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%version 1.0,\ \   24.01.2018 
%version 1.1,\ \   24.01.2018 Sent the file to KA and VR,
%    and found lots of errors immediately
% version 1.2, \ \   24.01.2018, Katya sent lots of corrections
% version 1.3, \ \   14.02.2018, stalks and germs exchanged; exact sequence for 
%                    sheaves reworded
% version 1.4, \ \ \ 17.02.2018, soft sheaves were wrong.

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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{1}{Hodge theory handout 1: 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 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{Sheaves and manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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 covering 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 covering 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

\exercise
Let $M$ be a topological space equipped
with a presheaf ${\cal F}$. Let $U\subset M$
be an open subset and $\{U_i\}$ 
its covering. Define maps
${\cal F}(U) \arrow \prod_{i} {\cal F}(U_i)
\arrow \prod_{i\neq j} {\cal F}(U_i\cap U_j)$
in appropriate way, and
prove that the conditions (B) an (C)
are equivalent to the exactness of the 
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 all open $U\subset M$, 
and any covering $\{U_i\}$ of $U$.
\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
$\Phi_{U,V}\in C^\infty M$ equal to 0 outside of the closure of $V$
and equal to 1 on $U$.
\ez

\exercise
Show that the following spaces of functions on
open subsets of $\Bbb R^n$ define presheaves, but not sheaves
\enum
\ite Space of constant functions.
\ite Space of bounded functions.
\ite Space of functions vanishing outside of a bounded set.
\ite Space of continuous functions with finite $\int |f|$.
\ee
\ez


\definition
Let $(M,{\cal F})$ be a topological manifold
equipped with a sheaf of functions. It is said to be a {\bf smooth
manifold} of {\bf class} $C^\infty$ or $C^i$ if every point in
$(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space
$(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ is a ring of 
functions on $\Bbb R^n$ of this class.
\ed

\exercise
State the usual definition of a manifold (with
charts and atlases). 
Prove that this definition is equivalent to the one with 
charts and atlases. 
\ez



\definition
A {\bf ringed space} $(M,{\cal F})$ is a
topological space equipped with a sheaf of rings,
such that all restriction maps are ring homomorphisms.
We shall only consider ringed spaces equipped with
a sheaf of functions, that is, with a subsheaf ${\cal F}$ of the sheaf of all
functions on $M$, closed under multiplication. A~{\bf morphism}
$(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces
is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for
every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the
function $f\circ\Psi$ belongs to the ring
${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf isomorphism} of ringed spaces
is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ 
are morphisms of ringed spaces.
\ed

\exercise
Show that any smooth map of manifolds defines a 
morphism of the corresponding ringed spaces.
\ez

\exercise
Let $f:\; M \arrow N$ be a map of manifolds which defines
a morphism of the corresponding ringed spaces.
Prove that it is smooth.
\ez

\exercise[*]
Describe all morphisms of ringed spaces from
$(\R^n,C^{i+1})$ to $(\R^n,C^i)$.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sheaf morphisms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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  $\phi:\; A \arrow B$ 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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Germs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf The space of germs}, or {\bf stalk} 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. Its elements
are called {\bf germs}.
\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 stalks 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 soft}
if for any finite subset $x_1, ..., x_n` \in M$, the natural restriction map
 ${\cal F}(M) \arrow \oplus_i {\cal F}_{x_i}$
from the space of global sections 
to the space of germs is surjective.
\ed

\exercise[*]
Let ${\cal F}$ be a soft 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 soft.
\ez

\hint
Use a partition of unity.
\eh

\definition
A free sheaf of modules  ${\cal F}^n$ 
over a ring sheaf  ${\cal F}$ is a sheaf such that
the space of sections over an open set $U$ 
is ${\cal F}(U)^n$. 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


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

\exercise[!]
Given a smooth manifold $M$,
define the tangent bundle $TM$, and prove that it is a locally
free sheaf of $C^\infty M$-modules.
\ez

\exercise Prove that the tangent bundle
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[!]
Let $B$ be a vector bundle on a manifold $(M, C^\infty M)$.
Prove that $B$ is soft (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


\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}
