\documentclass[10pt]{article} %version 1.0,\ \ 18.02.2013 (started translation) %version 1.1,\ \ 04.03.2013 4.21 ochepyatka %version 1.2,\ \ 08.04.2013 4.27 zvezdochka (ne nado) \newcommand{\version}{version 1.2,\ \ 08.04.2013} \newcommand{\firstdate}{25.02.2013} \addtolength{\topmargin}{-5mm} \addtolength{\textheight}{10mm} %\addtolength{\oddsidemargin}{-10mm} %\addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \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 % % %