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\begin{center}\begin{large}\textbf{Math 524 - Problem Set 2\\
Due Wednesday, Feb 23}\end{large}\end{center}



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Let $X$ be a topological space.

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\item Let $\FF$ and $\GG$ be sheaves on $X$. Define the \textit{\textbf{direct sum}} of $\FF$ and $\GG$, denoted $\FF \oplus \GG$, to be the presheaf which takes $U$ to $\FF(U) \oplus \GG(U)$. Show that $\FF \oplus \GG$ is a sheaf.


\item Let $k$ be a field, and $\mathcal{F}$ the sheaf of locally constant $k$-valued functions. Let $P,Q \in X$, and define $\GG = k^{(P)} \oplus k^{(Q)}$, where $k^{(P)}$ and $k^{(Q)}$ are skyscraper sheaves with stalk $k$ at $P$ and $Q$, respectively. Define $\phi: \mathcal{F} \to \mathcal{G}$ by $\phi(U)(f) = (f(P), f(Q))$, where $f(P)=0$ if $P\notin U$.

\begin{enumerate}
\item Verify that $\phi$ is a morphism of sheaves.
\item Show that $\phi$ is surjective.
\item Show that the map on sections $\phi(U)$ is not always surjective.
\item Describe the kernel of $\phi$.
\end{enumerate}


\item Let $\FF'$ be a subsheaf of a sheaf $\FF$. Show that the natural map $\FF \to \FF/\FF'$ is surjective and has kernel $\FF'.$\footnote{You can use the facts from problem II1.2 in Hartshorne that surjectivity and injectivity can be checked on stalks.} Thus, there is an exact sequence $$0 \to \FF' \to \FF \to \FF/\FF'\to 0.$$

\item Let $\FF$ be a sheaf on $X$, and $s \in \FF(U)$ a section over an open set $U$. The \textit{\textbf{support}} of $s$, denoted $\Supp s$ is defined to be $\{P \in U | s_P \neq 0\}$, where $s_P$ is the germ of $s$ in the stalk $\FF_P$. Show that $\Supp s$ is a closed subset of $U$.



\item Let $R$ be a ring. Recall that we defined $$\str_{\Spec R}(U) = \varprojlim_{D(b) \subseteq U} R_b.$$
\begin{enumerate}
\item Let $a \in R$. Verify the claim that $\str(D(a)) = R_a$. (In particular, $\str(\Spec R) = \str(D(1)) = R$.)
\item Prove that $\str$ as defined above is actually a sheaf.\footnote{You can use the fact we discussed in class that it suffices to show that it satisfies the presheaf and sheaf axioms on a basis.}
\end{enumerate}

\item Let $R = \C[x_1, \ldots, x_n]$ and let $U$ be the complement of the origin in $\mathbb{A}^n_{\C}$. That is, $U = \Spec R \setminus \{(x_1, x_2, \ldots, x_n)\}.$ What is $\str(U)$?\footnote{Hint: Remember that we gave a nice description of $\str(U)$ when $R$ is an integral domain.}  Note that your answer may depend on $n$.

\item Recall that a scheme $X$ is \textit{\textbf{reduced}} if $\str(U)$ has no nilpotents for each open $U \subset X$. Show that $X$ is reduced if and only if for every $P\in X$ the stalk $\str_P$ has no nilpotents.\footnote{Hint: First show that you can reduce to the affine case.}

\item Let $A$ and $B$ be rings, $\varphi : A \to B$ a ring homomorphism, and set $X = \Spec A,$ $Y=\Spec B$. Let $f:Y \to X$ be the morphism induced by $\varphi$.
\begin{enumerate}
\item Show that $f \in A$ is nilpotent if and only if $D(f) \subseteq X$ is empty.
\item Show that $\varphi$ is injective if and only if $f^\# : \str_X \to f_*\str_Y$ is injective. Show that in this case $f$ is dominant (i.e. $f(Y)$ is dense in $X$).
\item Show that $\varphi$ is surjective if and only if $f$ is a homeomorphism of $Y$ onto a closed subset of $X$ and $f^\#$ is surjective.
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