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Course, academic year 2019/2020
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Categories of Modules and Homological Algebra - NMAG434
Title in English: Kategorie modulů a homologická algebra
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English
Teaching methods: full-time
Additional information: http://karlin.mff.cuni.cz/~shaul/halg.html
Guarantor: Liran Shaul, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG029
Interchangeability : NALG029
Annotation -
Last update: T_KA (09.05.2013)
Category theory of modules (covariant and contravariant Hom functors, projective and injective modules, tensor product, flat modules, adjointness of Hom functors and tensor product, Morita equivalence of rings and its characterization, a generalization: tilting modules and tilted algebras), introduction to homological algebra (complexes, projective and injective resolutions, Extn and Torn functors, connections between Ext1 and extensions of modules.
Course completion requirements - Czech
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (11.06.2019)

Předmět je zakončen ústní zkouškou.

Literature - Czech
Last update: T_KA (09.05.2013)

F.W.Anderson, K.R.Fuller: Rings and Categories of Modules, Springer, New York 1992.

J. J. Rotman, An Introduction to Homological Algebra, Academic Press, San Diego, 1979.

C.Weibel: An Introduction to Homological Algebra, Cambridge Univ.Press, Cambridge, 1994.

Requirements to the exam - Czech
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (11.06.2019)

Zkouška má ústní formu. Její požadavky odpovídají rozsahu přednesené látky.

Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (17.12.2018)

1. Category theory of modules:

1.1 Covariant and contravariant Hom functors, projective and injective modules,

1.2.Tensor product, flat modules,

1.3 Adjointness of Hom functors and tensor product,

1.4 Morita equivalence of rings and its characterization,

1.5 A generalization: tilting modules and tilted algebras).

2. Introduction to homological algebra:

2.1 Complexes, projective and injective resolutions,

2.2 Extn and Torn functors,

2.3 Long exact sequences for Ext and Tor

2.4 Connections between Ext1 and extensions of modules.

Entry requirements -
Last update: T_KA (09.05.2013)

Basics of ring and module theory.

 
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