Homological Algebra
A.Y. 2020/2021
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 (In case of multiple editions, please check the period, as it may vary)
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
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.
1) 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...
Remark: the complexes (and not just their homotopy groups) are useful!
2) 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.
1) 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...
Remark: the complexes (and not just their homotopy groups) are useful!
2) 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 for admission
We assume known the basic notions from undergraduate algebra & topology. The students should be familiar with the language of categories & functors.
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).
Professor(s)