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\centerline{\bf HW \#8, DUE 3-3}
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\noindent 1. Let $A$ be a commutative ring.  Show that $p \in A[x],p\ne0$ is a zero divisor in $A[x]$ if and only if that is $a\in A,a\ne0$ with $ap = 0$.  (Look at a polynomial $q$ of least possible degree such that $pq = 0$).  Generalize this result to $A[x_1,\ldots,x_n]$.
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\noindent 2. Let $K$ be an algebraically closed field of characteristic 0 and let $V\subset K^2$ be our old friend the variety with equation $x^2 = y^3$.  Show that the maximal ideal $M_{(1,1)}$ of the coordinate ring $K[V]$ at the point $(1,1)$ is not the radical of any principal ideal.
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\noindent 3. Again let $K$ be algebraically closed (but of arbitrary characteristic).  Show that the subvariety of $K^2$ with equation $y^3 = x^5$ has a singularity at the origin that is resolved by two blowups.
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\noindent 4. Let $C$ be the rational quartic curve in $\Bbb P^3$, consisting of all points with projective coordinates $(s^4,s^3t,st^3,t^4)$ for some $s,t\in K,s,t$ not both 0.  Find a set of four generators of the (homogeneous) ideal $I$ of $C$ and prove that $I$ cannot be generated by three elements. 
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\noindent 5. Let $A$ be a commutative ring, $I$ an ideal of $A$.  Give $A$ the standard $I$-filtration, taking $A_n = I^n$, and assume that $\cap_n I^n = 0$ and that the graded ring $G=G(A)$ is an integral domain.  Show that $A$ is also an integral domain.
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