Sumários

Test 5

3 fevereiro 2022, 10:00 • Pedro Resende

Fifth continuous assessment test.

Lecture 32

1 fevereiro 2022, 10:00 • Pedro Resende

Primary components of a module over a PID. Primary Decomposition Theorem. Uniqueness of the invariant factors and of the elementary divisors of a finitely generated module over a PID. Applications of the classification theorem of finitely generated modules over PIDs: (1) examples of classification of finite abelian groups both in terms of invariant factors and of elementary divisors; (2) proof of the existence of a Jordan canonical form for any complex square matrix; (3) brief reference to the rational canonical form.

Bibliography: Dummit&Foote, chapter 12, and these notes (same as yesterday's).

Lecture 31

31 janeiro 2022, 08:30 • Pedro Resende

Torsion submodules. Noetherian modules. Classification of finitely generated modules over PIDs (existence theorems): Betti number (= free rank), invariant factors and elementary divisors. Corollary: classification of finitely generated abelian groups.

Bibliography: Dummit&Foote, section 12.1 and these notes (sections I, II, III, IV.1, V).

Lecture 30

27 janeiro 2022, 10:00 • Pedro Resende

More on the structure of group rings and their representations. Character theory.

Bibliography: these expanded notes  (from §22 onwards); Dummit&Foote, sections 18.2, 18.3 and 19.1.

Lecture 29

25 janeiro 2022, 10:00 • Pedro Resende

Rings that are completely reducible modules. Idempotents in rings. Matrix rings. Semisimple Artinian rings. Artin–Wedderburn theorem. Representations of finite groups. Maschke's theorem. Corollary: complex group algebras are products of finite families of rings of matrices over the complex numbers.

Bibliography: these  notes; and Dummit&Foote, sections 18.1 and 18.2.