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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#6\hspace{220pt}Due: 22 Oct 2021}
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\subsection*{Description}

Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as a pdf file via \href{https://www.gradescope.com/courses/281407}{Gradescope} by midnight on the due date.
Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty.

Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing \textbf{Sources consulted:\ none} at the top of your submission.

\medskip

\noindent
\begin{enumerate}
	\item Prove this alternate form of the Nullstellensatz: Let $k$ be a field of characteristic zero, and let $B$ be a domain that is finitely generated $k$-algebra. If $B$ is a field, then $[B:k] < \infty$.

	\item Let $A$ be a finite type domain over $\C$, $R = \C[t]$, $X = \Spec(A)$, and $Y = \Spec R$. Let $\varphi^{*}: A \to R$ be a homomorphism whose image is not a subset of $\C$, and let $\varphi: Y \to X$ be the corresponding morphism.
		Show that $R$ is a finite $A$-module, and that the image $\varphi$ is a closed subset of $X$.

	\item Let $\alpha$ be an element of a domain $A$, and set $\beta=\alpha^{-1}$. Prove that if $\beta$ is integral over $A$, the it is an element of $A$.

	\item Let $k$ be a field of characteristic zero. Let $A \subset B$ be finite type domains over $k$ with fraction fields $K \subset L$ of characteristic zero, and let $Y \to X$ be the corresponding morphism of affine varieties. Prove the following:
\begin{enumerate}
	\item There is a nonzero element $s$ in $A$ such $A_s$ is integrally closed.
	\item There is a nonzero element $s$ in $A$ such that $B_s$ is a finite $A$-module over a polynomial ring $A_s[y_1, \ldots, y_d]$.
	\item Supposed that $L$ is a finite extension of $K$ of degree $d$. There is a nonzero element $s \in A$ such that all fibres of the morphism $Y_s \to X_s$ consist of $d$ points.
\end{enumerate}
	\item Identify the normalization of $A= \C[x,y,z]/(y^2 - x z^2)$.

\end{enumerate}


\end{document}
