Remarks on Ass 1 by Marc Remarks on Ass 2 by Marc |
Course Assistant:
Marc Masdeu - Recall of the basic theory.
- Modules over PIDs (Recall only).
- Localizaton of rings and modules
(mostly through exercises)
- Tensor products.
- Projective, injective and flat modules
and resolutions.
- The snake lemma and other trivial, but
useful, diagrams.
2. BASIC NOTIONS OF CATEGORY THEORY - Categories and functors: the basic
examples.
- Universal objects: products,
coproducts, pullback and pushout, injective and projective limits.
- Adjoint functors. Equivalence of
categories.
3. SEMISIMPLE RINGS AND MODULES - Noetherian and Artinian rings and
modules. Hilbert's theorem.
- Semisimple rings and modules - the
basics.
- Nakayama's lemma and further study of
Artinian modules.
- Jacobson's density theorem and the
Artin-Weddrnburn theorem.
- The Brauer group.
4. REPRESENTATIONS OF FINITE GROUPS. (14 hrs) - Definition and basic operations (sum,
tensor product, dual, induction, symmetric and exterior products,
symmetric square).
- Maschke's theorem and the structure of
the group ring over an algebraically closed field.
- Character theory; behaviour under the
basic operations; orthogonality of characters, class functions.
- Frobenius reciprocity.
- The representations of groups of small
order and of dyhedral groups.
- Representations of S_n.
- There will be a midterm worth 25% of
the final grade.
- The assignments will be marked
(selected exercises only) and worth 20% of the grade.
- There will be a final exam (in class)
worth 55% of the final grade.
In accord with Syllabus and Grade Calculation. In the event of
extraordinary circumstances beyond the University’s control, the
content and/or evaluation scheme in this course is subject to change. |