Commutative algebra
Code: 100112 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| 2500149 Mathematics | OT | 4 |
Contact
- Name:
- Dolors Herbera Espinal
- Email:
- dolors.herbera@uab.cat
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
The knowlege of all the compulsory courses in algebra, specially the course Estructures Algebraiques.
Objectives and Contextualisation
The aim of this course is to give an introduction to the basic tools of the theory of commutative rings while using the category and functor class.
This means working on the basic concepts of categories and functors, also of rings, the structure of their ideals and the modules on it, delving into specific topics of each of these aspects.
At the end of the course, the student is expected to know the basic concepts of category and functor theory. General constructions on rings and modules, chain conditions, and some ideas about the prime spectrum of commutative rings. From here and depending on the topics in which you have gone deeper during the course, the objectives to be achieved may vary.
Competences
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
- Demonstrate a high capacity for abstraction.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
Learning Outcomes
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Understand in-depth demonstrations of some theorems of advanced algebra and assimilate the definition of new algebraic structures and constructions, relating them with other knowledge and deducing their properties.
- Use algebraic tools in different fields.
Content
Broadly speaking, the course will be structured following the contents of the classic reference book "M.F. Atiyah and I.G. Macdonald, Introducción al Álgebra Conmutativa" but emphasizing more the use of Category Theory language and also in the examples. We will also consider Dummit and Foote's book "Abstract Algebra"
The topics to be discussed will be
1. First approach to the language of categories and functors.
2. Categories of rings and categories of modules on them. Basic results.
3. Free modules and module presentations. Classification of finitely generated modules over a domain of principal ideals.
4. Some functors: Hom and tensor product.
5. Location. Specter of a ring.
6. Artinian rings and Noetherian rings. Hopkins theorem.
7. Whole extensions.
8. Noether's Normalization Lemma and Hilbert's Nullstellensatz.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Theory classes | 30 | 1.2 | |
| Type: Supervised | |||
| Problem classes | 15 | 0.6 | |
| Seminars | 6 | 0.24 | |
| Type: Autonomous | |||
| Solving problems | 60 | 2.4 | |
| Study of the theory | 36 | 1.44 |
This course has two hours of theory per week. The recommended bibliography is interesting; sometimes during the course the students should complement and complete the content of the lectures using this bibliography.
There are problem classes (one hour per week). Every student should present the solutions of some lists of problems on the blackboard or in paper to the lecturer. To solve questions about the course the student can approach the lecturer during the class or in the office (during office hours). The solution of these problems will be based in general on the theory: the theorems and their proofs.
Seminars will be dedicated to develop collaborative problem-solving techniques.
The course has a web “campus virtual” where the lecturer will post the problems list and other relevant information about the course.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Final exam | 40% | 3 | 0.12 | 8, 2, 7, 6, 4, 5 |
| Seminar attendance | 10% | 0 | 0 | 1, 2, 6, 4, 5, 3, 9 |
| Solved problems | 50% | 0 | 0 | 1, 2, 6, 4, 3, 9 |
The evaluation of the course will be a 50% of continued evaluation, and a 50% of exams.
- The continued evaluation will consist of the presentation of solutions of problems (solved individually), the attendance to the seminars and the presentation of solutions of the exercises proposed in the seminars (solved in group).
- The mark of the exams will be obtained from a final exam.
The "matrícules d'honor" will be decided taken into account the results of the continued evaluation and the exams.
The "non-evaluable" qualification will be awarded to students who do not turnout at the final exam.
Single assessment: Those who opt for single assessment will take a single exam where all course contents will be assessed. This exam will take place on the same day as the final exam.
In case of doubt about the interpretation of the evaluation method, the Catalan written version remains as the reference.
Bibliography
W. A. Adkins, S. H. Weintraub, Algebra, An Approach via Module Theory. Springer, New York, 1992.
A. Altman, S. Kleiman, A Term of Commutative Algebra. Worldwide Center of Mathematics, LLC, 2012.
M. Atiyah, I. Macdonald, Introducción al álgebra conmutativa. Ed. Reverté, Barcelona, 1968.
P. M. Cohn, Algebra, vol 2. Second Ed. John Wiley and Sons, New York, 1989.
David S. Dummit and Richard M. Foote, Abstract Algebra, Third edition, John Wiley & Sons, 2004
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry. Springer, New York, 2004.
B. Hartley, T. O . Hawkes, Rings, modules and linear algebra. Chapman and Hall, London 1983.
N. Jacobson, Basic Algebra I, Basic Algebra II. W. H. Freeman and Company, New York, 1989.
E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, New York, 2013.
S. Lang, Algebra. Aguilar, Madrid, 1977.
B. A. Magurn, An algebraic introduction to K-Theory, Encyclopedia of Mathematics and its applications, 87, Cambridge, 2002.
J.S. Milne, A Primer of Commutative Algebra, 2009.
O. Zariski, P. Samuel, Commutative Algebra I, II, Van Nostrand, Princeton (1958, 1960).
Software
There is an entire branch of commutative algebra devoted to developing computational methods. There is free software that allows you to work with ideal rings and modules, we will try to learn a little about some of these software.
Language list
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PAUL) Classroom practices | 1 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | second semester | afternoon |
| (TE) Theory | 1 | Catalan | second semester | morning-mixed |