The goals of this course are to learn basic tools, theorems, and calculations in homological algebra. A second goal is to refine participants’ skills at reading, writing, and presenting mathematics in a clear, careful, formal, and professional manner.

Mathematical maturity, some abstract algebra

Students will be evaluated via the following three tools, each of which will count for one third of the final
evaluation.
• weekly homework: problem sets combining conceptual exercises and explicit computations to help promote mastery of key tools and ideas. To encourage the development of mathematical prose skills, these homework sets should be written out formally and typeset using TeX or a similar tool.
• in-class presentations: one or more lecture-style presentation(s) of proofs, applications, computations, or reading to encourage deep engagement with the course material and development of a clear and organized mathematical presentation style.
• survey article: a longer piece of mathematical prose that presents a theorem or application. The topic should be directly related to the main focus of the course but not explicitly covered during classtime. This should include all important background material, properly referenced, and be professionally organized and typeset. No original research will be required or expected.
More detailed rubrics for these evaluative instruments will be distributed near the beginning of the semester.

• C. Weibel, An introduction to homological algebra, Cambridge
An Introduction to Homological Algebra (2/E), J. Rotman, Springer
• The nlab: https://ncatlab.org/nlab

Categories, Functors, Natural transformations of functors, Functor category, Limits and col- imits, Adjoint functors, Monads/Comonads, Universal morphisms, Yoneda lemma and representable functors, Yoneda embeddings, Ends and coends, Left and right Kan extension. Monoidal categories, Enriched categories, 2-category machine, Abelian categories, Category of modules over a ring, Chain and cochain complexes over abelian categories, Triangulated categories, Homology and cohomology functors, Localizations of triangulated category, Verdier quotient, tensor triangulated categories, Projective and injective resolutions. Ext and Tor, Homotopy category of chain complexes, Gabriel-Zysman localizations and, Derived category of abelian category, Derived functors, Sheaf cohomologies, Closed symmetric monoidal structure of derived category of quasi-coherent sheaves, , t-structures in triangulated category, Yoga of Grothendieck’s six functors, etc.

Classtime will be used for the following purposes:
• traditional lecture
• presentations by students • discussion and questions

In order to get the most out of class, you will all have to:
• do assigned reading in advance, so that we can isolate potential areas of difficulty and use class time effectively,
• complete homework assignments on time in order to verify and maintain your level of comprehension,
• engage actively with both reading and homework so that you are ready with questions and answers
during class time,
• carefully and diligently prepare for in-class presentations so that other students can benefit from
your presentations,
• actively provide useful critical feedback to your fellow students about their presentations so that
they can improve their presentation styles, and
• begin work on your survey article well in advance in order to have enough time to read, understand,
write, and revise.
This class aims to be an inclusive environment. Harassment will not be tolerated.