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\listok{6}{Geometry 6: Vector bundles and sheaves}

{\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{Sheaves of modules.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\remark
Now I will give a new definition of a sheaf.
The old definition (``sheaf of functions'') becomes
a special case of this one.
\er

\definition
Let $M$ be a topological space.
{\bf A sheaf}  ${\cal F}$ on $M$
is a collection of vector spaces  ${\cal F}(U)$
defined for each open subset $U\subset M$,
with the {\bf restriction maps}, which are linear 
homomorphisms  ${\cal F}(U) \stackrel{\phi_{U,U'}}\arrow {\cal F}(U')$,
defined for each $U'\subset U$, and satisfying the following
conditions.
\begin{description}
\item[(A)] Composition of restrictions is again a restriction:
for any open subsets $U_1\subset U_2 \subset U_3$,
the corresponding restriction maps
\[
{\cal F}(U_1) \stackrel{\phi_{U_1,U_2}}\arrow {\cal
F}(U_2) \stackrel{\phi_{U_2,U_3}}\arrow {\cal F}(U_3)
\]
give
$\phi_{U_1,U_2}\circ \phi_{U_2,U_3}=\phi_{U_1,U_3}$.\footnote{
If (A) is satisfied, ${\cal F}$ is called {\bf a presheaf}.}

\item[(B)] 
Let $U\subset M$ be an open subset, and $\{U_i\}$ 
a cover of $U$. For any  $f\in {\cal F}(U)$ 
such that all restrictions of $f$ to $U_i$ vanish, one has
$f=0$.

\item[(C)]  Let $U\subset M$ be an open subset, and $\{U_i\}$ 
a cover of $U$. Consider a collection $f_i \in {\cal F}(U_i)$
of sections, defined for each $U_i$, and satisfying
\[ f_i\restrict{U_i\cap U_j} = f_j\restrict{U_i\cap U_j}
\]
for each $U_i, U_j$. Then there exists $f\in {\cal F}(U)$ 
such that the restriction of $f$ to $U_i$ is $f_i$.
\end{description}
The space ${\cal F}(U)$ is called {\bf the space
of sections of the sheaf ${\cal F}$ on $U$}.
The restriction maps are often denoted
$f \arrow f\restrict U$
\ed

\remark
For a sheaf of functions, the conditions (A)
and (B) are satisfied automatically.
\er

\exercise
Let $M$ be a topological space equipped
with a presheaf ${\cal F}$.
Prove that the conditions (B) an (C)
are equivalent to exactness of the following
sequence.
\[
0 \arrow {\cal F}(U) \arrow \prod_{i} {\cal F}(U_i)
\arrow \prod_{i\neq j} {\cal F}(U_i\cap U_j) 
\]
for any open $U\subset M$
an open subset, and any cover $\{U_i\}$ of $U$.
\ez

\exercise
Let $f, g\in C^\infty M$  be functions which are 
equal on an open subset  $U\subset M$, and $D\in \Der_\R
C^\infty M$ a derivation on a ring of smooth functions. Prove that
$D(f)\restrict U = D(g)\restrict U$.
\ez

\definition
Let  $U\subset V$ be open subsets in $M$.
We write $U\Subset V$ if the closure of $U$ is contained in $V$.
\ed

\exercise
Let $U\Subset V$ be open subsets in a smooth metrizable
manifold. Prove that there exists a smooth function
$\Phi_{U,V}\in C^\infty M$ supported on $V$
and equal to 1 on $U$.
\ez

\exercise
Let  $D\in \Der_\R
C^\infty M$ be a derivation, and  $U\Subset V$ 
open subsets in $M$. Given $f\in \C^\infty V$, 
define $D(f)\restrict U$ using the formula
$D(f)\restrict U = D(\Phi_{U,V}\cdot f)$.
Prove that  $D(f)\restrict U$ satisfies the Leibnitz
rule, and is independent from the choice of $\Phi_{U,V}$.
\ez

\exercise[!]
Let  $D\in \Der_\R
C^\infty M$ be a derivation, and  $V\subset M$ an
open subset in $M$. \enum
\ite Prove that $D$ can be extended
to a derivation $D_V\in \Der_\R
C^\infty V$, in such a way that 
$D_V\left(f\restrict V\right)=D(f)\restrict V$.
\ite Prove that such an extension is unique.
\ee
\ez


\hint Use the previous exercise.
\eh



\exercise[!]
Show that this construction makes $\Der_\R(C^\infty M)$ into 
a sheaf of modules over $C^\infty M$.
\ez

\definition
{\bf A sheaf homomorphism} $\psi:\; {\cal F}_1 \arrow {\cal F}_2$
is a collection of homomorphisms
\[ \psi_U:\; {\cal F}_1(U) \arrow{\cal F}_2(U),\]
defined for each $U\subset M$, and commuting with the
restriction maps. {\bf A sheaf isomorphism} is a homomorphism
$\Psi:\; {\cal F}_1 \arrow {\cal F}_2$, for which there exists
an homomorphism $\Phi:\; {\cal F}_2 \arrow {\cal F}_1$,
such thate $\Phi\circ \Psi =\Id$ and 
$\Psi\circ \Phi =\Id$.
\ed

\exercise
Let $\psi:\; {\cal F}_1 \arrow {\cal F}_2$ 
be a sheaf homomorphism. 
\enum
\ite Show that $U \arrow \ker\psi_U$
and $U \arrow \coker\psi_U$ are presheaves.

\ite Prove that $U \arrow \ker\psi_U$
is a sheaf (it is called {\bf the kernel} of 
a homomorphism $\psi$).

\ite[*] Prove that $U \arrow \coker\psi_U$
is not always a sheaf (find a counterexample).
\ee
\ez

\definition
{\bf A subsheaf} ${\cal F'}\subset {\cal F}$
is a sheaf associating to each $U\subset M$
a subspace ${\cal F}'(U)\subset {\cal F}(U)$.
\ed


\exercise 
Find a non-zero sheaf ${\cal F}$ on $M$ such that
${\cal F}(M)=0$.
\ez

\remark
\label{_module_pullback_Remark_}
Let  $A:\; \phi \arrow B$ be a ring homomorphism, and
$V$ a  $B$-module. Then $V$ is equipped with a natural
$A$-module structure: $a v:= \phi(a) v$.
\er

\definition
{\bf A sheaf of rings} on a manifold $M$ is a sheaf
${\cal F}$ with all the spaces ${\cal F}(U)$
equipped with a ring structure, and all restriction
maps ring homomorphisms.
\ed


\definition
Let  ${\cal F}$ be a sheaf of rings on
a topological space $M$, and 
 ${\cal B}$ another sheaf.
It is called {\bf a sheaf of  ${\cal F}$-modules}
if for all $U\subset M$ the space of sections
${\cal B}(U)$ is equipped with a structure of ${\cal
  F}(U)$-module, and for all  $U'\subset U$, 
the restriction map 
${\cal B}(U) \stackrel{\phi_{U,U'}}\arrow {\cal B}(U')$
is a homomorphism of ${\cal F}(U)$-modules
(use Remark \ref{_module_pullback_Remark_}
to obtain a structure of ${\cal F}(U)$-module
on  ${\cal B}(U')$).
\ed

\exercise
Let ${\cal F}_1$ 
be a sheaf of rings and ${\cal F}$ its subsheaf.
Prove that ${\cal F}$ is a sheaf of modules over
 ${\cal F}$.
\ez

\definition
{\bf The space of germs} of a sheaf ${\cal F}$
at $x\in M$ is the limit $\lim\limits_\arrow {\cal F}(U)$,
where $U$ is the set of all neighbourhoods of $x$,
and the maps are restriction maps.
\ed

\exercise
Let ${\cal F}$ be a ring sheaf on $M$.
Prove that the space of germs of a sheaf of
${\cal F}$-modules is a module over the
ring of germs of  ${\cal F}$ in $x$.
\ez

\exercise
Let ${\cal B}$ be a sheaf with 
all germs equal 0. Prove that ${\cal B}=0$.
\ez

\exercise[*]
Find a sheaf ${\cal F}$ on $M$ with all germs 
non-zero, and ${\cal F}(M)$ zero.
\ez

\definition
A sheaf is called {\bf globally generated}
if for any $x\in M$, the natural restriction map
 ${\cal F}(M) \arrow {\cal F}_x$
from the space of global sections 
to the space of germs is surjective.
\ed

\exercise[*]
Let ${\cal F}$ be a globally generated sheaf
on $M$, and $U\subset M$ an open subset.
Prove that the map ${\cal F}(M)\arrow {\cal F}(U)$
is always surjective, or find a counterexample.
\ez

\exercise[*]
Let $M$ be a smooth, metrizable manifold, and
${\cal F}$ be a sheaf of modules over
$C^\infty(M)$. Prove that ${\cal F}$  is globally generated.
\ez


\definition
A free sheaf of modules  ${\cal F}^n$ 
over a ring sheaf  ${\cal F}$ maps an open set $U$ to 
the space ${\cal F}(U)^n$. A sheaf of ${\cal F}$-modules
is {\bf non-free} if it is not isomorphic to a free sheaf.
\ed

\exercise[!]
Find a subsheaf of modules in $C^\infty M$ which is
non-free in the sense of this definition.
\ez



\definition
{\bf Locally free sheaf of modules}
over a sheaf of rings  ${\cal F}$ is a sheaf 
of modules ${\cal B}$ satisfying the following
condition. For each $x\in M$ there exists a neighbourhood $U\ni x$
such that the restriction ${\cal B}\restrict U$ is free.
\ed

\exercise
Prove that a sheaf of $C^\infty M$-modules
$\Der_\R(C^\infty M)$ is locally free, for each
manifold $M$.
\ez

\exercise Prove that $\Der_\R(C^\infty M)$ 
is a free sheaf for the following manifolds.
\enum
\ite $M=\R$
\ite $M= S^1$ (a circle)
\ite $M=\R^2/\Z^2$ (a torus)
\ite[*] $M=S^3$ (a three-dimensional sphere)
\ee
\ez

\exercise[*]
Find a manifold for which the sheaf
$\Der_\R(C^\infty M)$ is not free.
\ez


\definition
{\bf A vector bundle} on a ringed space
$(M, {\cal F})$ is a locally free sheaf of ${\cal F}$-modules.
\ed

\definition
The sheaf of $C^\infty$-modules
$\Der_\R(C^\infty M)$ is called {\bf a tangent bundle}
to $M$.
\ed

\exercise[!]
Let $B$ be a vector bundle on a manifold $(M, C^\infty M)$.
Prove that $B$ is globally generated (as a sheaf).
\ez

\exercise[**]
Let $B_1$, $B_2$ be vector bundles on $(M, C^\infty)$
such that the spaces of sections $B_1(M)$ and
$B_2(M)$ are isomorphic as $C^\infty (M)$-modules.
Prove that the bundles  $B_1$ and $B_2$ are isomorphic.
\ez


\exercise[!]
Let ${\cal F}$ be a locally free sheaf of $C^\infty M$-modules.
Prove that ${\cal F}$ is soft.
\ez


\exercise[**]
Let ${\cal F}$ be a sheaf of $C^\infty M$-modules.
Prove that ${\cal F}$ is soft, or find a counterexample.
\ez


\definition
Let ${\cal F}$ be a  sheaf of $C^\infty M$-modules,
and ${\cal F}_x$ its germ in $x$. Denote the quotient
${\cal F}_x/{\goth m}_x {\cal F}_x$
by ${\cal F}\restrict x$. This space is called
{\bf the fiber} of ${\cal F}$ in $x$.
A morphism of sheaves induces a linear
map on each of its fibers.
\ed

\exercise[**]
Let ${\cal F}$ be a  sheaf of $C^\infty M$-modules
such that all its fibers ${\cal F}\restrict x$ vanish. 
Prove that ${\cal F}$ is zero, or find a counterexample.
\ez


\end{document}
