Homological Algebra
A.Y. 2024/2025
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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
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.
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
Lectures.
Teaching Resources
1) "An Introduction to Homological Algebra" by Charles A. Weibel;
2) "Categories and Sheaves" by Masaki Kashiwara and Pierre Shapira;
3) "Introduction to Étale Cohomology" by Günter Tamme.
2) "Categories and Sheaves" by Masaki Kashiwara and Pierre Shapira;
3) "Introduction to Étale Cohomology" by Günter Tamme.
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 consisting in:
- a discussion on the solutions of the homework
- questions on the contents and the proofs presented in the course
- a brief presentation on a topic correlated to the contents of the course, to be selected out of a list
The final grade will be given on a scale up to 30/30.
- a discussion on the solutions of the homework
- questions on the contents and the proofs presented in the course
- a brief presentation on a topic correlated to the contents of the course, to be selected out of a list
The final grade will be given on a scale up to 30/30.
Professor(s)