SUBJECT
Title
Commutative algebra
Type of instruction
lecture + practical
Level
master
Faculty
Part of degree program
Credits
3+3
Recommended in
Semesters 1-4
Typically offered in
Autumn/Spring semester
Course description
- Ideals. Prime and maximal ideals. Zorn's lemma. Nilradical, Jacobson radical. Prime spectrum.
- Modules. Operations on submodules. Finitely generated modules. Nakayama's lemma. Exact sequences. Tensor product of modules.
- Noetherian rings. Chain conditions for mudules and rings. Hilbert's basis theorem. Primary ideals. Primary decomposition, Lasker-Noether theorem. Krull dimension. Artinian rings.
- Localization. Quotient rings and modules. Extended and restricted ideals.
- Integral dependence. Integral closure. The 'going-up' and 'going-down' theorems. Valuations. Discrete valuation rings. Dedekind rings. Fractional ideals.
- Algebraic varieties. 'Nullstellensatz'. Zariski-topology. Coordinate ring. Singular and regular points. Tangent space.
- Dimension theory. Various dimensions. Krull's principal ideal theorem. Hilbert-functions. Regular local rings. Hilbert's theorem on syzygies.
Readings
Atiyah, M.F.–McDonald, I.G.: Introduction to Commutative Algebra. Addison–Wesley, 1969.