
COMMUTATIVE ALGEBRA
Prerequisites:

Basic notions of algebra (groups, rings, ideals, fields, quotients, etc.), as acquired in the "Algebra 1" course. 
Examination methods:

Written exam 
Course unit contents:

Commutative rings with unit, ideals, homomorphisms, quotient rings. Fields, integral domains, zero divisors, nilpotent elements. Prime ideals and maximal ideals. Local rings and their characterization. Operations on ideals (sum, intersection, product). Extension and contraction of ideals w.r.t. homomorphisms. Annihilator, radical ideal, nilradical and Jacobson radical of a ring. The Zariski topology on the prime spectrum Spec(R). Spec(R/I) as closed subset of Spec(A). Direct product of rings.
Modules, submodules and their operations (sums, intersection). Annihilator of a module. Faithful modules. Direct sums and direct products of modules. Exact sequences of modules, snake lemma. Projective and injective modules. Finitely generated and finitely presented modules, free modules. CayleyHamilton theorem and Nakayama's lemma.
Tensor product and its properties. Extension of scalars for modules. Algebras over a ring and their tensor product. Adjunction and exactness of the Hom and tensor product functors. Flat modules. Kahler differentials
Rings of fractions and localisation. Exactness of localisation. of rings and modules. Localisation and open subsets of Spec(R). Local properties. faithfully flat modules and descent theory. Projective and locally free modules.
Integral elements, integral extension of rings and integral closure. Going Up, Going Down and geometric translation. Norm, trace, discriminant. Valuation rings. Overview of completions.
Chain conditions, Artinian and Noetherian rings and modules. Hilbert's basis theorem. Normalization Lemma and Nullstellensatz.
Discrete valuation rings. Fractional ideals and invertible modules. Cartier and Weil divisors, Picard group, cycle map. Dedekind domains and their extensions. Decomposition of ideals, inertia, ramification.
Krull dimension, height of a prime ideal. Principal ideal theorem. Characterisation of factorial domains. Regular local rings. Finiteness of dimension for local noetherian rings. 

