Homological Algebra

A.Y. 2021/2022
6
Max ECTS
42
Overall hours
SSD
MAT/02
Language
Italian
Learning objectives
The task of this course is to give an introduction to the main tools of homological algebra.
Expected learning outcomes
Ability to make computations using derived functors and spectral sequences in various contexts of algebra and geometry.
Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
Remote classes using the Zoom platform. Teaching materials will be available for students on ARIEL.
Course syllabus
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.
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, resolutions, derived functors, spectral sequences.
Prerequisites for admission
We assume known the basic notions from undergraduate algebra & topology. The students should be familiar with the language of categories & functors.
Teaching methods
Traditional lectures
Teaching Resources
C. Weibel: An introduction to Homological Algebra Cambridge Studies in Advanced Mathematics Vol. 38, 1995.
Assessment methods and Criteria
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). The final grade will be given on a scale up to 30/30.
MAT/02 - ALGEBRA - University credits: 6
Lessons: 42 hours