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\begin{center} {\large \textbf{Math 721 -- Homework 9}} \medskip \\ Due Friday, April 10 at 5pm \\ \ \\ 
 \end{center}
\bigskip


\noindent Good practice problems (do not turn in solutions): \\
DF 17.1:   12, 14, 19,  27  \\
DF 17.2:   4, 5, 6, 8, 20\\

\bigskip \bigskip 



\begin{prob}[DF 17.1.21] Let $R = k[x,y]$ where $k$ is a field and let $I$ denote the maximal ideal $\langle x,y\rangle$ in $R$. 
\begin{itemize}
\item[(a)] Show that the following is a free resolution of $k$ as an $R$-module: 
\[
0 \to R \overset{\alpha}{\longrightarrow} R^2 \overset{\beta}{\longrightarrow} R \overset{\pi}{\longrightarrow} k \to 0, 
\]
where 
\[
\alpha(f) = (yf, -xf), \ \ \beta(f,g) = xf + yg, \ \ \text{ and } \ \  \pi(f) = f + I \in R/I\cong k. 
\]
\item[(b)] Use the resolution in (a) to show that $\Tor_2^R(k,k)\cong k$. \medskip 
\item[(c)] Prove that $\Tor_1^R(k,I)\cong k$. (Hint: Use the long exact sequence corresponding to the short exact sequence $0\to I\to R \to k \to 0$ and (b).) \medskip 
\item[(d)] Conclude that the torsion-free $R$ module $I$ is not flat.  
\medskip 

\end{itemize}
\end{prob}


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\begin{prob}[DF 17.2.9 +] Let $G$ be an infinite cyclic group with generator $\sigma$.
\begin{itemize}
\item[(a)] Show that the map $\aug: \Z G \to \Z $ defined by 
\[
\aug\left(\sum_{i\in \Z} a_i \sigma^i\right) = \sum_{i\in \Z} a_i
\]
is a $(\Z G)$-module homomorphism, taking the trivial action of $G$ on $\Z$. \medskip 
\item[(b)] Prove that multiplication by $\sigma-1$ in $\Z G$ gives the following free resolution 
of $\Z$ as a $(\Z G)$-module:
\[
0 \to \Z G \overset{\sigma-1}{\longrightarrow} \Z G \overset{\aug}{\longrightarrow} \Z \to 0. \medskip
\] 
\item[(c)] Let $A$ be a $G$-module. Show that 
 $H^0(G,A) \cong A^G$,
 $H^1(G, A) \cong A/(\sigma -1)A$, and 
 $H^n(G,A) = 0$ for $n\geq 2$. 
 \medskip 
\item[(d)] Show that $H^1(G, \Z G) \cong \Z$.\\ 
(This shows that free modules can have nontrivial cohomology groups.) \end{itemize}
\end{prob}



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