COURSE AIMS AND OBJECTIVES:
To familiarize students with the categorial methods of homological algebra on examples of working with modules over rings. Considered are the basic notions in homological algebra, like projective and injective resolutions, universal constructions, adjoint functors and derived functors.
COURSE DESCRIPTION AND SYLLABUS:
1. Rings and modules. Free and projective modules.
2. Projective modules over principal domain rings.
3. Dualization and injective modules. Injective modules over principal ideal domains. Cofree modules.
4. Categories and functors. Examples: Hom functor and tensor product of modules.
5. Natural transformation of a functor. Isomorphims and equivalence of categories.
6. Products and coproducts. Universal objects and universal constructions.
7. Adjoint functors. Various examples including reinterpretations of universal construction.
8. Abelian categories. Projective, injective and free objects in Abelian categories.
9. Complexes of modules. Morphisms of complexes. Exact sequencs. Homology and cohomology.
10. Homotopy. Motivation and examples from algebraic topology.
11. Resolution. Derived functors.
12. Ext i Tor functors are derived functors.
13. Computations of Ext i Tor functors. Some applications.
TEACHING AND ASSESSMENT METHODS:
Attending of at least 70% of lectures and examples classes, sloving at least 70% of homework assignments, and passing grade on two mid-term exams.
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