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\centerline{\bf HW \#1, DUE 4-7}
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\centerline{\bf MATH 506A}
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\noindent 1. Let $R$ be a (commutative) ring such the localization $R_M$ of $R$ at any maximal ideal $M$ is Noetherian and such that there are only finitely many maximal ideals containing any nonzero element of $R$.  Show that $R$ is Noetherian.  (Let $I\ne0$ be an ideal of $R$ and let $M_1,\ldots,M_r$ be the maximal ideals containing it.  Choose $x_0\in I,x_0\ne0$ and let
$M_1,\ldots,M_{r+s}$ be the maximal ideals containing $x_0$.  For $1\le j\le s$ pick $x_j\in I,x_j\notin M_{r+j}$.  The ideal $IR_{M_i}$ is finitely generated for $1\le i\le r$, so we can choose
$x_{s+1},\ldots,x_t\in I$ whose images generate $IR_{M_i}$ for $1\le i\le r$.  Let $I_0 = (x_0,\ldots,x_t)\subset R$.  Show that $I_0$ and $I$ generate the same ideal in $R_M$ for all maximal ideals $M$ of $R$ and deduce that $I_0 = I$.)
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\noindent 2. Let $R = K[x_1,\ldots]$ be the polynomial ring in infinitely many variables $x_i$ over a field $K$.  Let $m_1,m_2,\ldots$ be an increasing sequence of integers such that
$m_{i+1}-m_i>m_i-m_{i-1}$ for all $i$ (e.g. $m_i = 2^i$) and let $P_i = (x_{m_i+1},\ldots,x_{m_{i+1}})$ be the prime ideal of $R$ generated by the given variables.  Let $S$ be the complement of the union of the $P_i$.  Show that the localization $S^{-1}R$ satisfies the hypotheses of Problem 1, so that
$S^{-1}R$ is Noetherian.  Also show that each $S^{-1}P_i$ has height $m_{i+1}-\,m_i$, so the dimension of the Noetherian ring $S^{-1}R$ is infinite.
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\noindent 3. Let $P_1,\ldots,P_m$ be prime ideals in a polynomial ring $R=K[x_1,\ldots,x_n]$ (with $K$ algebraically closed) such that no $P_i$ contains another.  The intersection $I$ of the $P_i$ is then a radical ideal.  Show that the multiplicity of each $P_i$ in the $R$-module $R/I$ (as defined last quarter) is one, by setting $I_j = \cap_{i=j}^m P_i$ for $1\le j\le m, I_{m+1} = R$, filtering $R/I$ by the increasing intersections $I_1/I,I_2/I,\ldots,I_{m+1}/I$ and showing that, when localized at $P_i$, exactly one graded piece $I_j/I_{j-1}$ is isomorphic to $R/P_i$ while the others are 0.
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\noindent 4. Suppose that $R$ is complete with respect to an ideal $I$ and that $M$ is an $R$-module.  Call $M$ {\sl separated} with respect to $I$ if $\cap_n I^nM = 0$.  Show that if $M$ is separated and the images of $m_1,\ldots,m_n\in M$ generate $M/IM$, then the $m_i$ generate $M$.
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\noindent 5. The {\sl Jacobson radical} $J$ of a commutative ring $R$ is the intersection of all maximal ideals of $R$.  Show that $J$ consists exactly of those $r\in R$ such that $1+xr$ is a unit for all $x\in R$.  Deduce that if $R$ is complete with respect to an ideal $I$, then (the image of) $I$ lies in the Jacobson radical of $R$.
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