\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\Ker{\mathop{\rm Ker}} \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 4\hfill PJW\break} {\bf\noindent Date due: Monday December 2, 2019} \begin{enumerate} \item Let $P_S$ be an indecomposable projective module for a finite dimensional algebra over a field. Show that every non-zero homomorphic image of $P_S$ \begin{enumerate} \item has a unique maximal submodule, \item is indecomposable, and \item has $P_S$ as its projective cover. \end{enumerate} \item Let $A$ be a finite dimensional algebra. \begin{enumerate} \item Show that if $f:U\to V$ is a homomorphism of $A$-modules for which the restriction $f|_{\Soc U}:\Soc U\to V$ is one-to-one then $f$ is one-to-one. \item Show that the injective envelope of $A/\Rad A$ has the same dimension as $A$. (You may assume question 7 from homework 3.) \item Show that if $A/\Rad A \cong \Soc A$ as left $A$ modules then the left regular representation ${}_AA$ is injective as a left $A$ module; and also that the right regular representation $A_A$ is injective as a right $A$ module. (An algebra satisfying this condition is called \textit{self-injective}) \item Give an example of a self-injective algebra that is not semisimple. \end{enumerate} \item Let $A$ be a finite dimensional algebra and let $U$ be an $A$-module. \begin{enumerate} \item Prove that if $U$ is indecomposable then $\Rad_A(U,U) = \Rad_A^2(U,U)$. \item Find an example of an algebra $A$ and a module $U$ for which $\Rad_A(U,U) \ne \Rad_A^2(U,U)$. \end{enumerate} \item Let $A$ be a finite dimensional commutative algebra. Show that $A$ is a finite product of commutative local algebras. (The product of algebras $A$ and $B$ is often written as a direct sum $A\oplus B=\{(a,b)\bigm| a\in A,\,b\in B\}$.) \item Let $\alpha:U\to V_1\oplus V_2$ be a homomorphism of finite dimensional $A$-modules where $A$ is a finite dimensional algebra over a field, so that we can write $\alpha=\left[\begin{matrix}\alpha_1\cr \alpha_2\cr\end{matrix}\right]$ where $\alpha_i=p_i\circ\alpha:U\to V_i$ are the component maps of $\alpha$, the $p_i:V_1\oplus V_2\to V_i$ being the projections with respect to the direct sum decomposition. Suppose that $U$ is indecomposable, so that $\End_A(U)$ is a local ring. \begin{enumerate} \item Show that if $\alpha_1$ is split mono then $\alpha$ is split mono. \item Show that if $\alpha$ is split mono then one of $\alpha_1$ and $\alpha_2$ is split mono. \item Show that if $\alpha$ is an irreducible morphism then neither of $\alpha_1,\alpha_2$ is split epi. \item Show that if $\alpha$ is an irreducible morphism then each of $\alpha_1$ and $\alpha_2$ is an irreducible morphism \end{enumerate} \item Let $A$ be a finite dimensional $K$-algebra such that $\Rad_A^m(-,-)=0$ for some $m\ge 1$. Prove that any nonzero nonisomorphism between indecomposable modules in $A$-mod is a sum of compositions of irreducible morphisms. \item In this question, describe modules by showing their composition factors, in such a way that we can also see the composition factors if their radical and socle series. A diagrammatic notation (as done in class) is sufficient to achieve this. Let $P$ be the poset with four elements $\{1,2,3,4\}$ and partial order $1<2<4, 1<3<4$ so that the Hasse diagram is: \begin{center} \begin{tikzpicture}[xscale=.7,yscale=.7] %\draw[help lines] (0,0) grid (4,7); \draw (1,0)--(2,1)--(1,2)--(0,1)--(1,0); \draw[fill] (1,0) circle [radius=0.08]; \draw[fill] (2,1) circle [radius=0.08]; \draw[fill] (1,2) circle [radius=0.08]; \draw[fill] (0,1) circle [radius=0.08]; \end{tikzpicture} \end{center} We consider representations of this poset over $\QQ$, namely, modules for the category algebra $\QQ\calP$ of $P$ regarded as a category $\calP$, over $\QQ$, which are the same thing as functors from $\calP$ to $\QQ$-vector spaces. \begin{enumerate} \item Describe (in the sense just explained) the four indecomposable projective representations, and also the four indecomposable injective representations. \item For each of the four simple modules $S$, compute $DTr(S)$. \item Write down all almost split sequences that have as a middle term a module that is both projective and injective. \item Complete the calculation of the Auslander-Reiten quiver of $\QQ P$, giving a justification for each calculation made. \end{enumerate} \item Find the error in the following argument (perhaps showing by example what is wrong) and then give an example as requested at the very end: \begin{theorem} Let $0\to U\xrightarrow{\alpha} V\xrightarrow{\beta} W\to 0$ be an almost split sequence. If $V=V_1\oplus V_2$ is the direct sum of two non-zero submodules then the restriction of $\beta$ to each of $V_1$ and $V_2$ is one-to-one. \end{theorem} \begin{proof} Let $V=V_1\oplus V_2$ and let $p_i:V\to V_i$ be projection and $\iota_i:V_i\to V$ be inclusion with respect to this direct sum decomposition, $i=1,2$. Suppose one of the component maps $\beta\circ \iota_i=\beta|_{V_i}: V_i\to W$ is epi. Then it is not split epi, because otherwise $\beta$ would be split epi. Consider the commutative diagram with exact rows $$ \diagram{0&\to&U&\umapright{\alpha}&V&\umapright{\beta}&W&\to&0\cr &&\lmapup{\iota_i|_{ \Ker \beta|_{V_i}}}&&\lmapup{\iota_i}&&\|&&\cr 0&\to&\Ker \beta|_{V_i}&\umapright{}&V_i&\umapright{\beta|_{V_i}}&W&\to&0\cr } $$ Because $\iota_i$ is split by $p_i$, the restriction $\iota_i|_{ \Ker \beta|_{V_i}}$ is split by $p_i|_U$, so $\iota_i|_{ \Ker \beta|_{V_i}}$ is split mono. Now $U$ is indecomposable, so $\Ker \beta|_{V_i}$ is either isomorphic to $U$ or is 0. In the first case $\iota_i$ is an isomorphism, so $V$ does not have two summands. In the second case $V_i\cong W$ and $\beta$ is split epi. Both of these are contradictions, so each restriction $\beta|_{V_i}$ is one-to-one. \end{proof} Give an example of an almost split sequence where the middle term has two direct summands and the restriction of $\beta$ to one of them is epi. \end{enumerate} \end{document}