\documentclass[oneside,12pt]{article} \usepackage{latexsym,amscd, amsmath, amsthm, amssymb,epsfig,makeidx,tikz} \usepackage[nottoc,numbib]{tocbibind} \usepackage[top=1.4in, bottom=1.4in, left=1.4in, right=1.4in]{geometry} \def\calB{{\mathcal B}} \def\calC{{\mathcal C}} \def\calD{{\mathcal D}} \def\calE{{\mathcal E}} \def\calF{{\mathcal F}} \def\calG{{\cal G}} \def\calK{{\mathcal K}} \def\calQ{{\mathcal Q}} \def\calP{{\mathcal P}} \def\calS{{\mathcal S}} \def\RC{{R{\mathcal C}}} \def\RF{{R{\mathcal F}}} \def \Aut{\mathop{\rm Aut}\nolimits} \def\Cat{{\mathop{\rm Cat}\nolimits}} \def\cc{{\mathop{\rm cc}\nolimits}} \def \characteristic{\mathop{\rm char}} \def \End{\mathop{\rm End}\nolimits} \def \Ext{\mathop{\rm Ext}\nolimits} \def\Fun{\mathop{\rm Fun}} \def\Group{{\mathop{\rm Group}\nolimits}} \def \Hom{\mathop{\rm Hom}\nolimits} \def\Im{\mathop{\rm Im}} \def \Inn{\mathop{\rm Inn}\nolimits} \def \notdivide{\kern-2.1pt\not\kern2.1pt\bigm|} \def\Ob{{\mathop{\rm Ob}}} \def \Out{\mathop{\rm Out}\nolimits} \def \Rad{\mathop{\rm Rad}\nolimits} \def \rad{\mathop{\rm Rad}\nolimits} \def \rank{\mathop{\rm rank}} \def \restricted{\!\downarrow} \def \Soc{\mathop{\rm Soc}\nolimits} \def \soc{\mathop{\rm Soc}\nolimits} \def\Set{\mathop{\rm Set}\nolimits} \def \stab{\mathop{\rm Stab}} \def \Stab{\mathop{\rm Stab}} \def \Tor{\mathop{\rm Tor}\nolimits} \def\cod{\mathop{\rm cod}} \def\dom{\mathop{\rm dom}} \def\directlimit{{\lim\limits_{\textstyle \longrightarrow}}} \def\inverselimit{{\lim\limits_{\textstyle \longleftarrow}}} \def\Br{{\mathop{\rm Br}}} \def\res{{\mathop{\rm res}}} \def\tr{{\mathop{\rm tr}}} \def\AA{{\mathbb A}} \def\BB{{\mathbb B}} \def\CC{{\mathbb C}} \def\EE{{\mathbb E}} \def\FF{{\mathbb F}} \def\HH{{\mathbb H}} \def\NN{{\mathbb N}} \def\PP{{\mathbb P}} \def\QQ{{\mathbb Q}} \def\RR{{\mathbb R}} \def\SS{{\mathbb S}} \def\TT{{\mathbb T}} \def\ZZ{{\mathbb Z}} \makeindex %% ------------------------------------------------------------------------------- %% Document begins here %% ------------------------------------------------------------------------------- \begin{document} {\bf\noindent Math 8300\hfill Homework 3\hfill PJW\break} {\bf\noindent Date due: Friday November 1, 2019} \begin{enumerate} \item Let $Q$ be the quiver $\begin{matrix}&&2&&\cr &\nearrow&&\searrow&\cr 1&&\to&&3\cr \end{matrix} $ of type $\tilde A_2$. \begin{enumerate} \item Show that $(2,1,1)$ is a real root of $Q$, and describe explicitly an indecomposable representation of $Q$ over $K=\QQ$ with this dimension vector (specifying the maps between the three different vector spaces by matrices). \item Show that, over any field, $Q$ has infinitely many isomorphism classes of indecomposable representations. \end{enumerate} \item Describe all the abelian groups $M$ that can appear in an essential epimorphism $L\to M$ of abelian groups in each of the following three cases: \begin{enumerate} \item $L=\ZZ \to M$, \item $L=\ZZ/4\ZZ \to M$, \item $L=\ZZ/6\ZZ\to M$. \end{enumerate} \item Let $A$ be a ring with a 1, and let $V$ be an $A$-module. We write $\End_A(V)$ for the set of $A$-module homomorphisms $V\to V$. Two idempotents $e,f\in\End_A(V)$ are \textit{orthogonal} if $ef=fe=0$. The idempotent $e$ is \textit{primitive} if it cannot be written $e=e_1+e_2$ where $e_1,e_2$ are (non-zero) orthogonal idempotents. \begin{enumerate} \item Show that an element $e\in\End_A(V)$ is idempotent if and only if $e$ is the identity on restriction to the subspace $e(V)$ of $V$. \item Show that direct sum decompositions $V = W_1\oplus W_2$ as $A$-modules are in bijection with expressions $1 = e+f$ in $\End_A(V)$, where $e$ and $f$ are orthogonal idempotents. \item Show that an idempotent $e\in\End_A(V)$ is primitive if and only if the submodule $e(V)$ of $V$ is indecomposable as an $A$-module. \item Suppose that $V$ is semisimple with finitely many simple summands and let $e_1,e_2\in\End_A(V)$ be idempotent elements. Show that $e_1(V)\cong e_2(V)$ as $A$-modules if and only if $e_1$ and $e_2$ are conjugate by an invertible element of $\End_A(V)$ (i.e. there exists an invertible $A$-module homomorphism $\alpha:V\to V$ such that $e_2 = \alpha e_1\alpha^{-1}$). \end{enumerate} \item A module $U$ is said to be \textit{uniserial} if it has a unique composition series. \begin{enumerate} \item If $U$ is the direct sum of two non-zero submodules, show that $U$ is not uniserial. \item If $U$ is indecomposable and has just two composition factors, show that $U$ is uniserial. \item Give an example of a finite dimensional algebra $A$ with a finite dimensional indecomposable module that is not uniserial. (Yes, we have had some in class.) \end{enumerate} \item Show that the following conditions are equivalent for a module $U$ that has a composition series. \begin{enumerate} \item $U$ is uniserial (i.e. $U$ has a unique composition series). \item The set of all submodules of $U$ is totally ordered by inclusion. \item $\rad^r U/\rad^{r+1} U$ is simple for all $r$. \item $\soc^{r+1} U/\soc^r U$ is simple for all $r$. \end{enumerate} \item Let $A$ be a finite dimensional algebra over a field and let $U$ be an $A$-module. Write $\ell (U)$ for the Loewy length (radical length) of $U$. \begin{enumerate} \item Suppose $V$ is a submodule of $U$. Show that $\ell(V)\le \ell(U)$ and $\ell(U/V)\le \ell(U)$. Show by example that we can have equality here even when $0