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\begin{document}
\pagestyle{plain}
\title{Intro to Algebraic Geometry, Problem Set 1}
\date{September 14, 1999.}
\maketitle

{\parskip=12pt % closing bracket is just before the bibliography 

Due Tuesday September 21 in class.  No lates will be accepted, so Ana-Maria will be able to return it in time for
Thursday's class.

\begin{enumerate}

\item
Prove that the points of the form
$(t,t^2,t^3)$ in $\A^3$ form an algebraic set.  In other words, find a
set of functions that vanish on these points, and no others.

\item  If $I$ is an ideal, show that $\sqrt{I}$ is an ideal.

\item
(a)  Let $V$ be an algebraic set in $\A^n$, $P$ a point not in $V$. Show
that there is a polynomial $F$ in $\kbar[x_1,...,x_n]$ such that $F(Q) = 0$ for all $Q$ in
$V$, but $F(P) = 1$.  Hint:  $I(V) \neq I(V \cup {P})$.

\item[]
 (b) Let $\{ P_1, ..., P_r \}$ be a finite set of points in $\A^n(\kbar)$.  Show that
there are polynomials $F_1, ..., F_r \in \kbar [x_1,...,x_n]$ such that $F_i(P_j) =
0$ if $i \neq j$, and $F_i(P_i) = 1$.

\item  Show that for any ideal $I$ in $\kbar [x_1,...,x_n]$, $V(I) = V(\sqrt{I})$, and
$\sqrt{I}$ is contained in $I(V(I))$.  (You aren't allowed to use the Nullstellensatz.)

\item (a) If $I_1$ and $I_2$ are ideals of some ring, show that $\sqrt{I_1 I_2 } = \sqrt{ I_1 \cap I_2}$.

\item[] (b)
If $I_1$ and $I_2$ are radical ideals, show that their intersection is also
radical.

\item Find the ideal in $\kbar[x_1, x_2, x_3]$ corresponding to the $x_1$-axis union the point (1,1,1).

\item Without descending into horrific algebra, prove that there are three points $a, b, c \in \A^2$ such that
$$
\sqrt{  (x^2 - 2x y^4 + y^6, y^3-y)} = \fm_a \cap \fm_b \cap \fm_c.
$$ 
Hint:  interpret both sides geometrically.  

(The following part is not for credit, but the answer will give you
some insight into schemes.)  Can you think of some reason why you
would know that $(x^2-2xy^4 + y^6, y^3 -y)$ is not a radical ideal?


\item (a) If $S$ is a multiplicative subset of $R$ containing 0, show that 
$S^{-1}R$ is the zero-ring (i.e. the ring with one element, which you can
call 0 or 1).

\item[](b)
Let $A$ be a (commutative) ring.  Prove that an element $a \in A$ is nilpotent
(that is, $a^n = 0$ for some $n>0$) if and only if $a$ belongs to 
every prime ideal of $A$.


\item Consider the following two conditions on a ring $R$.
\begin{enumerate}
\item[(i)]  Every ascending chain of ideals of $R$ eventually
stabilizes (i.e. $R$ is Noetherian, by definition).
\item[(ii)]  Every ideal of $R$ is finitely generated.
\end{enumerate}
Prove that $R$ has condition (i) if and only if $R$ has condition (ii).
\end{enumerate}

} % end of parskip; it started just before the introduction
\end{document}

