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{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2016}

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\centerline{ Problem Set 7 (Last revised \number\month/\number\day/\number\year)}
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Let $K$ be an algebraically closed field unless otherwise noted.
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\item{1.}{We have seen that the regular functions on a {\it distinguished}
open $U$ in an algebraic variety $X$ can always be written as the ratio 
of polynomials $g/h$ where $h$ does not vanish on $U$.

\itemitem{a)}{Suppose that $X$ is an irreducible affine variety over an
  algebraically closed field $K$ and that the affine coordinate ring
  $A(X)$ is a unique factorization domain.  Show that for an arbitrary
  open set 
  $U \subset X$ a regular function on $U$ can always be written
  as the ratio of polynomials $g/h$ where $h$ does not vanish on $U$.}
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\itemitem{b)}{Use part a) to redo problem 3 in problem set 4.}
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\itemitem{c)}{Let $X$ be the cone $V(xz-yw) \subset \bA^4$.  Let
 $U=D(z) \cup D(w)$.  Show that the functions $f_w=x/w:D(w)\rightarrow
  K$ and $f_z=y/z:D(z)\rightarrow K$ are the restrictions of a regular
  function $f$ on $U$.  Prove that there is no polynomials $g,h$ such
that $h$ does not vanish on $U$ and $f(u)=g(u)/h(u)$ for  $u \in U$. 

  


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\item{2.}{Find explicit equations that describe the affine algebraic
group $PGL(2,K)$ as an affine variety in some affine space.}

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\item{3.}{Use geometry to find the radical of the ideal generated by 
$y^2-x^3,x^2+y$ in $K[x,y]$.}

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\item{4.}{Suppose that $Y$ is a topological space with a sheaf
  ${\cal F}_Y$ of
K-valued  functions.  Suppose that $U_i \subset Y$ is an open cover.
Suppose that $\phi:X \rightarrow Y$ is a continuous map from a space
$X$
with sheaf ${\cal F}_X$ of K-valued functions .  Show that
$\phi$ is a morphism if and only if it restricts to give a morphism 
$\phi_i$ of the spaces
$\phi^{-1}U_i$ with sheaf ${\cal F}_X|_{\phi^{-1}(U_i)}$ to $U_i$.  
Show that $\phi$ is an
isomorphism if and only if each $\phi_i$ is.}
 

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\item{5.}{(Harris 3.2)}

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\item{6.}{(Harris 3.8)}




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\bye



