Homological Algebra (Algebra Omologica)

Syllabus, Homological Algebra, A.A. 2020/21 (F4Y-25) 


Homological Algebra is a tool used in many branches of mathematics, especially in Algebra, Topology and Algebraic Geometry, and can be considered as an algebraic counterpart to the idea of homotopy on topological spaces. 
As a matter of fact, homological algebra gives the appropriate language to study algebraic invariants attached to “spaces” up to homotopies. Examples: Betti homology, de Rham cohomology, the Hodge filtration… 

In homological algebra one also studies the following type of general problem: given a non-exact functor (for example, the tensor product functor of modules over a ring, or the global section functor of a sheaf on a topological space, invariants of a group action), how can we measure its non-exactness?
In this course we will introduce some of the basic tools of homological algebra (such as abelian categories, simplicial sets, resolutions, derived functors, spectral sequences), with the specific aim to introduce and study Cech cohomology, sheaf cohomology, and de Rham Cohomology.


Prerequisites: This course is in the second semester. Some basic knowledge of category theory is necessary, together with some algebraic topology (e.g. Geometria 5). It is recommended to take this class after Commutative Algebra in the first semester.


Evaluations and grading: Homework will be assigned during the course, and solutions have to be submitted by the end of the course. The final assessment will be given by an oral exam (typically, a seminar on a subject picked from a list of themes suggested by us).

References

C. Weibel: An introduction to Homological Algebra Cambridge Studies in Advanced Mathematics Vol. 38, 1995. Find errata corrige and pdf file here.

More references: