Описание:Семестровый курс по гомологической алгебре, читаемый в рамках магистерской программы "Geometry and Quantum Fields" (2-й год) в Институте Теоретической и Математической Физики, МГУ.
Annotation:
A semester course introducing the basic constructions and techniques of homological algebra used in algebraic topology, algebraic geometry, and forming the basis of a number of geometric methods in mathematical physics.
The topics covered include chain complexes and differential graded al- gebras, quasi-isomorphisms, projective and injective modules, resolu- tions, homological dimension, Tor and Ext functors, regular sequences and Cohen–Macaulay rings, bicomplexes and filtered complexes, spec- tral sequences, A∞-morphisms.
Prerequisites:
a basic course in algebra (groups, rings, modules, vector spaces), basic concepts of topology (continuous maps, homotopy).
Course plan:
1. Algebras and modules, chain complexes, differential graded algebras, homology.
2. Chain homotopies, quasi-isomorphisms, long exact sequences.
3. Free, projective and injective modules, resolutions.
4. Tor and Ext.
5. Examples of resolutions: minimal resolution of graded modules, Koszul resolution, bar an cobar constructions.
6. Systems of parameters, regular sequences and Cohen–Macaulay algebras.
7. Projective dimension and depth of modules.
8. Multiplications: Eilenberg–Zilber and Kuenneth theorems.
9. Formality and Massey products.
10. Spectral sequences of a filtered complex and bicomplex.
10. A∞-morphisms, extended functoriality of Tor, the Eilenberg– Moore spectral sequence.