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Course, academic year 2023/2024
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Commutative algebra - NMAG538
Title: Komutativní algebra
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:4/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: Liran Shaul, Ph.D.
Class: M Mgr. MSTR > Povinně volitelné
Annotation -
This is a Master level introductory course to a standard theory of commutative noetherian rings. In includes basics on lovalization and completion of rings, Krull dimension and homological conditions on rings.
Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (29.04.2021)
Literature -

M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.

H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.

W. Bruns, J. Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.

D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (29.04.2021)
Syllabus -

1. Noetherian and Artinian rings.

2. Prime and maximal ideals, the Jacobson radical and the nilradical, Nakayama's lemma.

3. Localization, flatness, integral extensions, going up and going down theorems.

4. Adic completion, the Artin-Rees lemma and the Krull's intersection theorem.

5. Dimension theory of rings and Krull's principal ideal theorem.

6. Regular sequences and Koszul complexes.

7. Basics on Regular, Gorenstein and Cohen-Macaulay rings.

8. Serre theorem about localization of regular rings.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (29.04.2021)
Entry requirements -

Basics of homological algebra (to the extent of lecture NMAG434).

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (08.11.2021)
 
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