Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Algebra and Number Theory

Code

komalg1u0um17gm

Title

Commutative algebra (p)

Usual semester

Spring

ECTS

3

Language

Learning outcomes

Knowledge: Knowledge of basic concepts, results and methods in the field. Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyse mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Ideals. Prime and maximal ideals. Zorn's lemma. Nilradical, Jacobson radical. Modules. Operations on submodules. Finitely generated modules. Nakayama's lemma. Exact sequences. Noetherian rings. Chain conditions for modules and for rings. Hilbert's basis theorem. Primary ideals. Primary decomposition, Lasker--Noether theorem. Artinian rings. Localization. Quotient rings and modules. Extended and contracted ideals. Integral dependence. Integral closure. The 'going-up' and 'going-down' theorems. Valuations. Discrete valuation rings. Noether normalization lemma.  'Nullstellensatz'. Algebraic varieties. Zariski topology. Coordinate ring. Prime spectrum. Singular and smooth points. Tangent spaces. Regular local rings. Dimension theory. Krull dimension and its equivalent characterizations, Krull's principal ideal theorem, Hilbert functions. Hilbert's theorem on syzygies. Necessary prior knowledge: basic algebraic structures

Assessment method

(written or oral) exam plus term mark

Bibliography

lecture notes

Recommended bibliography

Atiyah, M.F.–McDonald, I.G.: Introduction to Commutative Algebra. Addison Reid, M.: Undergraduate Commutative Algebra, Cambridge University Press Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry, Springer

Programmes of the course