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%opening

\title{18.706 Homework 2}


\author{Due Sep.16,2020}


\begin{document}

\maketitle


%\section{Plan (Fall 2020)}
%
%Total: 25 lectures.
%
%Lecture 1, 9/2 (W),   Review of rings, ideals, modules. Examples of noncommutative rings. 
%
%No lecture, 9/7 (M),
%
%Lecture 2, 9/9 (W),  Simple/Semisimple rings. Wedderburn and W-Artin theorem. HW 1 due.
%
%Lecture 3, 9/14 (M), 
%
%Lecture 4, 9/16 (W), HW 2 due.
%
%Lecture 5, 9/21 (M),
%
%Lecture 6, 9/23 (W), HW 3 due.
%
%Lecture 7, 9/28 (M),
%
%Lecture 8, 9/30 (W), HW 4 due.
%
%Lecture 9, 10/5 (M),
%
%Lecture 10, 10/7 (W), HW 5 due.
%
%Lecture 11, 10/13 ({\bf T}),
%
%Lecture 12, 10/14 (W), HW 6 due.
%
%Lecture 13, 10/19 (M), 
%
%Lecture 14, 10/21 (W), HW 7 due.
%
%Lecture 15, 10/26 (M),
%
%Lecture 16, 10/28 (W), HW 8 due.
%
%Lecture 17, 11/2 (M),
%
%Lecture 18,11/4 (W), (no HW due because of the election)
%
%Lecture 19, 11/9 (M), HW 9 due.
%
%No lecture,11/11 (W), 
%
%Lecture 20, 11/16 (M),
%
%Lecture 21,11/18 (W), HW 10 due.
%
%Thanksgiving break.
%
%Lecture 22, 11/30 (M),
%
%Lecture 23,12/2 (W), HW 11 due.
%
%Lecture 24, 12/7 (M)
%
%Lecture 25, 12/9 (W)

\section*{Theorems}

In this part you will see theorems that we studied in the class. Please supply a detailed proof of each.


\begin{theorem} Let $M$ be an $R$-module equipped with two finite composition series $0\subset M_{1}\subset M_{2}\subset\cdots\subset M_{a}=M$, and $0\subset M'_{1}\subset M'_{2}\subset\cdots\subset M'_{b}=M$ (namely $M_{i}/M_{i-1}$ and $M'_{j}/M'_{j-1}$ are simple $R$-modules for $1\le i\le a, 1\le j\le b$). Then $a=b$. Moreover, there are {\em canonical} permutation $\s: \{1,\cdots ,a\}\isom \{1,\cdots, b\}$ and {\em canonical} isomorphisms $M_{i}/M_{i-1}\isom M'_{\s(i)}/M'_{\s(i)-1}$, for all $1\le i\le a$.
\end{theorem}



\section*{Exercises}

\begin{prob}\begin{enumerate}
\item Construct a $\CC$-algebra $S$ such that an $S$-module is the same as a $\CC$-vector space $V$ together with an antilinear involution $\s: V\to V$ (antilinear means $\s(av)=\ov{a} v$ for $a\in \CC, v\in V$).
\item Show that a $\CC$-vector space $V$ together with an antilinear involution $\s: V\to V$ is the same structure as a vector space $V_{0}$ over $\RR$.
\item Combining (1) an (2), we get an equivalence of categories between $S$-modules and $\RR$-vector spaces. How do you see this equivalence directly? What is $S$ in more familiar terms?
\end{enumerate}
\end{prob}

\begin{prob} Let $(V,q)$ be an $n$-dimensional quadratic space over a field $k$. Recall the Clifford algebra $\Cl(V,q)=T(V)/I$ where $I$ is the two-sided ideal generated by $v\cdot v-q(v)$ for $v\in V$.
\begin{enumerate}
\item Let $\{v_{1},\cdots, v_{n}\}$ be a basis of $V$. For a subset $I=\{i_{1},\cdots, i_{r}\}\subset\{1,2\cdots, n\}$ such that $i_{1}<i_{2}<\cdots<i_{r}$, let $v_{I}$ be the image of $v_{i_{1}}\cdots v_{i_{r}}$ in $\Cl(V,q)$. When $I=\vn$ set $v_{\vn}=1$. Show that $\{v_{I}\}_{I\subset \{1,2\cdots, n\}}$ form a $k$-basis of $\Cl(V,q)$.
\item Let $k=\RR$ and $n=2$. Assume $q$ is nondegenerate. Describe $\Cl(V,q)$ in more familiar terms according to the signature of $q$.
\item Suppose $V$ has an orthogonal decomposition $V=V_{1}\op \j{u,v}$ such that $q(u)=0=q(v)$ and $q(u+v)=1$. Show that $\Cl(V,q)$ is isomorphic to the ring of $2$-by-$2$ matrices with entries in $\Cl(V_{1},q_{1})$ (where $q_{1}$ is the restriction of $q$ to $V_{1}$). 

Hint: Find a left ideal $I\subset \Cl(V,q)$ that is a free module of rank two under the right multiplication by $\Cl(V_{1},q_{1})$.

\end{enumerate}

\end{prob}

\end{document}
`