Álgebra Comutativa

Objectives

To introduce commutative algebra and its relation to algebraic geometry.

Syllabus

Operations on ideals. Algebraic sets. Groebner bases. Localization. Noetherian and Artinian rings and modules. Hilbert's basis theorem. The Cayley-Hamilton theorem and Nakayama lemma. Associated primes. Algebras over rings. Going-up and going-down for prime ideals. Integral and finite extensions and Noether normalization theorem. Nullstellensatz and the spectrum of a ring. Affine varieties. Rational functions and applications on affine varieties. Krull dimension. Krull's principal ideal theorem. Systems of parameters and local regular rings. Discrete valuation rings and Dedekind domains. Fractional and invertible ideals. Filtrations and completions. Krull intersection theorem. Hensel and Artin-Rees lemmas. Possible additional topics: graded rings and modules; the Hilbert polynomial; characterization of dimension via the Hilbert function; Koszul complex; Cohen-Macaulay rings.

Prerequisites

Modules and Representations; some knowledge of general topology.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.