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\begin{center}\begin{large}\textbf{Math 524 - Problem Set 4\\
Due Wednesday, April 6}\end{large}\end{center}



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\begin{enumerate}




\item Let $\FF$ and $\GG$ be sheaves of abelian groups on a scheme $X$. 
\begin{enumerate}
\item For any open $U \subseteq X$, show that the set $\Hom(\FF|_U, \GG|_U)$ of morphisms of the restricted sheaves has the natural structure of an abelian group.
\item Show that the presheaf $U \mapsto \Hom(\FF|_U, \GG|_U)$ is a sheaf. It's denoted $\mathscr{H}\hspace{-.1cm}om(\FF, \GG)$, and is called ``sheaf hom."
\end{enumerate}

\item Let $X$ be a scheme and $\EE$ a locally free sheaf of finite rank $n$. Define the \textit{\textbf{dual}} of $\EE$, denoted $\check{\EE}$, to be the sheaf $\shom(\EE, \str_X)$
\begin{enumerate}
\item Show that the stalk $\EE_x$ at each point $x \in X$ is a free $\str_x$-module of rank $n$. 
\item Describe the natural morphism $f: \EE \to (\check{\EE})\check{ }$ by defining it on open sets. Make sure that your maps are compatible with restriction.
\item Show that $f$ is an isomorphism.

\end{enumerate}

\item Let $X$ be a Noetherian scheme and $\FF$ a coherent sheaf.
\begin{enumerate}
\item Suppose the stalk $\FF_x$ is a free $\str_x$-module for some point $x \in X$. Show that there is a neighborhood $U$ of $x$ such that $\FF|_U$ is free.
\item Show that  the converse of 2(a) holds. That is, $\FF$ is locally free if and only if $\FF_x$ is a free $\str_x$-module for every $x \in X$.
\item If $\FF$ is invertible (locally free of rank one), show that $\FF \otimes \check{\FF} \cong \str_X$.\footnote{Hint: Give a map on each open set, and show that the maps are compatible with restriction so that it defines a morphism. Then show that the morphism induces an isomorphism on stalks.}
\item Challenge/optional (requires additional knowledge about modules over local rings): Show that $\FF$ is invertible if and only if there is a coherent sheaf $\GG$ such that $\FF \otimes \GG \cong \str_X$.
\end{enumerate}

\item Let $f:X \to Y$ be a morphism of schemes and $\FF$ a coherent sheaf on $X$. Recall that we gave an example in class showing that $f_*\FF$ is not necessarily coherent, even if $X$ and $Y$ are both affine. Show that if $X$ and $Y$ are Noetherian and $f$ is finite, then $f_*\FF$ is coherent.

\item Let $X$ be a Noetherian scheme and $\FF$ a coherent sheaf on $X$. Recall that the residue field at a point $x \in X$ is $k(x) = \str_x/m_x$. Define a function $$\varphi(x) = \dim_{k(x)} (\FF_x \otimes_{\str_x} k(x)).$$ Use Nakayama's Lemma to prove the following results about $\varphi$.
\begin{enumerate}
\item $\varphi$ is upper semi-continuous. That is, for any integer $n$, the set $$\{ x \in X \mid \varphi(x) \geq n\}$$ is closed.\footnote{Hint: Reduce to the affine case so that $\FF \cong \tilde{M}$ and find a $k(x)$-basis at a point $x$. Show that there is some distinguished open $D(f)$ such that the same basis (after lifting to $M$) generates the $k(y)$-vector space at each $y \in D(f)$.}
\item If $\FF$ is locally free and $X$ connected, then $\varphi$ is constant.
\item If $X$ is reduced and $\varphi$ is constant, then $\FF$ is locally free.
\end{enumerate}

\end{enumerate}

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