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\title[CAAG Course Syllabus]{A course on commutative algebra for
algebraic geometers}
\author{Kapil Hari Paranjape}
\maketitle

\section*{About the course}
The aim is to introduce the fundamentals of commutative algebra as
applicable in other areas; notably algebraic geometry and algebraic
number theory.

\subsection*{Books}\label{books}
Auditors are encouraged to examine the following books:
\begin{itemize}
        \item Atiyah and McDonald: Commutative Algebra
        \item David Eisenbud: Commutative Algebra
        \item J. P. Serre: Algebre Locale (Local Algebra)
        \item O. Zariski and P. Samuel: Commutative Algebra
\end{itemize}

\subsection*{Topics}
The following topics are to be lectured; approximately one topic per
week.
\begin{enumerate}
   \item Finite rings and rings that are finite dimensional over a field.
   Artinian Rings. Hensel's lemma.

   \item Polynomial rings and their ideals. Hilbert's basis theorem and
   Nullstensatz. Notherian Rings and Modules.

   \item Associated primes, minimal primes and primary
   decomposition.

   \item Integral extensions. Noether normalisation. Proof of Nullstellensatz.
   Dimension and transcendence degree. 

   \item Localisation, gradation and Completion. Revisit earlier theorems
   for these rings.

   \item Filtrations. Artin-Rees Lemma. Krull Intersection Theorem.

   \item Tangent vectors and derivations. Tensors and Tensor Products.
   Module of derivations. Differentials.

   \item Categorical constructions. Tensor and Hom and their exact-ness
   properties. Basic homological algebra.
   
   \item Flatness and fibre-bundles. Generic flat-ness. Elmination
   theory.

   \item Dimension defined in different ways. Hilbert-Samuel polynomial.
    Height of ideals, number of generators. Krull's principal ideal
    theorem.

    \item The Koszul complex and minimal resolutions. Depth vs.
    dimension.
    
    \item Regular local rings. Smoothness. Etale-ness. Hensel's lemma
    revisited.

    \item Connections with complex analysis and number theory. Counting
    points. The statement of Weil's conjectures.
\end{enumerate}


\subsection*{Grading}
There will be one assignment per topic. This will carry 5 marks. The
end-semester examination will carry 35 marks.

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