset F_2\subset \ldots \subset End_k R$ is diferencial filtration if $[f,r] \in F_k$ for each $f\in F_{k+1}$ and for all $k\in\mathbb{N}_0$. Ring $End_k(R)$ has a maximal differential filtration in the sense of inclusion at each filtered level; its colimit is by definition the subring of regular differential operators $Diff(R)\subset End_k(R)$.

Regular diferential operator of order $s\in\mathbb{N}_0$ is a $k$-linear map $D\in\operatorname{End}_k(R)$ such that for all $r_0, r_1,\dots,r_s\in R$ we have $[\ldots[D,r_0],r_1],\ldots,r_s]=0$. Regular diferential operators of the order $s$ form a subspace $Diff_s(R)\subset End_k R$; regular diferential operator is an element of the space $Diff(R) = \cup_{s=0}^\infty Diff_s(R)$. The left multiplication $l(r)$ by the element $r\in R$ is a regular diferencial operator of order $0$; thus, if $R$ is an integral domain, then $R\subset\operatorname{Diff}_0(R)$. Diferential filtration $Diff_0(R)\subset Diff_1 \subset\ldots\subset Diff_k(R) \subset\ldots\subset End_k(R)$ is typically not cocomplete, i.e. its union
$Diff(R)\neq End_k(R)$. $C^\infty$-diferential operator on $C^\infty$-manifold $M$ is a regular diferencial operator over $C^\infty(M)$. They have the tensoriality property, i.e. if $f|_U=0$ on open subspace $U$, then $(D f)|_U = 0$.

Literature

• Émile Borel, Sur quelques points de la théorie des fonctions, Annales scientifiques de l’École Normale Supérieure, Sér. 3, 12 (1895), p. 9-55.

• A. Cannas da Silva, A. Weinstein, Geometric models for noncommutative algebras}, Berkeley Math. Lec. Notes Series, AMS 1999 http://math.berkeley.edu/~alanw/Models.pdf

• J. Dieudonné, Introduction to the theory of formal groups, Marcel Dekker, New York 1973.

• J. Dieudonné, Linearly compact spaces and double vector spaces over sfields, Amer. J. Math. 1951

• N. Durov, S. Meljanac, A. Samsarov, Z. \v{S}koda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309:1, str. 318-359 (2007) math.RT/0604096.

• S. Helgason, Differential geometry, Lie groups and symmetric spaces, Acad. Press 1978; Amer. Math. Soc. 2001.