Teaching plan for the course unit
Course unit name: Local Algebra
Course unit code: 568184
Academic year: 2018-2019
Coordinator: Santiago Zarzuela Armengou
Department: Department of Mathematics and Computer Science
Single program: S
Estimated learning time
Total number of hours 150
Face-to-face learning activities
- Problem-solving class
In order to be able to achieve the proposed objectives of the course, students are advised to have some basic knowledge of commutative algebra, equivalent to the optional subject Introduction to Commutative Algebra from the bachelor’s degree in Mathematics. Notwithstanding, this basic material is reread at the beginning of the course.
Competences to be gained during study
This course introduces the concepts, results and techniques of local algebra, which allow students to study local properties in various areas of algebra, combinatorics and geometry.
Referring to knowledge
— To understand the process of completion from an algebraic point of view; To analyse Hensel’s lemma.
— To learn to manipulate resolutions of ideals and modules, and their numerical invariants.
— To study degree theory and compare it with dimension theory.
— To learn to calculate Hilbert functions and study their applications.
— To acquire the knowledge and problem-solving skills to help assimilate the theory.
Referring to abilities, skills
— To perform introductory mathematical research in this subject area.
— To use basic bibliographic reference material on the subject.
Elements of commutative rings, ideals and modules
Graded rings and modules
Tor and Ext functors
Module categories; Limits and colimits
Tensor product and Hom functors
Projective and injective resolutions
Tor and Ext functors
Applications for commutative ring theory
Homological dimension; Syzygy theorem
Global dimension of a ring
Regular local rings; Characterization: The Auslander-Buchsbaum and Serre theorem
Factoriality of regular local rings: Auslander-Buchsbaum theorem
Completion of abelian topological groups
Adic completion of a ring
Hensel’s lemma and some applications of complete rings
Koszul complex associated to a linear form
Koszul complex associated to a family of elements
Regular sequences and the Koszul complex
The grade of a module
Ideals generated by regular sequences
Complete intersection ideals
Cohen-Macaulay rings and modules
Transfer of Cohen-Macaulay properties for morphisms
Computation and applications of the Hilbert function
Multiplicity of local rings
Examples and geometrical interpretation
Teaching methods and general organization
The teaching methodology for the subject includes:
• Face-to-face theory classes (lectures);
• Face-to-face practical classes;
• Individual tutored work based on the submission of answers to problem-solving exercises;
• Independent learning of some topics included in the program.
Teaching of the subject is structured around two weekly classes of two hours each held over one semester.
In these classes the teacher presents lectures on the theoretical content of the course; For each topic a list of problem-solving exercises is handed out, allowing students to practice acquired skills and the application of concepts; Completed answers to these exercises should be submitted to the teacher, who will then make annotated corrections; Students are required to present and discuss some of these exercises in class.
Throughout the semester each student is assigned a personal project related to the course content, which must be presented to the class in dedicated sessions that take place once all the face-to-face lectures have been completed; These projects are intended to be completed using bibliographic reference material provided by teachers and independently sourced.
Official assessment of learning outcomes
To pass the subject, students must demonstrate proficiency in the subject through continuous assessment. This includes the evaluation of answers to problem-solving exercises completed independently, which are used to assess the capacity and ability to apply acquired knowledge. The individual presentation of a personal project allows teachers to assess the capacity for independent learning and mathematical research. The final grade depends on both the quantity and quality of exercises handed in and the presentation of the personal project and corresponding written report.
Students who wish to opt out of continuous assessment must notify the teacher in charge within fifteen days from the commencement of face-to-face classes for the subject.
Reading and study resources
Consulteu la disponibilitat a CERCABIB
Bourbaki, N. Commutative algebra. Paris : Hermann ; Reading (Mass) : Addison-Wesley, 1972.
Bourbaki, N. Algèbre commutative : chapitres 8 et 9. Berlin [etc.] : Springer, 2006.
Bourbaki, N. Algèbre. Chapitre 10, Àlgebre homologique. Paris [etc.] : Masson, 1980.
Bourbaki, N. Algèbre commutative, Chapitre 10,/ Springer-Verlag, Berlin, 2007.
Bruns, W. ; Herzog, J. Cohen-Macaulay rings. Cambridge University Press, 2005.
Eisenbud, D. Commutative algebra with a view toward Algebraic Geometry. Springer, 1996.
Giral, J. M. Anillos locales regulares, teoría del grado, anillos de Cohen-Macaulay. Barcelona : Universidad de Barcelona. Departamento de Álgebra y Fundamentos, 1981.
Kaplansky, I. Commutative rings. Chicago [etc.] : University of Chicago Press, 1974.
Kunz, E. Introduction to commutative algebra and algebraic geometry. Boston : Birkhäuser, 1991.
Lafon, J. P. Les formalismes foundamentaoux de l’algèbre commutative. Paris : Herrmann, 1974.
Lang, S. Algebra. New York : Springer, 2005.
Matsumura, H. Commutative ring theory. New York : Cambridge University Press, 2008.
Osborne, M. S. Basic homological algebra. New York [etc.] : Springer, 2000.
Rotman, J. J. An introduction to homological algebra. Second edition, Universitext, Springer, New York, 2009.
Sally, J. D. Numbers of generators of ideals in local rings. New York : Marcel Dekker, 1978.
Serre, J. P. Local algebra. Berlin : Springer, 2000.
Zariski, O. ; Samuel, P. Commutative algebra. New York : Springer, [1975-1976]