Commutative algebra 1000-135ALP

1. Commutative rings. Prime ideals, maximal ideals, primary ideals.
Nilradical and its characterisation as the intersection of prime
ideals; Jackobson radical. Examples: polynomials, formal series, rings
of continuous functions.
2. Localization and local rings. Ideals in localized rings.
3. Modules over commutative rings. Exact sequences of modules, free and
projective modules. Nakayama's lemma. Tensor product and flat modules.
4. Noetherian rings and modules. Ascending chain condition and finite
generation. Decomposition into product of indecomposable elements.
Hilbert's basis theorem. Localization preserves notherianity.
5. Finite and integral extensions of rings. Equivalent characterizations
of integral extensions, tower theorems for extensions. Integral closure
of domains and normal rings. Noether's normalization theorem.
6. The Krull dimesion. Krull dimension of polynomial rings and of finitely
generated k-algebras. Dedekind rings.
7. Hilbert's Nullstellensatz, weak and strong versions. Algebraic sets in
affine space and decomposition to components. Zariski topology.
Spectrum of a noetherian ring, Spec of a finitely generated k-algebra,
Spec ZZ.
8. Graded rings and modules, filtrations. Homogeneous ideals. Hilbert
function and Poincare series. Relation to noetherianity.
9. Krull's itersection theorem, Artin-Rees' lemma, I-adic topology,
completions, p-adic numbers.
10. Discrete valuations and basic properties of discrete valuation
rings. Normal local domains of dimension 1 are discrete valuation
rings. Normal noetrian domain is intersection of discrete valuation
rings.
11. Prime ideal associated to a module. Primary decomposition of modules
and ideals in noetherian rings.

Note: the exposition of topics 8-11 is at the discretion of the lecturer.