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\begin{center}\begin{large}\textbf{Math 524 - Problem Set 3\\
Due Wednesday, March 16}\end{large}\end{center}



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\begin{enumerate}
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\item Let $k$ be a field and $X$ a scheme. Show that if a morphism $f: X \to \Spec k$ is finite, then $X$ consists of finitely many points. 


\item Let $A \to B$ be a morphism of rings. Recall that we defined $\P^n_A = \Proj A[x_0, \ldots, x_n]$. Show that we have the following isomorphism (which I assumed without proof in class): $$\P^n_B \cong \P^n_A \times_{\Spec A} \Spec B.$$

\item Let $X$ be a scheme. 
\begin{enumerate}
\item Show that there is a unique morphism $X \to \Spec \Z$. 
\item Show there is a unique morphism from the zero ring to $X$.\footnote{In the zero ring, $0=1$.}
\end{enumerate}

\item Let $X$ be an integral scheme and $Z \in X$ the generic point
\begin{enumerate}
\item Show that the local ring $\str_Z$ is a field. This is called the \textit{\textbf{function field}} of $X$ and is denoted $K(X)$. 
\item Show that if $U = \Spec A$ is any open affine in $X$, then $K(X)$ is isomorphic to the field of fractions of $A$.
\end{enumerate}

\item Choose (at least) two of the following facts about morphisms of finite type to prove.
\begin{enumerate}
\item Closed immersions are of finite type.
\item Quasi-compact\footnote{$f:X\to Y$ is \textbf{\textit{quasi-compact}} if there is a cover of $Y$ by open affines so that their preimages in $X$ are quasi-compact. Equivalently, if the preimage of every open affine is quasi-compact.}  open immersions are of finite type.
\item A composition of two morphisms of finite type is of finite type.
\item Morphisms of finite type are stable under base extension.

\end{enumerate}

\item Let $X$ and $Y$ be schemes over an algebraically closed field $k$. Show that there is a one-to-one correspondence between closed points of $X\times_k Y$ and pairs $(x,y)$ where $x\in X$ and $y \in Y$ are closed points.

\item Show that finite morphisms are proper.

\item Let $X$ be separated over an affine scheme $S$. Show that the intersection of any two affine open subsets of $X$ is affine. (Challenge/optional: Show that this fails if $X$ is not separated.)



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