COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY
Academic year and teacher
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- Versione italiana
- Academic year
- 2018/2019
- Teacher
- ALBERTO CALABRI
- Credits
- 9
- Didactic period
- Primo Semestre
- SSD
- MAT/03
Training objectives
- Knowledge of basic notions of commutative algebra, like commutative rings, prime and maximal ideals, localization, modules, noetherian property, and of the first notions of algebraic geometry, like affine algebraic varieties, coordinate ring, Zariski topology.
At the end of the course, the student will be able to know commutative rings and modules, their properties, and he/she will be able to construct new rings and modules by means of the tensor product, the localization and more generally the fractions. Prerequisites
- First year Algebra and elementary notions of topology.
Course programme
- Rings and ring homomorphisms. Ideals. Quotient rings. Zero-divisors. Nilpotent elements. Units. Prime ideals and maximal ideals. Nilradical and Jacobson radical.
Operations on ideals. Extension and contraction of ideals (12 hours).
Affine algebraic varieties. Coordinate ring of a variety. Zariski topology (4 hours).
Modules and module homomorphisms. Submodules and quotient modules. Operations on submodules. Direct sum and product. Finitely generated modules. Exact sequences. Tensor product of modules. Restriction and extension of scalars. Exactness properties of the tensor product. Algebras (12 hours).
Rings and modules of fractions. Local properties. Extended and contracted ideals in rings of fractions (6 hours).
Hilbert basis theorem. Hilbert theorem of zeroes and applications (6 hours).
Primary decomposition. Chain conditions. Noetherian Rings. Artin Rings (12 hours).
Integral dependence. The going-up theorem. Integrally closed integral domains. The going-down theorem.
Discrete valuation rings. Dedekind domains (6 hours).
Graded rings and modules. The associated graded ring.
Dimension theory. Hilbert functions and polynomial (5 hours). Didactic methods
- Lessons at the blackboard.
Learning assessment procedures
- Oral exam consisting of three questions on general aspects of the course: for each question a proof with all the details is required.
The answer to a question is evaluated with a score from 0 to 10, depending on the clarity and precision of the response, and the final score is the sum of the three scores.
In order to pass the exam, the score must be at least 18. Reference texts
- M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
