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\DeclareMathOperator{\Spec}{Spec}
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\DeclareMathOperator{\inn}{in}

\begin{document}


\begin{center}\begin{large}\textbf{Math 523 - Problem Set 6\\
Due Wednesday, April 27}\end{large}\end{center}



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\begin{enumerate}
\item Let $R$ be a ring and $M$ an $R$-module. Let $\mathcal{J}$ be the filtration $$M = M_0 \supset M_1 \supset \cdots .$$ 
\begin{enumerate}
\item Give an example to show that the function $\inn : M \to \text{gr}_{\mathcal{J}}M $ sending $m$ to $\inn(m)$ is not necessarily a group homomorphism.
\item Let $f,g \in M$. Suppose for each $i$, $f$ is in $M_i$ if and only if $g \in M_i$. Show that either $ \inn(\inn(f) + \inn(g)) = \inn(f+g) $ or $\inn(f) + \inn(g) = 0$.
\item Suppose $M = R$ and that $\mathcal{J}$ is multiplicative, in the sense that $M_i M_j \subset M_{i+j}$. Show that either $\inn(f)\inn(g) = \inn(fg)$ or $\inn(f)\inn(g)=0$.
\end{enumerate}

\item Let $R$ be a Noetherian ring and $I \subset R$ an ideal. Let $M$ be a finitely generated $R$-module.

\begin{enumerate}
\item Show there is a largest submodule $N \subset M$ such that $N$ is annihilated by an element of the form $1-r$ with $r \in I$.
\item Show that $\bigcap_{j = 1}^\infty I^j M = N.$
\end{enumerate}

\item Characterize the $p$-adic expansions in $\Z_p$ which are images of integers.

\item Let $R = k[x_1,\ldots, x_n]/I$, $I$ some ideal. Let $m = (x_1, \ldots, x_n)$. Show that $\hat{R}_m =  k[[x_1,\ldots, x_n]]/Ik[[x_1,\ldots, x_n]]$

\item Let $R = \R[x,y]/(y^2-x^2-x^3)$. Let $m=(x,y)$.
\begin{enumerate}
\item Verify that $R$ is an integral domain.
\item Show that $\hat{R}_m$ is not an integral domain.\footnote{Hint: Can you find a square root of $1+x$?}
\item Draw a picture of the curve in $\R^2$ cut out by $y^2-x^2-x^3=0$.
\item There is a sense in which $\Spec \hat{R}_m$ can be considered a very small neighborhood around $m \in \Spec R$. How is this reflected in your picture?
\end{enumerate}
\end{enumerate}



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