\documentclass[10pt]{article} \usepackage{url} %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. \newcommand{\version}{version 1.4,\ \ 17.02.2018} \newcommand{\firstdate}{24.01.2018} \addtolength{\topmargin}{-5mm} \addtolength{\textheight}{10mm} \addtolength{\oddsidemargin}{-5mm} \addtolength{\textwidth}{10mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \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}