\documentclass[12pt]{article} % version 1.0 -- April 16, 2013, started the file (from pde-8.tex) \usepackage{diagrams} \usepackage{amscd} \newcommand{\version}{version 1.0,\ \ 16.04.2013} \newcommand{\firstdate}{22.04.2013} %\addtolength{\topmargin}{-10mm} %\addtolength{\textheight}{20mm} %\addtolength{\oddsidemargin}{-10mm} %\addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{10}{Geometry 10: De Rham algebra} {\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. 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{K\"ahler differentials} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $R$ be a ring over a field $k$, and $V$ an $R$-module. A $k$-linear map $D:\; R \arrow V$ is called {\bf a derivation} if it satisfies {\bf the Leibnitz identity} $D(ab) = a D(b) + b D(a)$. The space of derivations from $R$ to $V$ is denoted $\Der_k(R, V)$. \ed \exercise Consider an action of $R$ on $\Der_k(R, V)$, with $rd$ acting as $a \arrow rd(a)$. Prove that this defines a structure of $R$-module on $\Der_k(R, V)$. \ez \exercise Let $[K:k]$ be a finite extension of a field of characteristic 0, and $V$ a vector space over $K$. Prove that $\Der_k(K, V)=0$. \ez \exercise[!] Let $M$ be a smooth manifold, $x\in M$ a point, $R = C^\infty M$, and ${\goth m}_x\subset R$ the maximal ideal of $x$. Consider an $R$-module $V:=R/{\goth m}_x$. Find $\dim_\R \Der_R(R, V)$. \ez \exercise[**] Let $R=C^0 M$ be a ring of continuous functions on a manifold $M$, and $V$ an $R$-module of dimension 1 over $\R$. Find a non-trivial derivation $\nu \in \Der_k(R, V)$, or prove that it does not exist. \ez \definition Let $R$ be a ring over a field $k$. Define an $R$-module $\Omega^1_k R$, sometimes denoted simply as $\Omega^1 R$, with the following generators and relations. The generators of $\Omega^1_k R$ are indexed by elements of $R$; for each $a\in R$, the corresponding generator of $\Omega^1_k R$ is denoted $da$. Relations in $\Omega^1_k R$ are generated by expressions $d(ab) = a db + b da$, for all $a, b \in R$, and $d\lambda =0$ for each $\lambda\in k$. Then $\Omega^1_k R$ is called {\bf the module of K\"ahler differentials} of $R$. \ed \exercise Prove that the natural map $R \arrow \Omega^1_k R$, with $a \mapsto da$ is a derivation. \ez \exercise Let $R$ be a quotient of $k[r_1, ..., r_k]$ by an ideal. Prove that $\Omega^1_k R$ is generated as an $R$-module by $dr_1, ..., dr_k$. \ez \exercise[!] \label{_diffe_dual_Zadacha_} Prove that $\Der_k(R) = \Hom_R(\Omega^1_k R, R)$. \ez \exercise \label{_diffe_katego_Zadacha_} Let $V$ be an $R$-module, and $D\in \Der_k(R, V)$ a derivation. Prove that there exists a unique $R$-module homomorphism $\phi_D:\; \Omega^1_k R \arrow V$ making the following diagram commutative. \begin{diagram} R & \rTo^d & \Omega^1_k R \\ &\rdTo~D &\dTo~{\phi_D}\\ & & V \end{diagram} \ez \remark This property is often taken as a definition of $\Omega^1_k R$. \er \exercise[!] Let $R= k[t_1, ..., t_n]$ be a polynomial ring over a field of characteristic 0. Prove that $\Omega^1_k R$ is a free $R$-module generated by $d t_1, dt_2, ..., dt_n.$ \ez \exercise[*] Let $I\subset R$ be an ideal. Construct an exact sequence \[ I/I^2 \arrow \Omega^1(R)\otimes_R R/I\arrow \Omega^1(R/I)\arrow 0. \] \ez \exercise Let $R\stackrel \phi \arrow R'$ be a ring homomorphism. Consider $\Omega^1 R'$ as an $R$-module, using the action $r, a \arrow \phi(r)a$. \enum \ite Prove that there exists an $R$-module homomorphism $\Omega^1 R \arrow \Omega^1 R'$, mapping $dr$ to $d\phi(r)$. \ite Prove that it is unique. \ee \ez \definition \label{_induced_Kahle_Definition_} In this case, we say that the homomorphism $\Omega^1 R \arrow \Omega^1 R'$ {\bf is induced by $\phi$}. \ed \exercise[*] Let $R$ be a ring of continuous functions on a manifold, and ${\goth m}_x$ a maximal ideal of a point. Prove that ${\goth m}_x\Omega^1 R=\Omega^1 R$. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Cotangent bundle} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $A, B$ be $R$-modules, and $\nu:\; A\times B \arrow R$ a bilinear pairing. It is called {\bf non-degenerate} if for each $a\in A$ there exists $b\in B$ such that $\nu(a, b)\neq 0$, and for each $b\in B$ there exists $a\in A$ such that $\nu(a, b)\neq 0$ \ed \exercise Let $A, B$ be vector spaces over $k$, and $\nu:\; A\times B \arrow k$ a non-degenerate pairing. Prove that $A$ is isomorphic to $B^*$, or find a counterexample \enum \ite When $A, B$ are finite-dimensional. \ite When $A, B$ are infinite-dimensional. \ee \ez \definition Let $V$ be an $R$-module. A {\bf dual $R$-module} $\Hom_R(V,R)$ is denoted $R^*$. \ed \exercise Let $V$ be an $R$-module. Consider the natural pairing $V\times V^*\arrow R$. Prove that it is non-degenerate, or find a counterexample, in the following cases: \enum \ite when $R$ is a field, and $V$ a (possibly infinite-dimensional) vector space \ite[!] when the natural map $V\arrow V^{**}$ is injective \ite when $V$ is a free $R$-module \ite[*] when $R$ is a ring which has no zero divisors. \ee \ez \exercise[*] Let $A, B$ be finitely-generated $R$-modules, and $\nu:\; A\times B \arrow R$ a non-degenerate pairing. Prove that $A$ is isomorphic to $B^*$, or find a counterexample. \ez \exercise[!] \label{_free_pairing_dual_Exercise_} Let $A$ be a free, finitely generated $R$-module, and $\nu:\; A\times B \arrow R$ a non-degenerate pairing. Prove that $B$ is also free, and isomorphic to $A^*$. \ez \definition Let $A$, $B$ be finitely generated $R$-modules, and $\nu:\; A\times B \arrow R$ a bilinear pairing. Define {\bf the annihilator of $\nu$ in $B$} as a submodule consisting of all elements $b \in B$ for which the homomorphism $\nu(\cdot, b):\; A \arrow R$ vanishes. \ed \definition Let $M$ be a smooth manifold, $R:= C^\infty M$ the ring of smooth finctions, and $\nu:\; \Der(R)\times \Omega^1 R \arrow R$ the pairing constructed in Exercise \ref{_diffe_dual_Zadacha_}. Consider its annihilator $K\subset \Omega^1 R$. Define {\bf the cotangent bundle} as $\Lambda^1 M:= \Omega^1 R /K$. For the purpose of this definition, $\Lambda^1 M$ is a $C^\infty M$-module. \ed %\exercise[!] %Prove that the natural pairing %$\nu:\; \Der(C^\infty M)\times \Lambda^1 M \arrow R$ %is bilinear and non-degenerate. %\ez \exercise Let $R:= C^\infty \R^n$, and $t_1, ..., t_n\in R$ be coordinate functions. Consider an element in $\Lambda^1 \R^n$, written as $P= \sum_{i=1}^n P_i dt_i$, let $Q= \sum_{i=1}^n Q_i \frac{d}{dt_i}\in\Der_k(R)$ -- be a vector field, and $\nu:\; \Der(R)\times \Lambda^1 \R^n \arrow R$ the natural pairing. Prove that $\nu(P, Q) = \sum_i P_i Q_i$. \ez \exercise[!] In these assumptions, prove that $\Lambda^1 R$ is a free $\R$-module, generated by $dt_1, ..., dt_n$. \ez \hint Prove that $\Der(R) = \Hom_R(\Omega^1 R, R)$, and $\Der(R)$ is a free $R$-module. Use exercise \ref{_free_pairing_dual_Exercise_}. \eh \exercise Let $A, B$ be finitely-generated projective $R$-modules, and $\nu:\; A\times B \arrow R$ a non-degenerate pairing. Prove that $B\cong A^*$. \ez \exercise[!] Let $M$ be a smooth, metrizable manifold. Prove that \[ \Lambda^1 M = \Hom_{C^\infty M}(\Der(C^\infty M), C^\infty M).\] \ez \hint Use the previous exercise and apply the Serre-Swan theorem. \eh \exercise[*] Let $K$ be the kernel of the natural projection \[ \Omega^1 (C^\infty M) \arrow \Lambda^1 M.\] Prove that ${\goth m}_x K =K$ for each maximal ideal of a point $x\in M$. \ez \exercise[**] Show that $K$ is non-empty. \ez %\exercise %Let $B$ be a smooth manifold, % $M \stackrel \Delta \arrow M \times M$ %the diagonal embedding, and $I\subset C^\infty M\times M$ %an ideal of all functions vanishing in $\Delta$. %\enum %\ite Show that the ring %$C^\infty (M\times M)/I$ is isomorphic to $C^\infty M$. %\ite[*] Prove that the natural map %$C^\infty M\times M\arrow \End_{C^\infty M\times M}(I/I^2)$ %can be factorized through the projection %\[ C^\infty (M\times M)\arrow C^\infty (M\times M)/I = C^\infty M. %\] %Deduce that the space $I/I^2$ is equipped with a %natural structure of a module over %$C^\infty M$. % %\ite[*] %Show that $I/I^2$ is isomorphic to $\Lambda^1 M$ %as a $C^\infty M$-module. %\ee %\ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{De Rham algebra} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a smooth manifold. {\bf A bundle of differential $i$-forms on $M$} is the bundle $\Lambda^i T^* M$ of antisymmetric $i$-forms on $TM$. It is denoted $\Lambda^i M$. \ed \definition Let $\alpha\in (V^*)^{\otimes i}$ and $\alpha\in (V^*)^{\otimes j}$ be polylinear forms on $V$. Define the {\bf tensor multiplication} $\alpha \otimes \beta$ as \[ \alpha \otimes \beta(x_1, ..., x_{i+j}):= \alpha(x_1, ..., x_j) \beta(x_{i+1}, ..., x_{i+j}). \] \ed \exercise Let $\bigotimes_k T^*M \stackrel \Pi \arrow \Lambda^k M$ be the antisymmetrization map, \[ \Pi(\alpha)(x_1, ..., x_n):= \frac 1{n!} \sum_{\sigma\in \Sym_n} (-1)^\sigma \alpha(x_{\sigma_1}, x_{\sigma_2}, ..., x_{\sigma_n}). \] Define the ``exterior multiplication'' $\wedge:\; \Lambda^i M\times \Lambda^j M \arrow \Lambda^{i+j} M$ as $\alpha \wedge \beta := \Pi (\alpha \otimes \beta)$, where $\alpha \otimes \beta$ is a section $\Lambda^i M\otimes \Lambda^j M \subset \bigotimes_{i+j} T^*M$ obtained as their tensor multiplication. Prove that this operation is associative and satisfies $\alpha \wedge \beta = (-1)^{ij} \beta\wedge \alpha$. \ez \definition The algebra $\Lambda^* M := \oplus_i\Lambda^i M$ with the multiplicative structure defined above is called {\bf the de Rham algebra} of a manifold. \ed \exercise[*] Let $M$ be an oriented manifold. Prove that all bundles $\Lambda^iM$ are oriented, or find a counterexample. \ez \exercise Prove that de Rham algebra is multiplicatively generated by $C^\infty M =\Lambda^0M$ and $d(C^\infty)\subset \Lambda^1 M$. \ez \exercise Prove that a derivation on an algebra is uniquely determined by its values on any set of multiplicative generators of this algebra. \ez \definition {\bf De Rham differential} $d:\; \Lambda^*M \arrow \Lambda^{*+1}M$ is an $\R$-linear map satisfying the following conditions. \begin{description} \item[(i)] For each $f \in \Lambda^0M=C^\infty M$, $d(f)\in \Lambda^1 M$ is equal to the image of the K\"ahler differential $df\in \Omega^1 M$ in $\Lambda^1 M = \Omega^1 M/K$. \item[(ii)] (Leibnitz rule) $d(a\wedge b) = da \wedge b + (-1)^j a\wedge db$ for any $a\in \Lambda^i M, b \in \Lambda^j M$. \item[(iii)] $d^2=0$. \end{description} \ed \exercise[!] Prove that de Rham differential is defined uniquely by these axioms. \ez \hint Use the previous exercise. \eh \exercise \label{_de_Rham_locally_Zadacha_} Let $t_1, ..., t_n$ be coordinate functions on $\R^n$, and $\alpha\in \Lambda^* \R^n$ a monomial obtained as a product of several $dt_i$, \[ \alpha = dt_{i_1}\wedge dt_{i_2} \wedge ... \wedge dt_{i_k}, \] $i_1 < i_2 < ...< i_k$ (such a monomial is called {\bf a coordinate monomial}). \enum \ite Prove that $\Lambda^*\R^n$ is a trivial bundle, and coordinate monomials are free generators of $\Lambda^*\R^n$. \ite Show that the de Rham differential, if it exists, satisfies $d(f\alpha)=\sum_i \frac {df}{dt_i} dt_i \wedge \alpha$ for any $f\in C^\infty \R^n$. \ite Prove that this formula defines the de Rham differential on $\Lambda^* \R^n$ correctly. \ee \ez \NewVedomost \exercise \enum \ite Prove that de Rham differential $d:\; \Lambda^*M \arrow \Lambda^{*+1}M$ commutes with restrictions to open subsets. \ite Show that de Rham differential (if it exists) defines a sheaf morphism. \ee \ez \hint Use uniqueness of de Rham differential. \eh \exercise[!] Prove that de Rham differential exists on any manifold. \ez \hint Locally, de Rham differential is constructed in exercise \ref{_de_Rham_locally_Zadacha_}. To go from local to global, use the previous exercise, and apply the sheaf axioms. \eh \exercise[*] Let $R$ be a ring over a field, and $\Omega^i R:= \Lambda^i_R \Omega^1 R$ an exterior algebra generated by K\"ahler differentials. Prove that there exists the de Rham differential $d:\; \Omega^* R\arrow \Omega^{*+1} R$ satisfying the axioms above. \ez \end{document}