\documentclass[11pt]{article} \usepackage{amsmath,amssymb,amsthm,amsfonts,verbatim} \usepackage{enumitem} \usepackage{microtype} \usepackage{url} \usepackage{hyperref} \usepackage[margin=1in]{geometry} \usepackage{xypic} \usepackage[normalem]{ulem} \usepackage{mathtools} \theoremstyle{definition} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{claim}[theorem]{Claim} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{question}{Question} \newtheorem{innerquestionst}{Question} \newenvironment{questionst}{\renewcommand\theinnerquestionst{{\textrm \thequestion\st}} \addtocounter{question}{1}\innerquestionst}{\endinnerquestionst} %\newtheorem{innernamedquestion}{Question} %\newenvironment{namedquestion}[1]{\renewcommand\theinnernamedquestion{{\textrm #1}}\innernamedquestion}{\endinnernamedquestion} \newtheorem{namedquestion}{Question} \renewcommand\thenamedquestion{\arabic{question}\Alph{namedquestion}} \newlist{compactitem}{itemize}{3} \setlist[compactitem]{nosep} \setlist[compactitem,1]{label=\textbullet} \setlist[compactitem,2]{label=--} \setlist[compactitem,3]{label=\ensuremath{\ast}} \newlist{compactenum}{enumerate}{3} \setlist[compactenum]{nosep} \setlist[compactenum,1]{label=\arabic*.} \setlist[compactenum,2]{label=(\alph*)} \setlist[compactenum,3]{label=\roman*.} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \title{\vspace{-50pt}Math 210A: Modern Algebra\\ \large Thomas Church ({\tt tfchurch@stanford.edu})\\ \href{http://math.stanford.edu/~church/teaching/210A-F17/}{\nolinkurl{http://math.stanford.edu/~church/teaching/210A-F17}}} \author{} \date{} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathbb{F}} \newcommand{\abs}[1]{\left\vert #1\right\vert} \newcommand{\m}{\mathfrak{m}} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Gal}{Gal} \newcommand{\into}{\hookrightarrow} \newcommand{\onto}{\twoheadrightarrow} \newcommand{\coloneq}{\mathrel{\mathop:}\mkern-1.2mu=} \newcommand{\st}{$^{\ast}$} \renewcommand{\phi}{\varphi} \newcommand{\ZG}{\Z G} \renewcommand{\baselinestretch}{1.1} \begin{document} \maketitle \vspace{-50pt} \begin{center} {\huge \bf Homework 7}\\\vspace{8pt} {\Large Due Thursday night, November 9} {\footnotesize(technically }{\huge \textbf{2am} }{\footnotesize Nov.\ 10)}\\ \end{center}\vspace{5pt} \begin{question} Recall that in class we used the free resolution from HW4 Q4(g) to compute for $G=\Z/2=\{1,s\}$ that \[ H^k(\Z/2;\,M) =\begin{cases} M^G & k=0\\ \frac{\{m\in M|sm+m=0\}}{\{sn-n|n\in M\}} & k=1,3,5,\ldots\\ \frac{\{m\in M|sm=m\}}{\{sn+n|n\in M\}} & k=2,4,6,\ldots\\ \end{cases} \] For $G=\Z/n=\{1,s,\ldots,s^{n-1}\}$, find a similar description of $H^k(\Z/n;\,M)$ for a $\Z G$-module $M$. {\small (Hint: find a free resolution of $\Z$ as a $\Z G$-module; note that $\Z G\cong \Z[s]/(s^n-1)$.\newline The resolution will again be 2-periodic just like for $\Z[s]/(s^2-1)$.} \end{question} \vfill \begin{question} Let $G$ be a group. \begin{enumerate}[label=(\alph*)] \item Prove that $H^0(G;\Z G)\cong \Z$ if $G$ is finite, and $H^0(G;\Z G)=0$ if $G$ is infinite. \item Prove that $H^1(G;\Z G) \neq 0$ if $G=\Z=\{\ldots,t^{-1},1,t,\ldots\}$. \item (Hard, very optional) Can you find another group for which $H^1(G;\ZG)\neq 0$? \end{enumerate} \end{question} \vfill \begin{question} Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$. The unit group $L^\times$ is an abelian group with an action of $G$, so we may consider the group cohomology $H^k(G;L^\times)$. A theorem of Noether states that $H^1(G;L^\times)=0$; you may assume this without proof. \begin{enumerate}[label=(\alph*)] \item Use Noether's theorem to prove that if $\Gal(L/K)$ is generated by a single element $s$, then every element $\ell\in L$ with norm 1 has\footnote{Recall that for a Galois extension $L/K$ the norm $N^L_K\colon L\to K$ is given by $N^L_K(\ell)=\prod_{g\in \Gal(L/K)} g\cdot \ell$.} the form $s(z)/z$ for some $z\in L$. \item Use part (a) to give a parametrization in two rational parameters of the rational points on the unit circle: \[S^1(\Q)=\{(x\in \Q,y\in \Q)\,|\,x^2+y^2=1\}.\] That is, give two rational functions $x(a,b)\in \Q(a,b)$ and $y(a,b)\in \Q(a,b)$ such that the resulting function $f\colon \Q^2\to \Q^2$ given by $(a,b)\mapsto \big(x(a,b),y(a,b)\big)$ has image $S^1(\Q)$. {\small NOTE: $f$ might not not actually be a function $\Q^2\to \Q^2$ because a rational function like $x(a,b)$ might take the value~$\infty$ sometimes. So formally this should say\newline ``the resulting function $f\colon \{\text{subset of $\Q^2$ where both $x$ and $y$ are not $\infty$}\}\to \Q^2$ has image $S^1(\Q)$.''} \end{enumerate} \end{question} \vfill \hfill (cont.) \newpage \noindent Given a chain complex $C_\bullet=\ \to \cdots C_2\to C_1\to C_0\to 0$ and a chain map $f\colon C_\bullet\to C_\bullet$: We call $f$ an \emph{involution} if $f\circ f = \id$. We call $f$ a \emph{weak involution} if there is a \emph{homotopy} $f\circ f\sim \id$. \begin{question} \label{q:chain-easy} Give an example of a chain complex $C_\bullet$ and a weak involution $f\colon C_\bullet\to C_\bullet$ that is not an involution. \end{question} \vfill \begin{question} (Optional, replaces Q\ref{q:chain-easy}) Give an example of a chain complex $C_\bullet$ and a weak involution $f\colon C_\bullet\to C_\bullet$ that is not \emph{homotopic} to an involution.\newline {\small (That is, there does not exist any involution $g\colon C_\bullet\to C_\bullet$ with $g\circ g = \id$ and $f\sim g$.)} \end{question} \vfill \begin{question} (Stupid hard, worth 0 points; only respect and admiration)\newline {\small OK to discuss with classmates and even turn in joint solution, but \emph{not} to discuss with others outside class. Can be turned in any time before the last day of class.}\\ Give an example of a chain complex $C_\bullet$ and a weak involution $f\colon C_\bullet\to C_\bullet$ that cannot be made homotopic to an involution \emph{even if we replace $C_\bullet$ by a homotopy equivalent complex}.\\ More precisely, note that given a homotopy equivalence $\phi\colon C_\bullet\to D_\bullet$ with homotopy inverse $\psi\colon D_\bullet\to C_\bullet$, then the map $\phi\circ f\circ \psi\colon D_\bullet\to D_\bullet$ will be a weak involution. Say that $f$ is ``fixable'' if for some such $\phi$ and $\psi$ this weak involution $\phi\circ f\circ \psi$ is homotopic to an involution $g\colon D_\bullet\to D_\bullet$.\\ Give an example of a chain complex $C_\bullet$ and weak involution $f\colon C_\bullet\to C_\bullet$ that is not ``fixable''. \end{question} \vfill\vfill \end{document}