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%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


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\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}
