Math 501. Abstract Algebra II

(1) Rings and modules II

(a) Rings, homomorphisms, ideals, commutative rings, polynomial rings, Hilbert basis theorem and nullstellensatz, power series rings and group rings, principal and factorial rings.
(b) Direct products and sums of modules, free modules, vector spaces, the dual.
(c) Complexes and homology, Euler characteristic, snake lemma.

(2) Categories and Functors

(a) Category theory, products and coproducts, universal objects, pullback and push-outs.
(b) Projective and injective modules, divisibility, Baer criterion.
(c) Functors, Natural transformations, adjoints, left/right exactness.
(d) Localization, direct limit, inverse limit.

(a) Tensor product, exterior product, symmetric product
(b) Schur functors
(c) Application: Representation theory of finite groups II

(a) Morphisms of complexes, induced map on homology, chain homotopy, short exact sequence of complexes yields long exact sequence in homology.
(b) Free, projective and injective resolutions.
(c) A first look at derived functors and delta-functors.