\documentclass[11pt]{article} \usepackage{url} %version 1.0,\ \ 26.01.2018 %version 1.1,\ \ 31.01.2018 ispravleniya ot Kati Amerik %version 1.2,\ \ 03.02.2018 2.9 and 2.11 %version 1.3,\ \ 17.02.2018 more problems with support %version 1.4,\ \ 21.02.2018 support: more corrrections from M. Temkin \newcommand{\version}{version 1.4,\ \ 21.02.2018} \newcommand{\firstdate}{31.01.2018} \addtolength{\topmargin}{-12mm} \addtolength{\textheight}{25mm} \addtolength{\oddsidemargin}{-10mm} \addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Hogde theory, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{2}{Hodge theory 2: Differential operators} {\scriptsize {\bf Rules:} You may choose to solve only ``hard'' exercises (marked with !, * and **) or ``ordinary'' ones (marked with ! or unmarked), or both, if you want to have extra stuff to work. To have a perfect score, a student must obtain (in average) a score of 10 points per week. If you have got credit for 2/3 of ordinary problems or 2/3 of ``hard'' problems, you receive $6t$ points, where $t$ is a number depending on the date when it is done. Passing all ``hard'' or all ``ordinary'' problems brings you $10t$ points. Solving of ``**'' (extra hard) problems is not obligatory, but each such problem gives you a credit for 2 ``*'' or ``!'' problems in the ``hard'' set. The first 3 weeks after giving a handout, $t=1.5$, between 21 and 35 days, $t=1$, and afterwards, $t=0.7$. The scores are not cumulative, only the best score for each handout counts. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %\subsection{Commutative algebra in the ring $C^\infty M$.} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %\exercise %Let $M$ be a positive-dimensional manifold. %Find an ideal in $C^\infty M$ which is not finitely generated. %\ez % %\hint %Consider an ideal of all functions %vanishing in a closure of an open set. %\eh % %\definition %A module $A$ over a ring $R$ is called {\bf projective} %if it is isomorphic to a direct summand of a free $R$-module, %that is, if $A\oplus B$ is free for some $R$-module $B$. %\ed % %\exercise[*] %Let $F$ be a vector bundle over a compact manifold %$M$. Consider the set of smooth sections of $F$ as %a $C^\infty M$-module. Prove that it is projective. %Prove that all projective $C^\infty M$-modules %are obtained this way. %\ez % %\definition %Let $R$ be a ring over a field $k$, and %$D:\; R \arrow A$ a $k$-linear map from %$R$ to an $R$-module. It is called %{\bf a derivation} if it satisfies the %Leibniz rule: $D(xy) = y D(x) + x D(y)$. %\ed % %\definition %Consider a free $R$- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Differential operators (after Grothendieck)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $R$ be a commutative ring over a field $k$. Given $a\in R$, consider the map $L_a:\; R \arrow R$ mapping $x$ to $ax$. Define $\Diff^k(R)\subset \Hom_k(R, R)$ inductively as follows. The $\Diff^0(R)$ is the space of all $R$-linear maps from $R$ to $R$, that is, the space of all $L_a$, $a\in R$. The space $\Diff^k(R)$, $k>0$ is \[ \Diff^k(R):= \{ D\in \Hom_k(R, R)\ \ |\ \ [L_a, D]\in \Diff^{k-1}(R) \ \ \ \forall a\in R.\} \] The union of all $\Diff^i(R)$ is called {\bf the space of differential operators on $R$}. Differential operators on the ring $C^\infty M$ is called {\bf differential operators on $M$,} denoted $\Diff^*(M)$. \ed \exercise\label{_filtra_diffe_Zadacha_} Let $D^i \in \Diff^i(R)$, $D^j \in \Diff^j(R)$ be differential operators. Prove that the composition $D^i D^j$ lies in $\Diff^{i+j}(R)$. \ez \hint Use induction and identity $[ v, D^i D^j] = [v, D^i] D^j + D^i [v, D^j]$ \eh \exercise\label{_filtra_commu_Zadacha_} Let $D^i \in \Diff^i(R)$, $D^j \in \Diff^j(R)$ be differential operators. Prove that the commutator $[D^i, D^j]$ lies in $\Diff^{i+j-1}(R)$. \ez \hint Use induction and Jacobi identity \[ [ v, [D^i, D^j]] = [[v, D^i], D^j] + [D^i, [v, D^j]]. \] \eh \definition Let $R$ be a $k$-algebra, and $D:\; R \arrow A$ a $k$-linear map from $R$ to an $R$-module. It is called {\bf a $k$-derivation}, or just {\bf derivation} if it satisfies the Leibniz rule: $D(xy) = y D(x) + x D(y)$. \ed \exercise \enum \ite Prove that $D(k)=0$ for any $k$-derivation on a $k$-algebra (we assume $\Char k=0$). \ite [!] Let $R$ be a finite extension of a field $k$ of characteristic 0. Prove that the space $\Der_k(R,R)$ of derivations vanishes. \ee \ez \exercise[**] Let $R$ be the ring of continuous functions on a manifold $M$. Prove that $\Der_\R(R,R)=0$, or find a counterexample. \ez \exercise[*] Let $x_1, ..., x_n$ be coordinates on $\R^n$. Prove that any derivation on $C^\infty \R^n$ is written as coordinates as $D(f) = \sum_{i=1}^n f_i \frac d {dx_i}$, where $f_i\in C^\infty M$. \ez \hint Use the Hadamard lemma and an inclusion $D(I^k)\subset I^{k-1}$ (Exercise \ref{_diff_ope_ideal_Exercise_}). \eh \exercise[!] Let $D\in \Diff^1(R)$ be a differential operator of first order. Prove that $D- D(1)$ is a derivation of $R$. Prove that $\Diff^1(R)/\Diff^0(R)$ is isomorphic to the space of derivations of $R$. \ez \exercise Let $R= k[t]$ be an algebra of polynomials over a field $k$ of characteristic 0, and $D\in \Diff^k(R)$. \enum \ite Prove that $D$ is uniquely determined by its restriction on polynomials of degree $\leq k$. \ite[*] Prove that $\Diff^k(R)$ is a free $k[t]$-module, generated by $\tau_0, \tau_1, ... \tau_k$, where $\tau_i$ maps all $1, t, t^2, t^3, ... t^k$ except $t^i$ to 0, and $t^i$ to 1. \ite[**] Prove that $\Diff^*(\R[t_1, ..., t_n])$ is an algebra freely generated by generators $t_1, ..., t_n$ and $\frac d {dt_1}, ..., \frac d {dt_n}$ and relations $[t_i, \frac d {dt_j}] =\delta_{ij}$. \ee \ez \exercise\label{_diff_ope_ideal_Exercise_} Let $I\subset R$ be an ideal, and $D\in \Diff^k(R)$. Prove that $D(I^{k+1}) \subset I$. \ez \hint Use induction in $k$ and identity $[D, L_a L_b]= [D, L_a] L_b + L_a[D, L_b]$. \eh %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Differential operators are local} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $f\in {\cal F}(M)$ be a section of a sheaf ${\cal F}$, and $U\subset M$ the union of all open subsets $V\subset M$ such that $f \restrict V=0$. The complement $M \backslash U$ is called {\bf support} of $M$, denoted $\Supp(f)$. \ed \exercise Prove that $\Supp(f)$ is the set of all $m\in M$ such that the germ of $f$ in $m$ is non-zero. \ez \definition Let $f, g\in C^\infty M$. We say that {\bf $f$ is divisible by $g$} if $f=f'g$ for some $f'\in C^\infty M$. \ed \exercise\label{_divisible_open_Exercise_} Let $K\subset U \subset M$ be subsets of a manifold $M$, where $U$ is open and $K$ compact. \enum \ite Prove that any function with support in $K$ is divisible by any function which is non-zero in $U$. \ite Let $f$ be a function on $M$ which is non-zero somewhere in $U\backslash K$. Prove that there exists a function $g$ with $\Supp(g)\supset K$ such that $f$ is not divisible by $g$. \ee \ez \hint If $f$ vanishes in $x\in U$, but support of $f$ contains $x$, construct $g$ which is equal to $f^2$ in a neighbourhood of $x$ and supported in $K\subset U$, and show that $f$ is not divisible by $g$. \eh \exercise\label{_divisible_power_Exercise_} Let $D\in \Diff^*(M)$ be a differential operator, and $f$ a function divisible by any power of $g$. Prove that $D(f)$ is divisible by any power of $g$. \ez \definition An operator $D:\; C^\infty M \arrow C^\infty M$ is called {\bf local} if it $\Supp(D(f))\subset \Supp(f)$ for any function $f\in C^\infty M$. \ed \exercise \enum \ite Let $K$ be a closed subset of $M$. Prove that $\Supp(f)\subset K$ if and only if $f$ is infinitely divisible by any function $g$ which is non-zero everywhere in $K$. \ite[!] Prove that any differential operator (in the sense of Grothendieck) is local. \ee \ez \hint Use Exercise \ref{_divisible_open_Exercise_} and Exercise \ref{_divisible_power_Exercise_}. \eh %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Jet spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a smooth manifold, $f\in C^\infty M$ a function, and $x\in M$ a point. Denote by $\goth m_x$ the maximal ideal of $x$, that is, the ideal of all functions vanishing in $x$. The {\bf $k$-jet of $f$ in $x$} is an image of $f$ in $C^\infty M/\goth m_x^{k+1}$. {\bf The space of $k$-jets of functions in $x$} is $C^\infty M/\goth m_x^{k+1}$, and the corresponding vector bundle $\Jet^k(M)$ with the fiber in $x$ equal to $C^\infty M/\goth m_x^{k+1}$ is called {\bf the bundle of $k$-jets}. \ed \exercise Let $\Delta$ be the diagonal in $M\times M$, and $J_\Delta\subset C^\infty(M\times M)$ the ideal of all functions vanishing in $M$. Denote by $\pi:\; M\times M\arrow M$ the projection to the first component \enum \ite[!] Prove that the cotangent bundle $T^*M$ is isomorphic as a sheaf of $C^\infty M$-modules to $J_\Delta/J_\Delta^2$, considered as a sheaf on $\Delta=M$, with $C^\infty M$-action induced by $\pi^* C^\infty M\restrict\Delta\cong C^\infty M$ (that is, we take the functions which are constant along the second component, and consider $J_\Delta/J_\Delta^2$ as a sheaf of modules over such functions). \ite[*] Prove that the bundle of $k$-jets is identified, in a similar way, with the sheaf $C^\infty (M\times M) / J_\Delta^k$, considered as a sheaf on $\Delta=M$, with $C^\infty M$-action induced by $\pi^* C^\infty M\restrict\Delta\cong C^\infty M$. \ee \ez \definition For any $f\in C^\infty M$, define {\bf the $k$-jet of $f$} as a section of $\Jet^k(M)$ which is equal to the $k$-jet of $f$ at each $x\in M$. \ed \exercise Let $D\in \Diff^k(M)$ be a differential operator (in the sense of Grothen\-dieck), $f\in C^\infty M$, and $J^k(f)$ its $k$-jet. Prove that for each $x\in M$, the number $D(f)(x)$ depends linearly on the $k$-jet of $f$ in $x$. \ez \hint Use Exercise \ref{_diff_ope_ideal_Exercise_}. \eh \exercise[!] In these assumptions, prove that there exists a $C^\infty M$-linear map $D_J:\; \Jet^k(M)\arrow C^\infty M$ such that $D(f)= D_J(J^k(f))$. \ez \hint Use the previous exercise. \eh \exercise[*] Prove that $\Diff^k(M)$ is a vector bundle. Prove that this vector bundle is dual to $\Jet^k(M)$. \ez \hint Use locality, dimension count and the previous exercise. \eh \exercise[*] Prove that all differential operators $D\in \Diff^k(M)$ (in the sense of Grothendieck) can be expressed in local coordinates as \[ D = f_0 + \sum_{i=1}^n f_i \frac \6 {\6x_i} + \sum_{i,j=1}^n f_{ij} \frac {\6^2} {\6x_i\6x_j} + \sum_{i,j,k=1}^n f_{ijk} \frac {\6^3} {\6x_i\6x_j\6x_k} + ... \] \ez \hint Use the previous exercise. \eh \exercise[**] Prove that any local operator on a compact manifold is a differential operator. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The ring of symbols of differential operators} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $R$ be an associative algebra. {\bf (Increasing) filtration} on $R$ is a collection of subspaces $R_0 \subset R_1\subset R_2 \subset ...$ such that $R_i R_j \subset R_{i+j}$. The natural product map $(R_k/R_{k-1}) \otimes (R_l/R_{l-1}) \arrow R_{k+l}/R_{k+l-1}$ defines an associative product structure on the space $\bigoplus_{i=0}^\infty R_i/R_{i-1}$. The algebra $\bigoplus_{i=0}^\infty R_i/R_{i-1}$ is called {\bf the associated graded algebra} of this filtration. \ed \exercise Consider the algebra $\Diff^*(R)$ with its filtration by $\Diff^i(M)$. Prove that its associated graded algebra is commutative. \ez \hint Use Exercise \ref{_filtra_commu_Zadacha_}. \eh \definition This ring is called {\bf the ring of symbols of differential operators}. For any $D\in \Diff^k(R)$, its representative in $\Diff^k(R)$ is called {\bf the symbol of $D$}. \ed \exercise Consider sections of $TM$ as differential operators of the first order. \enum \ite Prove that $TM= \Diff^1 M /\Diff^0 M$. \ite Prove that the multiplication in the ring of symbols defines a surjective, $C^\infty M$-linear map $\Sym^k TM \arrow \Diff^k M/\Diff^{k-1} M$. \ite[!] Prove that this map is an isomorphism. \ee \ez \end{document}