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Course, academic year 2018/2019
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MSTR Elective 1 - NMAG498
Title in English: Výběrová přednáška z MSTR 1
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Note: you can enroll for the course repeatedly
Guarantor: doc. RNDr. Jan Šťovíček, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Volitelné
Classification: Mathematics > Algebra
Annotation -
Last update: T_KA (22.04.2016)
Non-repeated universal elective course.
Literature -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (04.10.2018)

1. Robin Hartshorne. "Residues and duality: lectures notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64." Springer, 1966.

2. Amnon Yekutieli. "A Course on Derived Categories." arXiv preprint arXiv:1206.6632 (2012).

3. Amnon Yekutieli. "Derived Categories", book to be published by Cambridge University Press, 2018, preview version: arXiv preprint, arXiv:1610.09640.

4. Richard Thomas. "Derived categories for the working mathematician." arXiv preprint math/0001045 (2000).

5. Markus Brodmann and Rodney Y. Sharp. "Local cohomology: an algebraic introduction with geometric applications." Vol. 136. Cambridge university press, 2012.

Requirements to the exam -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (11.10.2017)

The course is completed with an oral exam. The requirements for the exam correspond to what is presented in lectures.

Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (04.10.2018)

Derived categories and their applications in commutative algebra

The aim of this course is to introduce modern homological algebra in the language of derived categories, and to use this language to study commutative rings. We will explain how classical notions in commutative algebra such as Gorenstein and Cohen–Macaulay rings have a very easy description using the derived category. Following Grothendieck work on the subject, we will study dualizing complexes, local cohomology, and aim to prove Grothendieck's local duality theorem.

Some motivation

Cochain complexes and the notions of a homotopy equivalence and a

quasi-isomorphism

Projective modules, projective resolutions

Injective modules, injective resolutions, Matlis duality

Classical derived functors

Triangulated and derived categories

Derived functors in the derived category

Dualizing complexes

Local cohomology

Local duality

 
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