M8350 Algebra IV

Faculty of Science
Spring 2024
Extent and Intensity
2/1/0. 5 credit(s). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
Teacher(s)
doc. John Denis Bourke, PhD (lecturer)
Guaranteed by
doc. Lukáš Vokřínek, PhD.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 19. 2. to Sun 26. 5. Thu 16:00–17:50 M3,01023
  • Timetable of Seminar Groups:
M8350/01: Mon 19. 2. to Sun 26. 5. Thu 18:00–18:50 M3,01023, J. Bourke
Prerequisites
Algebra I, II, III
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The goal is to finish the course of algebra a use the acquired knowledge to the following topics:
- homological algebra
- finiteness conditions in the theory of modules
- commutative algebra in connections with algebraic geometry
- reprezentations of groups.
Learning outcomes
After finishing the course, students will have an idea about special methods of algebra and their use. In particular they will be able:
- to use methods of homological algebra;
- to have an idea how finite dimensionality is extended from linear algebra to the theory of modules;
- to understand basic techniques of commutative algebra in connections with algebraic geometry;
- to appreciate the power of reprezentations of groups by matrices.
Syllabus
  • 1. Chain complexes: chain complexes, exactness, homology, projective and injective resolutions. 2. Derived functors: derived functors, Functor Ext, functor Tor, extensions of modules, the connection with the group Ext. 3. Homological dimension: projective and injective dimension of a module, global dimension of a ring. 4. Noetherian rings and modules: noetherian modules, artinian modules, radical of a ring, Najayama lemma. 5. Commutative algebra: Dedekind rings, local rings, localization, primary decomposition, spectrum of a ring, connections with algebraic geometry. 6. Linear representations of groups: group rings and group algebras and modules over them, irreducible reprezentations, Schur lemma, Maschke theorem. 7. Characters od groups: characters, orthogonality, applications to finite groups, induced reprezentations, Frobenious reciprocity theorem.
Literature
  • J. R. Rotman, Advanced Modern Algebra, AMS 2017
Teaching methods
The lecture offering the basic understanding of the subject, its mutual relationships and its applications. The exercise offering illustrative examples.
Assessment methods
oral exam + 3 marked assignments
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
The lessons and exam will be in English.
The course is also listed under the following terms Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/spring2024/M8350