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\begin{document}
%Proposed topics for the Ph. D. Qualifying Oral Examination.
{\par\centering \textbf{\Large 
Ph. D. Qualifying Oral Examination
}\Large \par}
{\centering\textbf{Candidate: Carlo Mazza}\par}

%Commitee: Charles Weibel, Wolmer Vasconcelos and Friedrich Knop.

\medskip{}
{\par\centering \textbf{\Large Homological Algebra.}\Large \par}
\medskip{}

\begin{enumerate}{}
{\setlength{\parsep}{0ex}\setlength{\partopsep}{0ex}
\setlength{\itemsep}{0ex}
\setlength{\topsep}{0ex}}
\item Chain Complexes

\begin{itemize}{}
{\setlength{\parsep}{0ex}\setlength{\itemsep}{0ex}
\setlength{\topsep}{0ex}}
\item Complexes of Modules.
\item Long Exact Sequences.
\item Chain Homotopies.
\item Mapping Cones and Cylinders.
\end{itemize}
\item Derived Functors

\begin{itemize}
\item Derived Functors.
\item Projective and Injective Resolutions.
\item Left and Right Derived Functors
\item Tor and Ext.
\end{itemize}
\item Homological Dimension

\begin{itemize}
\item Dimensions.
\item Local Rings.
\item Koszul Complexes.
\end{itemize}
\item Spectral Sequences

\begin{itemize}
\item Introduction and Terminology.
\item The Leray-Serre Spectral Sequence.
\item Spectral Sequence of a Filtration.
\item Convergence.
\item Spectral Sequences of a Double Complex.
\item Hyperhomology.
\item Grothendieck Spectral Sequences. 
\item Exact Couples.\smallskip{}
\goodbreak

\end{itemize}
\item Simplicial Methods

\begin{itemize}
\item Simplicial Objects.
\item Dold-Kan Correspondence.
\item The Eilenberg-Zilber Theorem.
\end{itemize}
\end{enumerate}
\medskip{}
\goodbreak

{\par\centering \textbf{\Large 
Commutative Algebra \& Algebraic Geometry
}\Large \par}
\smallskip{}

\begin{enumerate}
\item Basic Commutative Algebra

\begin{itemize}
\item Rings and Modules.
\item Rings and Modules of Fractions.
\item Primary Decomposition.
\item Integral Dependence and Valuations.
\item Chain Conditions.
\item Noetherian Rings.
\item Artin Rings.
\item DVR's and Dedekind Domains.
\item Completions.
\item Dimension Theory.
\end{itemize}
\item Varieties

\begin{itemize}
\item Affine and Projective Varieties.
\item Morphisms.
\item Rational Maps.
\end{itemize}
\item Schemes

\begin{itemize}
\item Sheaves.
\item Schemes.
\item First Properties of Schemes.
\item Separated and Proper Morphisms.
\item Divisors.
\item Projective Morphisms.
\end{itemize}
\goodbreak
\item Cohomology

\begin{itemize}
\item Derived Functors.
\item Cohomology of Sheaves.
\item Cohomology of a Noetherian Affine Scheme.
\item Cech Cohomology.
\item The Cohomology of Projective Space.
\item Ext Groups and Sheaves.
\item The Serre Duality Theorem.
\item Higher Direct Images of Sheaves.
\end{itemize}
\end{enumerate}
\bigskip{}%\vspace{10cm}
{\par\centering \textbf{\Large References}\Large \par}
\bigskip{}

C. A. Weibel, \emph{An Introduction to Homological Algebra}, \emph{}Cambridge
University Press, 1994.
\smallskip{}

M. F. Atiyah and I. G. MacDonald, \emph{Introduction to Commutative Algebra},
\emph{}Addison-Welsey, 1969.

\smallskip{}
R. Hartshorne, \emph{Algebraic Geometry}, Springer-Verlag, 1977.

\end{document}
