Commutative algebra

elective courses

Algebra I 1000-113bAG1a
Topology I 1000-113bTP1a

The student knows and understands the basic concepts of topology, in particular non-metric spaces. The student knows and understands basic objects of algebra (ring, integral domain, field, homomorphism, principal ideal domain) and is able to fluently operate them on basic examples: ZZ, K[x].

This lecture class provides an introduction to commutative algebra; it is required for algebraic geometry lecture. The topics concern commutative rings and modules over such rings. An important class of rings considered are noetherian rings.

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.

1. M.F. Atiyah, I.G. MacDonald. Introduction to commutative algebra.

2. J. Browkin. Teoria ciał. (Theory of fields, in Polish).

3. S. Balcerzyk, T. Józefiak. Commutative Noetherian and Krull rings,

and S. Balcerzyk, T. Józefiak. Commutative rings. Dimension, multiplicity and homological methods.

4. D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry. Springer 1995.

5. I. Kaplansky. Commutative Algebra.

6. S. Lang. Algebra, (both editions).

7. H. Matsumura. Commutative ring theory

8. M. Reid. Undergraduate commutative algebra.

The student knows and has practiced with all the objects of modern commutative algebra: localization, spectra, tensor product. The student knows and is able to apply the main results of the theory of finitely generated algebras: Noether's normalization and Nullstellensatz, dimension theory. The student knowsand is able to operate with important classes of rings, such as DVR and Dedekind domains and apply the results to number theoretic investigations.

Final exam