ELEMENTS OF ADVANCED ALGEBRA

Learning Objectives

• Topics of algebra

  The aim of this course is to deepen the study of coomutative ring theory, taking particular attention to polynomial rings and their quotients, with a view towards applications to allgebraic geometry and number theory.

  The student will also improve his skill of making abrstact argomentaions and will learn that a deep theoretical knowledge allows to develop signifinant applicative tools.

• Computational Algebra

  The objective of the second module of the course is to introduce the theory of Groebner bases , in order to begin the computational student to algebra and its applications

Detailed Course Content

• Topics of algebra

  I. Rings and ideals. Definitions and first properites. Prime and maximal ideals. Local rings. Nilradical and Jacobson radical. Ideals operations.Homomorphisms.

  II. Modules. Definitions and first properties. DIrect product and direct sum; free modules. Finite modules and Nakayama's lemma. Module homomorphisms. Algebras.

  III. Frctions rings and modules. Definition and properties. Localization and local properties. Ideals in fraction rings.

  IV. Noetherian rings. Affine varieties, affine K-algebras, correspondance betwwen algebra and algebraic-geometry concepts. Krull dimension. Noetherian rings and modules: definitions and first properties. Hilbert's basis theorem. Conditions for a sub-algebra to be finitely generated.

  V. Artinian rings. Artinian rings and modules. Composition series. Length. A ring is artinian if and only if it is noetherian and zero-dimensional.

  VI. Primary decomposition. Primary ideals and primary decomposition. Associated primes and their characterization. Zero-divisors. Unicity of isolated components. The noetherian case.

  VII. Hilbert Nullstellensatz: weak and strong formulations.

  VIII. Integral dependance. Definizions and first properties. Going Up theorem. Normal domains and Going Down theorem. Noether's normalization lemma.

  IX. First steps in dimension theory. Chain of primes, height, dimension. Krull's principal ideal theorem.Krull's height theorem.Dimension for polynomial rings with coefficient in a field. Local rings. System of parameters. Embedding dimension. Regular local rings (only definition and geometric relevance).

• Computational Algebra

  I. Basic Theory of Groebner Bases. The linear case. The case of a single variable. Monomial orders. The division algorithm. Definition of Groebner Bases. S - polynomials and Buchberger algorithm. Reduced Groebner bases .

  II . Applications of Groebner Bases. Elementary applications of Groebner Bases. Theory of elimination. Polynomial maps. Some applications to Algebraic Geometry .

  III . Modules. Groebner bases and Syzygies. Calculation of the module of syzygy of an ideal.

Textbook Information

• Topics of algebra

  1. M.F. Atiyah, I.G. Macdonald , Introduzione to commutative algebra.

  2. E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkauser 1985

• Computational Algebra

  W.W. Adams, P. Loustaunau, An introduction to Groebner Bases, American Math. Soc, 1994.