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\begin{document}

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\listok{5}{Geometry 5: Vector fields and derivations}

{\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 problems.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.
It's up to you to ignore handouts entirely,
because passing tests in class and having
good scores at final exams could compensate
(at least, partially) for the points obtained 
by grading handouts.

Solutions for the problems are to be explained to the 
examiners orally in the class and marked in the score sheet. 
It's better to have a written version of your solution with 
you. It's OK to share your solutions with other students, and use
books, Google search and Wikipedia, we encourage it.
The first score sheet will be distributed
February 11-th. 

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
(except at most 2) 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.

Please keep your score sheets until the final
evaluation is given.

}

\def\Der{\operatorname{Der}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Derivations of a ring}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\remark
All rings in these handouts are assumed to be 
commutative and with unit. Algebras are associative,
but not necessarily commutative (such as the matrix algebra).
{\bf Rings over a field $k$} are rings containing a field $k$.
\er

\definition
Let $R$ be a ring over a field $k$. A $k$-linear map
$D\; R \arrow R$ is called {\bf a derivation}
if it satisfies {\bf the Leibnitz equation}
$D(fg) = D(f) g + gD(f)$. The space of derivations
is denoted as  $\Der_k(R)$.
\ed

\exercise
Let $D\in \Der_k(R)$. Prove that $D\restrict k=0$.
\ez

\exercise
Let $D_1, D_2$ be derivations. Prove that the
commutator $[D_1, D_2]:= D_1 D_2 - D_2 D_1$
is also a derivation.
\ez

\exercise[!]
Let $K\supset k$ be a field which contains a field
$k$ of characteristic 0, and is finite-dimensional over $k$ 
(such fields $K$ are called {\bf finite extensions} of $k$).
Find the space $\Der_k(K)$.
\ez

\exercise[*]
Is it true if $\Char k =p$?
\ez

\exercise
Consider a ring $k[\epsilon]$, given by a relation
$\epsilon^2=0$. Find $\Der_k (k[\epsilon])$.
\ez

\exercise[*] 
Find all rings $R$ over $\C$ such that $R$
is finite-dimensional over $\C$, and $\Der_\C(R)=0$.
\ez

\exercise[**]
Let $D\in \Der_k(K)$ be a derivation of a field $K$ over $k$,
$\Char k =0$, and $[K':K]$ a finite field extension. Prove that
$D$ can be extended to a derivation $D'\in
\Der_k(K')$.
\ez


\exercise
\label{_I^n_Exercise_}
Let $D\in \Der_k(R)$ be a derivation, and $I\subset R$ --
an ideal. Prove that $D(I^k)\subset I^{k-1}$.
\ez



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Modules over a ring}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $R$ be a ring over a field $k$.
{\bf An $R$-module} is a vector space
$V$ over $k$, equipped with an algebra
homomorphism $R\arrow \End(V)$, where $\End(V)$ 
denotes the endomorphism algebra of $V$, that
is, the matrix algebra.
\ed

\exercise 
Let $R$ be a field. Prove that $R$-modules
are the same as vector spaces over $R$.
\ez

\remark
An $R$-module is a group, equipped with
an operation of ``multiplication by elements of $R$'',
and satisfying the same axioms of distributivity
and associativity as in the definition of a 
vector space.
\er

\remark
Homomorphisms, isomorphisms,
submodules, quotient modules, direct sums 
of modules are defined in the same way as for the
vector spaces. A ring $R$ is itself an $R$-module.
A direct sum of $n$ copies of
$R$ is denoted $R^n$. Such $R$-module is called 
{\bf a free $R$-module}.
\er


\remark
$R$-submodules in $R$ are the same as ideals in $R$.
\er

\definition
A ring $R$ is called {\bf a principal ideal ring},
if all non-zero submodules of $R$ are isomorphic to $R$.
\ed

\exercise 
Prove that $R$ is a principal ideal ring iff $R$
has no zero divisors, and all ideals in $R$
are {\bf principal}, that is, are of form
$Rx$, for some non-invertible $x\in R$.
\ez

\exercise
Are these rings principal ideal rings?
\enum
\ite $R=\C[t]$
\ite[!] $R=\C[t_1, t_2]$
\ite[*] $R:=\R[x, y]/(x^2 + y^2=-1)$. 
\ee
\ez


\definition
{\bf Finitely generated} $R$-module is a quotient
module of $R^n$.
\ed

\exercise
Find a finitely generated, non-free $R$-module
for $R=\C[t]$.
\ez

\definition
{\bf A Noetherian ring} is a ring $R$ with all
ideals finitely generated as $R$-modules.
\ed

\exercise[*] 
Let $R$ be a Noetherian ring. Prove that
any submodule of a finitely generated
$R$-module is finitely generated.
\ez

\exercise
Let $R$ be a ring
obtained as a direct limit of the following
diagram. The vertices of this diagram
are numbered by natural numbers. The corresponding
vector spaces are rings $\C[t]$. 
The arrows $\phi_{k,km}$ 
of this diagram are going from the
$k$-th vertex to the $km$-th.
The corresponding homomorphisms
of polynomial rings $\C[t]\stackrel{\phi_{k,km}}\arrow \C[t]$
are determined by the action of $\phi_{k,km}$
on the polynomial generators: $\phi_{k,km}(t)= t^m$.
Prove that $R$ is a set of formal linear combinations
$a_1 t^{\alpha_1}+a_2 t^{\alpha_2} + ... + a_2 t^{\alpha_2}$,
where $a_i \in \C$, and $\alpha_i$ -- non-negative rational numbers,
with an obvious formula for multiplication.
\ez

\exercise
Consider a ring $R$ defined in the previous exercise.
Prove that an ideal generated by the polynomials
$a_1 t^{\alpha_1}+a_2 t^{\alpha_2} + ... + a_2 t^{\alpha_2}$,
where all $\alpha_i$ are positive, is not finitely generated.
\ez

\exercise
Consider a ring of germs of smooth functions in 
a point, and let $K$ be an ideal of all functions
with all derivatives of all orders vanishing. Show that 
this ideal is not principal.
\ez

\exercise[*]
Prove that $K$ is not finitely generated.
\ez

\exercise[*]
Let $I$ be a finitely generated ideal in the
ring $R$ of germs of smooth functions on $\R$ at 0.
Prove that $I$ is principal, or find a counterexample.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Vector fields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\remark
Let $R$ be a ring over $k$. The space $\Der_k(R)$ 
of derivations is also an $R$-module, with multiplicative
action of $R$ given by $rD(f) = r D(f)$.
\er

%\exercise
%Prove that  $\Der_k(R)$ -- ������������� $R$-������.
%\ez

\exercise
Let $R= k[t_1, .., t_k]$ be a polynomial ring.
Prove that $\Der_k(R)$ is a free $R$-module
isomorphic to $R^n$, with generators $\frac d {dt_1}, 
\frac d{dt_2}, ...,\frac d{dt_n}$.
\ez

\hint
Construct a map
$\Der_k(R) \arrow R^n$, 
\[ D \arrow (D(t_1), D(t_2), ..., D(t_n))
\]
and prove that it is an isomorphism of $R$-modules.
\eh

\exercise[*]
Let $R= k(t_1, .., t_k)$ be a ring of rational 
functions, that is, the ring of functions $\frac {P}{Q}$,
where $P$ and $Q\in k(t_1, .., t_k)$ are arbitrary
polynomials, $Q\neq 0$.
Prove that $\Der_k(R)$ is a free $R$-module, isomorphic
to $R^n$.
\ez

\exercise[!]
Prove the {\bf Hadamard's lemma}: Let $f$ be a smooth function
$f$ on $\R^n$, and $x_i$ the coordinate functions. Then
$f(x)=f(0)+\sum_{i=1}^n x_i g_i(x)$, for some smooth 
$g_i\in C^\infty \R^n$.
\ez

\hint
Consider a function $h(t)\in C^\infty \R^n$, $h(t)=f(tx)$.
Then $\frac{dh}{dt}=\sum \frac{df(tx)}{dx_i}(tx)x_i$.
Integrating this expression over $t$, obtain
$f(x)-f(0)=\sum_i x_i  \int_0^1\frac{df(tx)}{dx_i}(tx)dt$.
\eh 

\definition
Consider coordinates  $t_1, ..., t_n$ on $\R^n$, and let
\[ 
  \Der(C^\infty\R^n) \stackrel \Pi \arrow(C^\infty\R^n)^n,
\]
map $D$ to $(D(t_1), D(t_2), ..., D(t_n))$.
\ed

\exercise 
Prove that $\Pi$ is surjective.
\ez

\exercise
Prove that $\Pi(D)=0$ $\Leftrightarrow$ $D(P)=0$
for each $P(t_1, ..., t_n)$.
\ez

\exercise
Let ${\goth m}_x\subset C^\infty\R^n$ 
be an ideal of all smooth functions vanishing at  
$x\in \R^n$. Prove that it is maximal.
\ez

%\exercise
%Let  $\R[[t_1, ..., t_n]]$ be a ring of formal
%power series. Prove that it is local.
%\ez
%
%\exercise
%Consider a natural map
%\[
%C^\infty\R^n\stackrel \Psi \arrow \R[[t_1, ..., t_n]],
%\]
%from the ring of smooth functions to the ring 
%of formal power series
%$\R[[t_1, ..., t_n]]$, mapping a function to its Taylor
%serie in 0. Let $\tilde {\goth m}$ be a maximal ideal of
%$\R[[t_1, ..., t_n]]$.
%Prove that
%$\Psi^{-1}(\tilde {\goth m})= {\goth m}_0$.
%\ez

\exercise
Let $f$ be a smooth function on $\R^n$
satisfying $f(x)=0$ and  $f'(x)=0$. Prove that
$f\in {\goth m}_x^2$.
\ez


\hint Use the Hadamard's Lemma.
\eh

\exercise[!]
Let $D\in \Der_\R(C^\infty\R^n)$ be a derivation, satisfying
$D \in \ker \Pi$ (that is, vanishing on coordinate
functions). Prove that for all
$f\in C^\infty\R^n$, and all $x\in \R^n$, one has
$D(f)\in {\goth m}_x$.
\ez

\hint
Use the previous exercise and Exercise \ref{_I^n_Exercise_}.
\eh


\exercise[!]
Prove that the map
\[ 
\Der(C^\infty\R^n) \stackrel \Pi \arrow(C^\infty\R^n)^n
\]
is an isomorphism
\ez

\hint
Use the previous exercise.
\eh


\exercise[**]
Find a non-trivial element 
$\gamma\in\Der_\R(C^0\R)$ in the space of derivations
of continuous functions, or prove that it is empty.
\ez

\exercise[**]
Find a non-trivial element 
$\gamma\in\Der_\R(C^1\R)$ in the space of derivations
of the ring of differentiable functions of class $C^1$, 
or prove that it is empty.
\ez


\end{document}
