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\title{MAT535 HW 9}
% \author{Ljudmila Kamenova}
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% \noindent
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\maketitle

{\it Due on 05/05 in class. Each problem is worth 10 points. 
You are required to do four of the problems of your choice. 
You can do the rest for extra credit.} \\


{\bf Problem 1.}
Show that $I ({\mathbb A}^n) = (0)$. \\



{\bf Problem 2.} 
If $I \subset R$ is any ideal, show that $\sqrt{I} := \text{rad} ~~ I$ 
is a radical ideal. \\



{\bf Problem 3.}  Prove that: 

(a) $S \subset I(Z(S))$, where $S$ is any subset of the polynomial ring
$k[x_1, \cdots , x_n]$ . \\

(b) $W \subset Z(I(W))$, where $W$ is any subset of the affine set
${\mathbb A}^n$. \\

(c) If $W$ is an algebraic set, then $W = Z(I(W))$. \\

(d) If $I \subset k[x_1, \dots, x_n]$ is any ideal, then $Z(I)=Z(\sqrt{I})$ 
and $\sqrt{I} \subset I(Z(I))$. \\



{\bf Problem 4.} 
(a) Show that the set $X = \{ (t, t^2, t^3) \in {\mathbb A}^3 | t\in k\} $ 
is closed in ${\mathbb A}^3$ and find $I(X)$. \\ 

(b) Same for the subset $Y= \{ (t^3, t^4, t^5) \in {\mathbb A}^3 | t\in k\} $ 
of ${\mathbb A}^3$. \\

(c) Show that $I(Y)$ can't be generated by less than three polynomials. \\
{\it Hint:} Is $I(Y)$ a graded ideal ? \\

{\bf Problem 5.}
Show that $W = \{ (x,y,z) \in {\mathbb A}^3 | x^2 = y^3, y^2 = z^3 \}$ is 
an irreducible closed subset of ${\mathbb A}^3$ and find $I(W)$. \\

{\bf Problem 6.} 
Find the radical $\sqrt{(y^2+2xy^2+x^2-x^4, x^2-x^3)}$. \\

{\bf Problem 7.} Let $X$ be a Noetherian topological space. Prove that: \\

(a) If an irreducible closed set $Y$ is contained in a union 
$\cup X_i$ of finitely many closed sets $X_i$, then $Y \subset X_i$ 
for some $i$. \\ 

(b) $X$ has finitely many components.





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