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Proposed topics for the Ph. D. Qualifying Oral Examination

Candidate : Jooyoun Hong

Committee : Barbara Osofsky, Charles Weibel and Wolmer Vasconcelos




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{\bf \Large Commutative Algebra}
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{\bf \large 1. Basic Commutative Algebra}

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1.1 \; Localization of rings and modules

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1.2 \; The Hilbert Nullstellensatz

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1.3 \; Associated primes and primary decomposition

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1.4 \; Integral dependence and valuations

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1.5 \; DVRs, Dedekind rings, and Krull rings

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1.6 \; Chain Conditions

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1.7 \; Noetherian and Artinian rings

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1.8 \; Completions

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{\bf \large 2. Hilbert Functions of Graded Modules}

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2.1 \; Graded rings and modules

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2.2 \; Hilbert functions over homogeneous rings

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2.3 \; Macaulay's theorem on Hilbert functions

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2.4 \; Gotzmann's regularity and persistence theorem

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2.5 \; Hilbert functions over graded rings

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2.6 \; Filtered rings

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2.7 \; The Hilbert-Samuel function and reduction ideals

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2.8 \; Multiplicity

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{\bf \large 3. Cohen-Macaulay Rings}

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3.1 \; Regular sequences

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3.2 \; Grade and depth

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3.3 \; Depth and projective dimension

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3.4 \; Cohen-Macaulay rings and modules

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3.5 \; Regular rings and normal rings

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3.6 \; Complete intersections

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{\bf \Large Homological Algebra and Gr\"{o}bner Bases}
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{\bf \large 1. Homological Algebra}

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1.1 \; Chain complexes

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1.2 \; Chain homotopies

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1.3 \; Mapping cones and cylinders

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1.4 \; Projective and injective resolutions

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1.5 \; Left and right derived functors

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1.6 \; Adjoint functors and left/right exactness

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1.7 \; Tor and Ext

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1.8 \; Dimensions

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1.9 \; Rings of small dimensions

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1.10 \; Change of ring theorems

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1.11 \; Koszul complexes

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1.12 \; Local cohomology

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{\bf \large 2. Gr\"{o}bner Bases}

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2.1 \; Orderings on the monomials in $k[x_1 , \cdots , x_n ]$

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2.2 \; Division algorithm in $k[x_1 , \cdots , x_n]$

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2.3 \; Monomial ideals and Dickson's lemma

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2.4 \; Hilbert basis theorem and Gr\"{o}bner bases

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2.5 \; Properties of Gr\"{o}bner bases

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2.6 \; Buchberger Algorithm

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2.7 \; Applications of Gr\"{o}bner bases

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\begin{thebibliography}{99}

\bibitem{AM}{ M.F. Atiyah and I.G. Macdonald, {\it Introduction to Commutative Algebra}, Addison-Wesley, 1969}

\bibitem{BH}{W. Bruns and J. Herzog, {\it Cohen-Macaulay Rings}, Cambridge University Press, 1993}

\bibitem{M}{H. Matsumura, {\it Commutative Ring Theory}, Cambridge University Press, 1986}

\bibitem{AL}{W.W. Adams and P. Loustaunau, {\it An Introduction to Gr\"{o}bner Bases}, AMS, 1994}

\bibitem{CLO}{D. Cox, J. Little, and D. O'Shea, {\it Ideals, Varieties and Algorithms }, Springer-Verlag, 2nd edition, 1997}

\bibitem{R}{J.J. Rotman, {\it An Introduction to Homological Algebra}, Academic Press, 1979}

\bibitem{V}{W.V. Vasconcelos, {\it Computational Methods in Commutative Algebra and Algebraic Geometry}, Springer-Verlag, 1998 }

\bibitem{W}{C.A. Weibel, {\it An Introduction to Homological Algebra}, Cambridge University Press, 1994}

\end{thebibliography}
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