Theorems: Preservation of limits/colimits by adjoint functors.
Linear Algebra II:
Notions: Tensor product, extension of scalars, symmetric and exterior powers, flat modules, projective modules, injective modules, Tor, Ext.
Theorems: A free module is projective. A projective module is flat. Existence and homotopy uniqueness of projective and injective resolutions of modules.
Notions: Localization of rings and modules, integral homomorphism of rings, integral closure, normal domains, spectra of rings and Zariski topology, transcendental field exensions, Krull dimension.
Theorems: Nakayama's Lemma, Going-up and Going-down, Noether normalization, Hilbert and Zariski Nullstellensatz, (transcendence degree equals Krull dimension for finitely generated algebras over a field).
Notions: Additive category, abelian category, complexes, homotopy category of complexes, derived category, derived functors and cohomology, spectral sequence.
NOTE: At the discretion of the instructor, the Homological Algebra section, and possibly the final topics in the Commutative Algebra section, could be replaced by: