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\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{}

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\DeclareMathOperator{\Hom}{Hom} 
\DeclareMathOperator{\coker}{coker} 
\DeclareMathOperator{\Tor}{Tor} 
\DeclareMathOperator{\Ext}{Ext} 
\newcommand{\consistent}{consist} 

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\begin{document}
\maketitle
\vspace{-50pt}
\begin{center}
{\huge \bf Homework 5}\\\vspace{8pt}
{\Large Due Thursday night, October 26} {\footnotesize(technically 5am Oct.\ 27)}\\
\end{center}

\begin{question}
Let $R=\Z[t]/(t^2-1)$. Regard $\Z$ as an $R$-module by letting $t$ act by the identity. Compute $\Tor^R_k(\Z,\Z)$ and $\Ext_R^k(\Z,\Z)$ for all $k\geq 0$.
\end{question}
\vfill


\begin{question}
Let $R=\Z[\sqrt{-30}]$. Regard $\F_2$ as an $R$-module by letting $\sqrt{-30}$ act by 0.\newline Compute $\Tor^R_k(\F_2,\F_2)$ and $\Ext_R^k(\F_2,\F_2)$ for all $k\geq 0$.
\end{question}
\vfill


\begin{question}
Let $R=\R[T]$. Let $M=\R^2$, with $R$-module structure where $T$ acts by $\begin{psmallmatrix}1&2\\0&1\end{psmallmatrix}$. Let $N=\R$ with $R$-module structure where $T$ acts by $0$.\newline
Compute $\Tor^R_k(M,N)$ and $\Ext_R^k(M,N)$ and $\Ext_R^k(N,M)$ for all $k\geq 0$.
\end{question}
\vfill

\begin{question}
Let $R=\C[T]$. Given $\lambda\in \C$, let $\C_\lambda$ denote $\C$ regarded as an $R$-module by letting $T$ act by $\lambda$. Compute $\Ext_R^k(\C_\lambda,\C_\mu)$ for all $k\geq 0$, for all $\lambda,\mu\in \C$.
\end{question}
\vfill

\begin{question}
Let $R=\C[x,y]$.
\begin{enumerate}[label=(\alph*)]
\item Regard $\C$ as an $R$-module by letting $x$ and $y$ act by $0$. Compute $\Tor^R_k(\C,\C)$  for all $k\geq 0$.
\item Let $I\subset R$ be the ideal $I=(x,y)$. We would like to understand $I\otimes_R I$, so:

Give a basis for $I\otimes_R I$ as a complex vector space.\newline If you can also describe the $R$-module structure without too much pain, please do.
\end{enumerate}
\end{question}

\vfill \hfill (cont.)
\newpage
{\small Recall that $R$ is a PID (principal ideal domain) if $R$ is a domain and every ideal in $R$ is principal (generated by one element). \newline\phantom{x} \hfill Remember that you can use earlier questions in later questions.}\newline

\noindent In the next questions, let $R$ be a PID, and let $M$ and $X$ be $R$-modules.\newline
\textbf{NOTE:} You {\Large \textbf{cannot}} use the structure theorem for modules over a PID on this homework.

{\small [Note: Q6 was intended to be a helpful intermediate step to help you solve Q7, and some of you did it this way. But for others, Q6$'$ below might be an easier intermediate step. You can do Q6$'$ in place of Q6 if you prefer. Or, if you already have a direct proof of Q7, you can just skip Q6/Q6$'$ entirely.]}
\begin{question}
Prove that if $M$ is torsion-free and finitely generated, then \[\Tor_k(M,X)=0\text{\qquad for all }k>0\qquad \text{ and any $X$}.\]
\end{question}

\begin{questionsixprime} (replaces Q6)
Prove that if $M$ is torsion-free, then \[\phantom{\emph{finitely generated}}\Tor_k(M,X)=0\text{\qquad for all }k>0\qquad \text{ and any \emph{finitely generated} $X$}.\]
\end{questionsixprime}

\begin{question}
Deduce from Q6, \emph{or} from Q6$'$, \emph{or} prove directly: for any torsion-free $M$, \[\Tor_k(M,X)=0\text{\qquad for all }k>0\qquad \text{ and any $X$}.\]
\end{question}
[If you give a self-contained direct proof for Q7, you will automatically get credit for Q6.]

\begin{question}
Deduce from the previous question that for \emph{any} $M$, \[\Tor_k(M,X)=0\text{\qquad for all }k>1\qquad \text{ and any $X$}.\]

\end{question}



\vfill
Do at least one of the following questions. If you've seen one of these questions before, please at least try to do one of the others.
\addtocounter{question}{1}
\begin{namedquestion}
Compute $\Ext^1_\Z(\Q,\Z)$.
\end{namedquestion}
\vfill
If $M$ is a $\Z$-module, note that $d|n$ implies $nM\subset dM$, so there is a quotient map\linebreak $\pi_n^d\colon M/nM\to M/dM$ (it descends from the identity $M\to M$, so in symbols it's just $\overline{m}\mapsto \overline{m}$).

 Define $\consistent(M)$ to be the submodule of $\prod_{n\in \N} M/nM$ defined by 
\[\consistent(M)\coloneq \big\{\ (m_n\in M/nM)_{n\in \N}\ \big\vert\ d|n \implies \pi_n^d(m_n)=m_d\ \big\}\]
This makes $\consistent$ an additive functor from $\Z$-modules to $\Z$-modules {\small (you do not have to prove this)}.\newline 
{\footnotesize (Note that $\N=\{1,2,3,\ldots\}$ here; it does not include 0.)}
\begin{namedquestion}
Is $\consistent$ an exact functor? Prove your answer is correct.
\end{namedquestion}
\vfill

\begin{namedquestion} $\consistent(\Z)$ has a natural ring structure (for example, it is a subring of $\prod_{n\in \N}\Z/n\Z$); you do not have to prove this.

Describe the commutative ring $\Q\otimes_\Z \consistent(\Z)$.\newline{\small (You have some flexibility here in what your ``description'' should be, but don't just rephrase the definition.)}
\end{namedquestion}



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