\documentclass[12pt]{article}

% version 1.0 -- April 16, 2013, started the file (from pde-8.tex)

\usepackage{diagrams}
\usepackage{amscd}


\newcommand{\version}{version 1.0,\ \   16.04.2013}
\newcommand{\firstdate}{22.04.2013}



%\addtolength{\topmargin}{-10mm}
%\addtolength{\textheight}{20mm}
%\addtolength{\oddsidemargin}{-10mm}
%\addtolength{\textwidth}{20mm}

\input{defs-listki-en.tex}


\begin{document}

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

\listok{10}{Geometry 10: De Rham algebra}

{\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.

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.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{K\"ahler differentials}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\definition
Let $R$ be a ring over a field $k$, and $V$ an $R$-module.
A $k$-linear map  $D:\; R \arrow V$ is called {\bf a derivation}
if it satisfies {\bf the Leibnitz identity} 
$D(ab) = a D(b) + b D(a)$. The space of derivations
from $R$ to $V$ is denoted  $\Der_k(R, V)$.
\ed

\exercise
Consider an action of $R$ on $\Der_k(R, V)$, with
$rd$ acting as $a \arrow rd(a)$.
Prove that this defines a structure of $R$-module on
$\Der_k(R, V)$.
\ez


\exercise
Let $[K:k]$ be a finite extension of a field of characteristic 0,
and $V$ a vector space over $K$. Prove that 
 $\Der_k(K, V)=0$. 
\ez

\exercise[!]
Let $M$ be a smooth manifold, $x\in M$ a point,
$R = C^\infty M$, and ${\goth m}_x\subset R$ the
maximal ideal of $x$. Consider an $R$-module $V:=R/{\goth m}_x$.
Find $\dim_\R \Der_R(R, V)$.
\ez

\exercise[**]
Let $R=C^0 M$ be a ring of continuous functions on a
manifold $M$, and $V$ an $R$-module of dimension 1 over $\R$.
Find a non-trivial derivation $\nu \in \Der_k(R, V)$,
or prove that it does not exist.
\ez

\definition
Let $R$ be a ring over a field $k$.
Define an $R$-module $\Omega^1_k R$, sometimes denoted 
simply as $\Omega^1 R$, 
with the following generators and relations.
The generators of $\Omega^1_k R$ 
are indexed by elements of $R$; for each
 $a\in R$, the corresponding generator of
$\Omega^1_k R$ is denoted $da$. 
Relations in $\Omega^1_k R$ are generated by 
expressions $d(ab) = a db + b da$, for all $a, b \in R$, 
and $d\lambda =0$ for each
$\lambda\in k$. Then $\Omega^1_k R$
is called {\bf the module of K\"ahler differentials}
of $R$.
\ed

\exercise
Prove that the natural map  $R \arrow \Omega^1_k R$, with
$a \mapsto da$ is a derivation.
\ez

\exercise
Let $R$ be a quotient of $k[r_1, ..., r_k]$ by an ideal.
Prove that $\Omega^1_k R$ is generated as an $R$-module by 
$dr_1, ..., dr_k$.
\ez

\exercise[!]
\label{_diffe_dual_Zadacha_}
Prove that $\Der_k(R) = \Hom_R(\Omega^1_k R, R)$.
\ez

\exercise
\label{_diffe_katego_Zadacha_}
Let $V$ be an $R$-module, and $D\in \Der_k(R, V)$ 
a derivation. Prove that there exists a unique 
$R$-module homomorphism 
$\phi_D:\; \Omega^1_k R \arrow V$ making the following 
diagram commutative.
\begin{diagram}
R & \rTo^d & \Omega^1_k R \\
&\rdTo~D &\dTo~{\phi_D}\\
& & V
\end{diagram}
\ez

\remark
This property is often taken as a definition of
$\Omega^1_k R$. 
\er

\exercise[!]
Let $R= k[t_1, ..., t_n]$ be a polynomial ring over a field
of characteristic 0. Prove that $\Omega^1_k R$ is a free $R$-module
generated by $d t_1, dt_2, ..., dt_n.$
\ez


\exercise[*]
Let $I\subset R$ be an ideal. Construct an exact sequence
\[
I/I^2 \arrow \Omega^1(R)\otimes_R R/I\arrow
\Omega^1(R/I)\arrow 0.
\]
\ez

\exercise
Let $R\stackrel \phi \arrow R'$ be a ring homomorphism.
Consider $\Omega^1 R'$ as an $R$-module, using the
action $r, a \arrow \phi(r)a$.
\enum
\ite
Prove that there exists an $R$-module
homomorphism $\Omega^1 R \arrow \Omega^1 R'$,
mapping $dr$ to $d\phi(r)$. 

\ite Prove that it is
unique.
\ee
\ez

\definition
\label{_induced_Kahle_Definition_}
In this case, we say
that the homomorphism $\Omega^1 R \arrow \Omega^1 R'$
{\bf is induced by $\phi$}.
\ed


\exercise[*]
Let $R$ be a ring of continuous functions on a manifold,
and  ${\goth m}_x$ a maximal ideal of a point. 
Prove that ${\goth m}_x\Omega^1 R=\Omega^1 R$.
\ez




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cotangent bundle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $A, B$ be $R$-modules, and
$\nu:\; A\times B \arrow R$ a bilinear pairing.
It is called {\bf non-degenerate} if for each
$a\in A$ there exists $b\in B$ such that 
$\nu(a, b)\neq 0$, and  for each
$b\in B$ there exists $a\in A$ such that 
$\nu(a, b)\neq 0$
\ed

\exercise
Let $A, B$ be vector spaces over $k$, and $\nu:\; A\times B \arrow
k$ a non-degenerate pairing. Prove that $A$ is isomorphic
to $B^*$, or find a counterexample
\enum
\ite When $A, B$ are finite-dimensional.
\ite When $A, B$ are infinite-dimensional.
\ee
\ez

\definition
Let $V$ be an $R$-module. A {\bf dual $R$-module}
$\Hom_R(V,R)$ is denoted $R^*$.
\ed

\exercise
Let $V$ be an $R$-module. Consider the natural pairing
$V\times V^*\arrow R$. Prove that it is non-degenerate,
or find a counterexample, in the following cases:
\enum
\ite when $R$ is a field, and $V$ a (possibly infinite-dimensional)
vector space
\ite[!] when the natural map $V\arrow V^{**}$ is
injective
\ite when $V$ is a free $R$-module
\ite[*] when $R$ is a ring which has no zero divisors.
\ee
\ez


\exercise[*]
Let $A, B$ be finitely-generated $R$-modules, and
$\nu:\; A\times B \arrow R$ a non-degenerate pairing.
Prove that $A$ is isomorphic to $B^*$,
or find a counterexample.
\ez

\exercise[!] 
\label{_free_pairing_dual_Exercise_}
Let $A$ be a free, finitely generated $R$-module, and
 $\nu:\; A\times B \arrow R$ a non-degenerate pairing.
Prove that $B$ is also free, and isomorphic to $A^*$.
\ez

\definition
Let $A$, $B$ be finitely generated $R$-modules,
and $\nu:\; A\times B \arrow R$ a bilinear pairing.
Define {\bf the annihilator of $\nu$ in $B$}
as a submodule consisting of all elements $b \in B$ for which
the homomorphism $\nu(\cdot, b):\; A \arrow R$ vanishes.
\ed

\definition
Let $M$ be a smooth manifold, 
$R:= C^\infty M$ the ring of smooth finctions, and
 $\nu:\; \Der(R)\times \Omega^1 R \arrow R$ 
the pairing constructed in Exercise
\ref{_diffe_dual_Zadacha_}. Consider
its annihilator $K\subset \Omega^1 R$.
Define  {\bf the cotangent bundle} as
$\Lambda^1 M:= \Omega^1 R /K$. For the purpose
of this definition, $\Lambda^1 M$ is 
a $C^\infty M$-module.
\ed


%\exercise[!]
%Prove that the natural pairing
%$\nu:\; \Der(C^\infty M)\times \Lambda^1 M \arrow R$ 
%is bilinear and non-degenerate.
%\ez

\exercise
Let $R:= C^\infty \R^n$, and
$t_1, ..., t_n\in R$ be coordinate functions. Consider
an element in $\Lambda^1 \R^n$, written as
$P= \sum_{i=1}^n P_i dt_i$, let  
$Q= \sum_{i=1}^n Q_i \frac{d}{dt_i}\in\Der_k(R)$ -- 
be a vector field, and $\nu:\; \Der(R)\times \Lambda^1
\R^n \arrow R$ the natural pairing. Prove that
$\nu(P, Q) = \sum_i P_i Q_i$. 
\ez


\exercise[!]
In these assumptions, prove that
$\Lambda^1 R$ is a free $\R$-module,
generated by $dt_1, ..., dt_n$.
\ez

\hint
Prove that
$\Der(R) = \Hom_R(\Omega^1 R, R)$,
and $\Der(R)$ is a free $R$-module.
Use exercise \ref{_free_pairing_dual_Exercise_}.
\eh



\exercise
Let $A, B$ be finitely-generated projective $R$-modules,
and  $\nu:\; A\times B \arrow R$ a non-degenerate pairing. 
Prove that $B\cong A^*$.
\ez

\exercise[!]
Let $M$ be a smooth, metrizable manifold. Prove that
\[ \Lambda^1 M = \Hom_{C^\infty M}(\Der(C^\infty M), C^\infty M).\]
\ez

\hint
Use the previous exercise and apply the Serre-Swan theorem.
\eh

\exercise[*]
Let $K$ be the kernel of the natural projection
\[ \Omega^1 (C^\infty M) \arrow \Lambda^1 M.\]
Prove that ${\goth m}_x K =K$ for each
maximal ideal of a point $x\in M$.
\ez

\exercise[**]
Show that $K$ is non-empty.
\ez



%\exercise
%Let $B$ be a smooth manifold, 
% $M \stackrel \Delta \arrow M \times M$  
%the diagonal embedding, and  $I\subset C^\infty M\times M$ 
%an ideal of all functions vanishing in $\Delta$.
%\enum
%\ite Show that the ring
%$C^\infty (M\times M)/I$ is isomorphic to $C^\infty M$.
%\ite[*] Prove that the natural map
%$C^\infty M\times M\arrow \End_{C^\infty M\times M}(I/I^2)$ 
%can be factorized through the projection
%\[ C^\infty (M\times M)\arrow C^\infty (M\times M)/I = C^\infty M.
%\]
%Deduce that the space $I/I^2$ is equipped with a
%natural structure of a module over
%$C^\infty M$.
%
%\ite[*]
%Show that  $I/I^2$ is isomorphic to $\Lambda^1 M$
%as a $C^\infty M$-module.
%\ee
%\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{De Rham algebra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M$ be a smooth manifold. {\bf A bundle of 
differential $i$-forms on $M$} is the bundle
$\Lambda^i T^* M$ of antisymmetric $i$-forms on $TM$.
It is denoted $\Lambda^i M$.
\ed

\definition
Let $\alpha\in (V^*)^{\otimes i}$ and $\alpha\in (V^*)^{\otimes j}$
be polylinear forms on $V$. Define the {\bf tensor multiplication}
$\alpha \otimes \beta$ as
\[ \alpha \otimes \beta(x_1, ..., x_{i+j}):= 
  \alpha(x_1, ..., x_j) \beta(x_{i+1}, ..., x_{i+j}).
\]
\ed

\exercise
Let $\bigotimes_k T^*M \stackrel \Pi \arrow \Lambda^k M$ 
be the antisymmetrization map,
\[ \Pi(\alpha)(x_1, ..., x_n):= \frac 1{n!} \sum_{\sigma\in \Sym_n} 
(-1)^\sigma \alpha(x_{\sigma_1}, x_{\sigma_2}, ..., x_{\sigma_n}).
\]
 Define the ``exterior multiplication''
$\wedge:\; \Lambda^i M\times \Lambda^j M \arrow \Lambda^{i+j} M$ as
$\alpha \wedge \beta := \Pi (\alpha \otimes \beta)$,
where $\alpha \otimes \beta$ is a section
$\Lambda^i M\otimes \Lambda^j M \subset \bigotimes_{i+j}
T^*M$ obtained as their tensor multiplication.
Prove that this operation is associative and
satisfies $\alpha \wedge \beta = (-1)^{ij} \beta\wedge \alpha$.
\ez

\definition
The algebra $\Lambda^* M := \oplus_i\Lambda^i M$ 
with the multiplicative structure defined above is called
{\bf the de Rham algebra} of a manifold.
\ed

\exercise[*]
Let $M$ be an oriented manifold. Prove that
all bundles $\Lambda^iM$ are oriented, or find
a counterexample.
\ez

\exercise
Prove that de Rham algebra is multiplicatively
generated by  $C^\infty M =\Lambda^0M$ and 
$d(C^\infty)\subset \Lambda^1 M$. 
\ez


\exercise
Prove that a derivation on an algebra is 
uniquely determined by its values on 
any set of multiplicative generators
of this algebra.
\ez

\definition
{\bf De Rham differential}  $d:\; \Lambda^*M \arrow \Lambda^{*+1}M$
is an $\R$-linear map satisfying the following conditions.
\begin{description}

\item[(i)] For each $f \in \Lambda^0M=C^\infty M$,
$d(f)\in \Lambda^1 M$ is equal to the image
of the K\"ahler differential $df\in \Omega^1 M$
in $\Lambda^1 M = \Omega^1 M/K$.

\item[(ii)] (Leibnitz rule)
$d(a\wedge b) = da \wedge b + (-1)^j a\wedge
db$ for any $a\in \Lambda^i M, b \in \Lambda^j M$.

\item[(iii)] $d^2=0$.
\end{description}
\ed

\exercise[!]
Prove that de Rham differential is defined uniquely by these
axioms.
\ez

\hint 
Use the previous exercise.
\eh



\exercise
\label{_de_Rham_locally_Zadacha_}
Let $t_1, ..., t_n$ be coordinate functions
on $\R^n$, and $\alpha\in \Lambda^* \R^n$ a monomial
obtained as a product of several $dt_i$,
\[ \alpha = dt_{i_1}\wedge dt_{i_2} \wedge ... \wedge dt_{i_k},
\]
$i_1 < i_2 < ...< i_k$
 (such a monomial
is called {\bf a coordinate monomial}).
\enum
\ite Prove that $\Lambda^*\R^n$ is a trivial bundle, 
and coordinate monomials are free generators of $\Lambda^*\R^n$.
\ite Show that the de Rham differential, if it exists, satisfies
$d(f\alpha)=\sum_i \frac {df}{dt_i} dt_i \wedge \alpha$
for any $f\in C^\infty \R^n$.
\ite Prove that this formula defines the de Rham differential on
$\Lambda^* \R^n$ correctly.
\ee
\ez

\NewVedomost

\exercise
\enum
\ite 
Prove that de Rham differential
$d:\; \Lambda^*M \arrow \Lambda^{*+1}M$
commutes with restrictions to open subsets.

\ite Show that  de Rham differential (if it exists)
defines a sheaf morphism.
\ee
\ez

\hint
Use uniqueness of de Rham differential.
\eh

\exercise[!]
Prove that de Rham differential exists on any manifold.
\ez

\hint
Locally, de Rham differential is constructed in
exercise \ref{_de_Rham_locally_Zadacha_}.
To go from local to global, use the previous exercise,
and apply the sheaf axioms.
\eh



\exercise[*]
Let  $R$ be a ring over a field,
and $\Omega^i R:= \Lambda^i_R \Omega^1 R$
an exterior algebra generated by K\"ahler differentials.
Prove that there exists the de Rham differential
$d:\; \Omega^* R\arrow \Omega^{*+1} R$
satisfying the axioms above.
\ez


\end{document}

