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\begin{document}
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\begin{center}

  \huge{The Department of Mathematics}\\[0.1\baselineskip]
  \Large{2021--22--A term}\\[0.2\baselineskip]

\end{center}

\begin{description}
  \item[Course Name]
    Homological Algebra

  \item[Course Number]
    \LRE{201‭.2‭.2091}

  \item[Course web page]\mbox{}\\
    \url{https://www.math.bgu.ac.il//~amyekut/teaching/2021-22/homol-alg/course_page.html}

  \item[Lecturer]
    Prof. Amnon Yekutieli,
    \nolinkurl{<amyekut@bgu.ac.il>},
    Office 202

\item[Office Hours]
  \url{https://www.math.bgu.ac.il/en/teaching/hours}
\end{description}

\section*{Abstract}

Course Topics: (as much as time permits)

\begin{enumerate}
\item{} Review of prior material. On rings, ideals and modules (including noncommutative rings).
\item{} Categories and functors. Emphasis on linear categories. (This topic will be introduced gradually, as we go along.)
\item{} Universal constructions. Free modules, products, direct sums, polynomial rings.
\item{} Tensor products. Definition, construction and properties.
\item{} Exactness. Exact sequences and functors.
\item{} Special modules. Projective, injective and flat modules.
\item{} Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.
\item{} Resolutions. Projective, flat and injective resolutions.
\item{} Left and right derived functors. Applications to commutative algebra.
\item{} Further applications of derived functors. Classification problems, extensions.
\item{} Morita Theory.
\end{enumerate}

(Some of the material might move to the subsequent course ``Commutative Algebra'')

For an updated syllabus and course requirements see the \href{https://www.math.bgu.ac.il/~amyekut/teaching/2021-22/homol-alg/course_page.html}{course web page}


\section*{Requirements and grading\footnote{Information may change during the first two weeks of the term. Please consult the webpage for updates}}

see 
\href{https://www.math.bgu.ac.il/~amyekut/teaching/2021-22/homol-alg/course_page.html}{course web page}


    \includepdf[pages=-]{syll-homol-alg-2021-22.pdf}



\section*{Course topics}

\begin{enumerate}
\item{} Recalling prior material. Rings (including noncommutative), ideals, modules and bimodules, exact sequences, infinite direct sums and products, tensor products of modules and rings.
\item{} Categories and functors. Morphisms of functors, equivalences. Linear categories and linear functors. Exactness of functors.
\item{} Special modules. Projective, injective and flat modules.
\item{} Morita Theory. Equivalences of module categories realized as tensor products.
\item{} Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.
\item{} Resolutions. Projective, injective and flat resolutions – existence and uniqueness.
\item{} Left and right derived functors. The general theory. Tor and Ext functors.
\item{} Applications to commutative algebra. Some local and global theorems, involving $Tor$ and $Ext$ functors. Derived completion and torsion functors.
\item{} Sheaf cohomology. A survey of the role of homological algebra in geometry.
\item{} Nonabelian cohomology. A survey of classification theorems: Galois cohomology, vector bundles.
\end{enumerate}

\paragraph*{Bibliography}

\begin{enumerate}
\item{} R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New-York, 1977.
\item{} P.J. Hilton and U. Stammbach, “A Course in Homological Algebra”, Springer, 1971.
\item{} S. Maclane, “Homology”, Springer, 1994.
\item{} J. Rotman, “An Introduction to Homological Algebra”, Academic Press, 1979.
\item{} L.R. Rowen, “Ring Theory” (Student Edition), Academic Press, 1991.
\item{} C. Weibel, “An introduction to homological algebra”, Cambridge Univ. Press, 1994.
\item{} M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
\item{} \href{http://stacks.math.columbia.edu}{The Stacks Project}, an online reference, J.A. de Jong (Editor).
(9) A. Yekutieli, “Derived Categories”, Cambridge Univ. Press, 2019. \href{https://arxiv.org/abs/1610.09640v4}{Free prepublication version}.
(10) Course notes, to be uploaded every week to the course web page
\end{enumerate}

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