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%version 1.2,\ \   08.04.2013 4.27 <s>zvezdochka</s> (ne nado)


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\begin{document}

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

\listok{4}{Geometry 4: Germs of functions}

{\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.

}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Direct limit}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf Commutative diagram} of vector spaces
is given by the following data. First, there is
a directed graph (graph with arrows). For each
vertex of this graph (also called a diagram)
one gives a vector space, and each arrow
corresponds to a homomorphism of the
associated vector spaces. These 
homomorphism are compatible, in the 
following way. Whenever there exist two ways of going from
one vertex to another, the compositions of the
corresponding arrows are equal.
\ed

\remark
A {\bf neighbourhood} of a subset $X\subset M$ is an
open subset containing $X$.
\er

\exercise
Let $(M, {\cal F})$ be a space ringed by 
a sheaf of functions, $x\in M$ a point,
 $\{U_i\}$ -- the set of all neighbourhoods of
$x$. Consider a diagram, with the set of vertices 
indexed by $\{U_i\}$, and arrows from $U_i$ to $U_j$
corresponding to inclusions $U_j \hookrightarrow U_i$. 
Prove that the space of sections  ${\cal F}(U_i)$ 
with homomorphisms given by restrictions form
a commutative diagram.
\ez

\definition
Let ${\cal C}$ be a commutative diagram of vector spaces
 $A, B$ -- vector spaces, corresponding to two vertices
of a diagram, and  $a\in A, b\in B$ elements of these
vector spaces. Write  $a\sim b$ if $a$ and $b$ are mapped
to the same element $d\in D$ by a composition of 
arrows from  ${\cal C}$. Let $\sim$ be an
equivalence relation generated by such $a\sim b$.
\ed

\exercise
\enum
\ite  Let $A\stackrel \phi \arrow B$ be a diagram
of two spaces and one arrow. Prove that $b\sim b'$
is equivalent to $b=b'$ for each $b, b' \in B$.
\ite
Let $A\stackrel \phi \arrow B$, $A\arrow 0$ 
be a diagram of three spaces, with $\phi$ injective.
Prove that for each $b, b'\in
B$, $b\sim b'$ is equivalent to $b- b' \in {\rm im} \phi$.
\ee
\ez

\definition
Let  $\{C_i\}$ be a set of vector spaces associated
with the vertices of a commutative diagram ${\cal C}$,
and $E\subset \bigoplus_i C_i$ a subspace generated by
the vectors $(x-y)$, where $x\sim y$.
A quotient $\bigoplus_i C_i/E$ is called
{\bf a direct limit} of a diagram $\{C_i\}$.
The same notion is also called {\bf colimit}
and {\bf inductive limit}. Direct limit is denoted
$\lim\limits_\rightarrow$.
\ed

\exercise
Let  $C_1 \arrow C_2 \arrow C_3
\arrow ...$ be a diagram with all arrows injective.
Prove that $\lim\limits_\rightarrow C_i$ 
is a union of all $C_i$.
\ez

\exercise
Let  $C_1 \arrow C_2 \arrow C_3
\arrow ... \arrow C_n$ be a diagram. Prove that
$\lim\limits_\rightarrow C_i= C_n$.
\ez

\exercise
Find an example of a diagram
$C_1 \arrow C_2 \arrow C_3 \arrow ...$
where all spaces  $C_i$ are non-zero, and the colimit
$\lim\limits_\rightarrow C_i$ vanishes.
\ez

\exercise[*]
Find an example of a diagram
$C_1 \arrow C_2 \arrow C_3 \arrow ...$
where all spaces  $C_i$ are non-zero, all arrows are also non-zero,
and the colimit $\lim\limits_\rightarrow C_i$ vanishes.
\ez

\definition
A diagram ${\cal C}$ is called {\bf filtered}
if for any two vertices $C_i, C_j$, there exists
a third vertex $C_k$, and sequences of arrows
leading from $C_i$ to $C_k$ and from $C_j$ to $C_k$.
\ed

\exercise
Let ${\cal C}$ be a commutative diagram of vector spaces 
$C_i$, with all $C_i$ equipped with a ring structure,
and all arrows ring homomorphisms. Suppose that 
the diagram ${\cal C}$ is filtered. Prove that 
$\lim\limits_\rightarrow C_i$ is a ring, equipped
with natural ring homomorphisms $C_i \arrow \lim\limits_\rightarrow C_i$.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A ring of germs of a sheaf of functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M, {\cal F}$ be a ringed space,
$x\in M$ its point, and $\{U_i\}$ the set of all
its neighbourhoods. Consider a commutative
diagram with vertices indexed by $\{U_i\}$,
and arrows from $U_i$ to $U_j$ 
corresponding to inclusions $U_j \hookrightarrow U_i$.
For each vertex $U_i$ we take a vector space
of sections  ${\cal F}(U_i)$, and for each
arrow the corresponding restriction map. 
The direct limit of this diagram is called
{\bf the ring of germs of the sheaf ${\cal F}$
in $x$}.
\ed

\remark 
This limit is indeed a ring, as follows
from the previous exercise.
\er

\remark
As a special case of this definition, 
we obtain rings of germs of 
smooth functions, real analytic functions,
continuous, $C^i$ and so on.
\er

\exercise
Let ${\cal F}$ be a sheaf of functions on a manifold,
such that all its germs are zero. Prove
that ${\cal F}$ is a zero sheaf.
\ez

\definition
{\bf A constant sheaf}  $\R_M$
is a sheaf of functions which are constant
on each connected $U\subset M$.
\ed

\exercise
Prove that a ring of germs of a constant
sheaf at each point is $\R$.
\ez

\exercise[*]
Let ${\cal F}$ be a sheaf of $\R$-valued functions
on $M$, such that all its germs are isomorphic to $\R$.
Prove that it is constant.
\ez

\definition
{\bf An ideal} in a ring $R$ is an abelian subgroup  $I\subsetneq R$,
such that for all $x\in R, a \in I$, the product $xa$ belongs to $I$.
\ed

\remark
A quotient space $R/I$ is a ring (prove this).
Also, for any ring homomorphism, its kernel is an ideal.
\er

\definition
{\bf A maximal ideal} is an ideal $I\subset R$, 
such that for any other ideal $I'\supsetneq I$,
$I'\ni 1$.
\ed

\exercise
Show that any ideal is contained in a maximal ideal
(use Zorn's lemma).\footnote{You are not required to prove
Zorn's lemma in this exercise.}
\ez


\exercise
Show that an ideal  $I\subset R$ 
is maximal if and only if the quotient $R/I$ is a field.
\ez

\exercise[*] Find all maximal ideals in the 
ring of smooth functions on a compact manifold.
\ez

\definition
A ring is called {\bf local} if it contains only
one maximal ideal.
\ed

\exercise
Prove that a ring of rational numbers $\frac m n$,
where $m,n$ are integer, and $n$ odd, is local.
Find its quotient by the maximal ideal.
\ez

\exercise
Let $F$ be a ring of rational functions
(functions $\frac P Q$, where $P$ and $Q\in \C[t_1, ...,
  t_n]$ are polynomials) without a pole in 0.
Show that this ring is local. Find its quotient by a maximal ideal.
\ez

\exercise[!]
Are the following rings local?
\enum
\ite The ring of germs of smooth functions.
\ite The ring of germs of polynomial functions
on $\R^n$.
\ite The ring of germs of functions of differentiability
class $C^i$, $i\geq 0$.
\ite The ring of germs of continuous functions.
\ite The ring of germs of real analytic functions on $\R^n$.
\ee
\ez

\exercise
Show that a ring with a maximal ideal $I$
is local iff each element $r\notin I$
is invertible.
\ez

\definition
{\bf Zero divisors} 
in a ring are non-zero elements
$r_1, r_2$, saisfying $r_1 r_2=0$.
{\bf Nilpotent} is $r\in R$ such that
 $r^n=0$ for some $n$. 
\ed

\exercise
Find whether the following rings have zero divisors.
\enum
\ite The ring of germs of smooth functions.
\ite The ring of germs of polynomial functions.
\ite The ring of germs of continuous functions.
\ee
\ez


\definition
A continuous function $f$ on $\R^n$ is called 
{\bf piecewise polynomial} if $\R^n$ is represented
as a union of polyhedra, and on each of these
polyhedra, $f$ is polynomial.
\ed


\exercise
Let  ${\cal F}$ -- a sheaf of piecewise polynomial
functions on $\R$, а $S$ -- a ring of its germs at 0. 
\enum
\ite Find out whether $S$ is a local ring.
\ite Show that $S$ is isomorphic to $\R[t_1, t_2]/(t_1t_2=0)$.
\ee
\ez

\exercise[!]
Let $R$ be a local ring, $\goth m$ its maximal ideal, and
$K(R):= \bigcap_i {\goth m}^i$. Prove that it is an ideal.
Find whether this ideal is zero for
\enum
\ite The ring of germs of smooth functions.
\ite The ring of germs of real analytic functions.
\ite The ring of germs of continuous functions.
\ee
\ez

\exercise[*]
Let $R=k[t_1,..., t_n]$ be a ring of polynomials over a field,
and $I\subset R$ an ideal.\footnote{The ideals in $R$ are tacitly
assumed to be $\neq R$.} Prove that
$\bigcap_i I^i=0$.
\ez

\exercise
Let $R$ be a ring of germs of smooth functions 
in $x$, $\goth m$ its maximal ideal, and
$K(R):= \bigcap_i {\goth m}^i$.
Prove that for all $f\in K(R)$,
all derivatives of $f$ in zero
(of any order) vanish.
\ez

\exercise
Let  $x_1, ..., x_n$ be coordinates on $\R^n$, 
and $f$ a function with all derivatives of any
order vanishing. Show that 
$\frac{f}{\left(\sum_i x_i^2\right)^p}$
is continuous for any $p>0$.
\ez

\exercise[!]
Under assumptions of the previous exercise,
prove that the function $\frac{f}{\sum_i x_i^2}$
is smooth.
\ez

\exercise[!]
Let $R$ be a ring of germs of smooth functions in $x\in \R^n$, 
а $K(R):=\bigcap_i {\goth m}^i$ the ideal defined
above. Prove that $K(R)$ is an ideal of functions
with vanishing derivatives of any order at $x$.
\ez

\hint
Use the previous exercise.
\eh

\exercise[*] 
Let $R/K(R)$ be the ring defined above.
\enum
\ite Are there non-zero nilpotents in $R/K(R)$?
\ite Are there zero divisors in  $R/K(R)$?
\ee
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Soft sheaves}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let  $(M, {\cal F})$ be a topological space ringed by
a sheaf of functions, and  $X\subset M$ its subset.
Consider a diagram indexed by open subsets 
$U_i\subset M$ containing $X$, with arrows 
corresponding to inclusions  $U_j \subset U_i$, 
and associate with each $U_i$ the corresponding
section space ${\cal F}(U_i)$.
A direct limit of this diagram is called
{\bf the ring of germs of  ${\cal F}$ in $X$,} and denoted
as ${\cal F}(X)$.
\ed

\exercise
Prove that for each open subset  $U\subset M$
the corresponding germ space coincides with the
space of sections ${\cal F}(U)$.
\ez

\exercise[*]
Let $(M, C^\infty M)$ 
be a manifold ringed by a sheaf of smooth functions,
and  $X\subset M$. Suppose that the
space of germs of $C^\infty M$ in $X$ is a local ring.
Prove that $X$ is a point.
\ez

\definition
A ring of functions ${\cal F}$ on $M$ 
is called {\bf soft} if for any closed subset 
 $X\subset M$, the natural map from the space of
global sections ${\cal F}(M)$ to the space of germs
${\cal F}(X)$ is surjective.
\ed

\exercise
Show that the sheaf of real analytic functions on $\R^n$
is not soft.
\ez


\exercise
Show that a constant sheaf on a manifold is not 
soft.\footnote{All manifolds are tacitly assumed to be 
of positive dimension.}
\ez


\NewVedomost


\exercise
Find a topological space $M$ and a sheaf of functions
${\cal F}$ on it such that the restriction map
from ${\cal F}(M)$ to the space of germs of ${\cal F}$ 
in a point is always surjective, but the sheaf 
${\cal F}$ is not soft.
\ez

\exercise
Let $N, N'\subset M$ be two closed subsets
of a metric space, $N\cap N'=\emptyset$. Prove that there
exist non-intersecting
neighbourhoods $U\supset N$, $U'\supset N'$.
\ez


\exercise[!]
Let $M$ be a manifold admitting a partition of unity,
$N\subset M$ a closed subset, and  $U\supset N$
its neighbourhood. Prove that $M$ has a locally
finite cover  $\{U_i\}$, such that all $U_i$
which intersect $N$ are contained in $U$.
\ez

\hint
Prove that $M$ admits a metric, and
use the previous exercise.
\eh

\definition
{\bf Support} of a function $f$ is the set
of all points where $f\neq 0$. A function is
called {\bf supported in $U$} if its support
is contained in $U$.
\ed

\exercise
Let $U\subset M$ be an open subset of a manifold,
$U'\Subset M$ an open subset satisfying $\bar U'\subset U$,
and $f$ a smooth function on $U$ with support in $U'$.
Prove that $f$ can be extended to a smooth function on $M$.
\ez

\exercise[*]
Let $M$ be a manifold admitting a partition of unity.
Prove that the sheaf of smooth functions on $M$ is soft.
\ez

\hint
Given a smooth function $f$ on $U\supset N$,
find a cover $\{U_i\}$, $i\in I$  as in previous exercise,
and let $\{\psi_i\}$ be a subordinate partition of unity.
Let $A\subset I$ be the set of indices  $\alpha \in I$ 
such that $U_\alpha \cap N \neq 0$. 
Prove that the function
$f':=\sum_{\alpha\in A} \psi_\alpha f$
is supported in $U'\Subset U$, can be extended smoothly 
to the whole $M$, and equal $f$ on $N$.
\eh

\definition
Let $f\in {\cal F}(M)$ be a section of a sheaf ${\cal F}$ on $M$.
{\bf Support} of $f$ is the set of all points $x\in M$ such that
there is no neighbourhood $U\ni x$ such that $f\restrict U=0$.
\ed

\exercise
Prove that support of any section is closed.
\ez

\definition
A sheaf ${\cal F}$ on $M$ is called {\bf fine}
if for any locally finite cover $\{U_\alpha\}$
of an open set $U\subset M$ indexed by $\alpha\in I$
and any section $f\in {\cal F}(U)$ 
there exists a collection of sections $f_\alpha\in {\cal
  F}(U)$ indexed by the same set $I$ such that a support 
of any $f_\alpha$ is contained in $U_\alpha$, and 
$\sum_I f_\alpha=f$.
\ed

\remark
Essentially the fine sheaves are sheaves which admit partition
of unity.
\er

\exercise[*]
Let $M$ be a smooth manifold. Prove that the sheaf of 
smooth functions is fine.
\ez

\exercise[*]
Let $M$ be a smooth manifold. Prove that the sheaf of 
smooth functions is soft.
\ez

\hint
Use the previous exercise.
\eh



\exercise[**]
Let $M$ be a metrizable topological space.
Prove that the sheaf of continuous functions is fine.
\ez

\exercise[**]
Let $M$ be a metrizable topological space.
Find a soft sheaf on $M$ which is not fine.
\ez

\exercise[**]
Let ${\cal F}$ be a soft sheaf of functions, with 
the rings of germs local at all points.
Prove that ${\cal F}$  is fine, or find
a counterexample.
\ez

\exercise[**]
Let $M$ be a  metrizable topological space.
Prove that any fine sheaf on $M$ is soft.
\ez



\end{document}

%\exercise
%Let  $(M, {\cal F}) \stackrel \phi 
%\arrow (M_1,{\cal F}_1)$ be a morphism of ringed spaces,
%$R_m$ the space of rings of ${\cal F}$ in $m$,
%and $R_{\phi(m)}$ the space of rings of ${\cal F}_1$ in $\phi(m)$.
%Given a function $f\in R_{\phi(m)}$ defined in a neighbourhood
% $U\supset \phi(m)$, consider its pullback
%$\phi \circ f$, defined on $\phi^{-1}(U)$.
%Prove that $f\arrow \phi \circ f$ defines a
%ring homomorphism $R_m\arrow R_{\phi(m)}$.
%\ez
%
%\definition
%This homomorphism is called
%{\bf induced by $\phi$}.
%\ed
%
%\exercise
%Let $M, M_1$ be smooth manifolds,  $\phi:\; M \arrow M_1$
%a continuous map inducing a homomorphism of the
%germs of continuous functions, mapping
%germs of smooth functions to germs of smooth
%functions. Prove that $\phi$ is smooth.
%\ez
%



%\exercise[ !]
%Пусть  $M, M_1$ многообразия, а $\phi:\; M \arrow M_1$
%непрерывное отображение, такое, что $\phi \circ f$ 
%гладко на $M$ для любой $f\in C^{\infty} M_1$.
%Докажите, что $\phi$ гладкое.
%\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Normal spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%\definition
%Топологическое пространство $M$ {\bf нормально},
%если у любых непересекающихся, замкнутых подмножеств
%$X, Y \subset M$, найдутся непересекающиеся окрестности
%$U\supset X$, $V \supset Y$.
%\ed
%
%\exercise
%Докажите, что любое метрическое пространство нормально.
%\ez
%
%\exercise
%Докажите, что любое компактное, хаусдорфово
%топологическое пространство нормально.
%\ez
%
%
%\exercise[*]
%Докажите, что любое 
%локально компактное, хаусдорфово пространство со счетной базой 
%нормально.\footnote{Из этого следует, что любое многообразие 
%со счетной базой нормально.}
%\ez
%
%\exercise[*] 
%Пусть $M$ - нормальное топологическое пространство.
%Докажите, что пучок непрерывных функций на $M$ мягкий.
%\ez
%
%\exercise[!]
%Пусть $M$ - нормальное гладкое многообразие.
%Докажите, что для любых двух непересекающихся
%замкнутых подмножеств $X, Y \subset M$
%найдется гладкая функция на $M$, которая равна
%1 на $X$ и 0 на $Y$.
%\ez
%
%
%
