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\centerline{\bf Oral Qualifying Examination Syllabus}\\
\centerline{\bf Thuy Pham (December, 2003)}
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\centerline{Committee: W. Vasconcelos, C. Weibel, V. Retakh, E. Taft}
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\section{Commutative Algebra}
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\label{Commutative Algebra}
\subsection{Basic Constructions}\label{sec:Localization}
\begin{itemize}
	\item {Localization}
  \item {Associated primes and primary decomposition}
  \item {Integral dependence and the Nullstellensatz}
  \item {Filtrations and the Artin-Rees lemma}
  \item {Completions and Hensel's lemma}
  \item {Valuation rings (DVRs - Dedekind rings - Krull rings)}
\end{itemize}
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\subsection{Dimension Theory}
\label{sec:DimensionTheory}
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\begin{itemize}
	\item {Dimension zero}
	\item {Principal Ideal Theorem and system of parameters}
	\item {Noether Normalization and finiteness of integral closure}
\end{itemize}

\subsection{Hilbert Functions of Graded Modules}
\label{sec:HilbertFunctionsOfGradedModules}
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\begin{itemize}
	\item {Graded rings and modules}
	\item {Hilbert functions over homogeneous rings}
	\item {Macaulay's theorem on Hilbert functions}
	\item {Gotzmann's regularity and persistence theorem}
	\item {Filtered rings}
	\item {The Hilbert-Samuel function and reduction ideals}
	\item {Multiplicity}
\end{itemize}

\subsection{Cohen-Macaulay Rings}
\label{sec:CohenMacaulayRings}
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\begin{itemize}
	\item {Regular sequences}
	\item {Grade and depth of a module}
	\item {Depth and projective dimension}
	\item {Cohen-Macaulay rings and modules}
	\item {Regular rings and normal rings}
	\item {Complete intersection}
\end{itemize}
\goodbreak

\subsection{Canonical Modules}
\label{sec:CanonicalModules}

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\begin{itemize}
	\item {Finite modules of finite injective dimension}
	\item {Injective hulls - Matlis duality}
	\item {The canonical module of a local ring}
	\item {The canonical module of a graded ring}
\end{itemize}

%\pagebreak
\section{Homological algebra}\label{Homological algebra}
\subsection{Category Theory Language}
\label{sec:CategoryTheoryLanguage}
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\begin{itemize}
	\item {Categories}
	\item {Functors}
	\item {Adjoint functors}
	\item {Abelian categories}
\end{itemize}

\subsection{Chain Complexes}
\label{sec:ChainComplexes}
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\begin{itemize}
	\item {Chain homotopies}
	\item {Mapping cones and cylinders}
\end{itemize}

\subsection{Derived Functors}
\label{sec:DerivedFunctors}
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\begin{itemize}
	\item {Projective and Injective resolutions}
	\item {Left and Right derived functors}
	\item {Tor and Ext}
\end{itemize}

\subsection{Homological Dimensions}
\label{sec:HomologicalDimensions}
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\begin{itemize}
	\item {Rings of small dimensions}
	\item {Change of ring theorems}
	\item {Koszul complexes}
	\item {Local cohomology}
\end{itemize}

\subsection{Spectral Sequences}
\label{sec:SpectralSequences}
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\begin{itemize}
	\item {Spectral sequence of a filtration}
	\item {Convergence}
	\item {Spectral sequence of a double complex}
\end{itemize}

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\begin{thebibliography}{99}
	\bibitem{AM}{Atiyah, M.F. and Macdonald, I.G., (1969).
  \textit{Introduction to Commutative Algebra}. Addison-Wesley.}
	\bibitem{BH}{Bruns, W. and Herzog, J. (1993). \textit{Cohen-Macaulay
Rings}.
	Cambridge University Press.}
	\bibitem{E}{Eisenbud, D. (1994). \textit{Commutative Algebra with a
	View Toward Algebraic Geometry}. Springer-Verlag.}
	\bibitem{M}{Matsumura, H. (1980). \textit{Commutative Algebra}.
	Pearson Benjamin Cummings.}
	\bibitem{R}{Rotman, J.J. (1979). 
\textit{An Introduction to Homological Algebra}.
	Academic Press.}
	\bibitem{V}{Vasconcelos, W.V. (1998). \textit{Computational Methods in
	Commutative Algebraic Geometry}. Springer-Verlag.}
\bibitem{W}{Weibel, C. (1994). \textit{An Introduction to Homological
Algebra}.	Cambridge U. Press.}
\end{thebibliography}
\end{document}
