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\begin{center}\begin{large}\textbf{Math 524 - Problem Set 1\\
Due Wednesday, Feb 2}\end{large}\end{center}



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\begin{enumerate}
%\addtocounter{enumi}{-1}

\item Describe the map $f: \Spec \C[x] \to \Spec \R[x]$ induced by the inclusion $\R[x] \hookrightarrow \C[x]$.

\item For each ring $A$ and point $q \in \Spec A,$ calculate the dimension of the tangent space to $\Spec A$ at $q$.
\begin{enumerate}
\item $A = \C[x,y,z]/(x^2 - y^3), q =(x,y),$
\item $A = \C[x,y]/(x,y^3), q = (x,y),$
\item $A = \Z[x], q = (3).$
\end{enumerate}

\item 
Let $R, S$ be rings.
\begin{enumerate}
\item Check that the Zariski topology on $\Spec R$ is indeed a topology and that the distinguished open sets $$D(f):= \Spec R \backslash V(f) $$ form a basis.
\item If $\phi: R \to S$ is a homomorphism, verify that the induced map $\Spec S \to \Spec R$ is continuous.
\end{enumerate}

\item Let $R$ be a ring and $I \subset R$ an ideal. Show that the natural map $f: \Spec(R/I) \to \Spec(R)$ is a homeomorphism onto its image $V(I)$.\footnote{The closed sets of $X \subseteq \Spec(R)$ are those of the form $V(J) \cap X$ for $J$ an ideal in $R$.}

\item Let $R$ be a ring. Show that a nonzero element $f \in R$ is a zerodivisor if and only if there are closed sets $X, Y \subseteq \Spec R$ with $Y \neq \Spec R$ such that $\Spec R = X \cup Y$ and $f$ evaluated\footnote{Via the natural evaluation map $A \to k(x)$ described in class.} at $x$ is zero for all $x \in X$.

\item
\begin{enumerate} 
\item For each open set $U\subseteq \R$ (given the standard topology), define $\mathcal{F}(U)$ to be the group of continuous functions $f:U \to \R$ that are nondecreasing. That is, for $x, y \in U$ with $x<y$, we require $f(x) \leq f(y)$. Is $\mathcal{F}$ a presheaf? Is it a sheaf?
\item What if you replace the word ``nondecreasing" with ``even"? That is, if $x, -x \in U$, then $f(x) = f(-x)$. In this case is $\mathcal{F}$ a (pre)sheaf?
\end{enumerate}

\item Let $X$ be a topological space and $\FF$ and $\GG$ presheaves on $X$. Let $\phi: \FF \to \GG$ be a morphism. Verify the following claim from class: there is an induced map on stalks $\phi_P: \FF_P \to \GG_P$ for each $P \in X$. Make sure you explicitly describe the map and check that it is well-defined.

\item Let $X$ be a topological space. Let $\mathcal{F}$ be the constant presheaf, defined $\mathcal{F}(U) = \Z$ for every nonempty $U\subseteq X $ open, where the restriction maps are the identity, and let $\mathcal{F'}$ be the sheaf of locally constant functions described in class.

\begin{enumerate}
\item Check that $\mathcal{F'}$ is indeed a sheaf.

\item Show that $\mathcal{F'}$ is the sheafification\footnote{We will get to this in a couple weeks -- see Prop-def 1.2 in Hartshorne for a definition.} of $\mathcal{F}$.

\end{enumerate}

\end{enumerate}

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