Lecturer: Christian Böhning

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Timetable:

Prerequisites: A thorough command of basic techniques and concepts in linear algebra and algebra, including groups, rings, modules, fields etc. Some familiarity with algebraic topology and algebraic geometry is not required, but may be useful to get more out of the course.

Content: The module will provide an introduction to modern techniques in homological algebra, in particular the language of derived and triangulated categories and some of their applications in neighbouring fields such as algebraic geometry, representation theory and topology (depending on the interests and background of the audience).

Possible topics comprise:

-Some basic notions from category theory, including additive and abelian categories and functors in abelian categories
-Simplicial sets
-Localisation and derived category of an abelian category
-Derived functors
-Spectral sequences and derived functor of a composition
-Triangulated categories and exact functors
-Exceptional sequences and semiorthogonal decompositions
-Cores and t-structures
-Some view towards dg- categories and A-infinity categories

We will also make every attempt to bring this material to life with lots of examples and applications in other fields of mathematics. Which areas these will mainly be taken from (algebraic geometry/coherent sheaves on varieties, representation theory, topology), will again depend on the background of the participants.

References:

-S.I. Gelfand, Y.I. Manin: Methods of Homological Algebra, 2nd ed., Springer Monographs in Math. (2003)
-D. Huybrechts: Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monographs (2006)
-A. Yekutieli, Derived categories, Cambridge studies in advanced math. 183, (2020)
-P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Math. Society (2008)