Commutative algebra

- Know the basic elements of the theory of commutative rings and their ideals, with special emphasis on the polynomial rings, as well as their quotients and localizations.
- Introduce the basic algebraic techniques to be used in Algebraic Geometry and Number Theory.
- Understand the role that commutative rings play in the various fields of mathematics.

1. Rings and ideals. (5 sessions)
Prime ideals and maximal ideals. Prime spectrum of a commutative ring. Local rings. Nilradical and Jacobson radical. Extension and contraction of ideals. Polynomial rings in several variables.

2. Modules. (6 sessions)
Exact sequences, split sequences, the snake lemma. Modules of homomorphisms and exact sequences. The tensor product, exactness of tensor product, flat modules. Algebras, algebras of finite type, the tensor product of algebras. Restriction and extension of scalars. Modules of finite type: Nakayama's lemma and its applications.

3. Localization. (4 sessions)
Fraction rings and modules. Ideals in fraction rings. Local properties. Support of a module.

4. Noetherian and Artinian Rings and Modules. (5 sessions)
Chain conditions. The Jordan-Hölder theorem. Modules of finite length. Noetherian Rings. Hilbert basis theorem. Artinian rings.

5. Associated primes and primary decomposition. (4 sessions)
Associated primes of a module, behaviour with respect to localization, relation to the support, the case of finitely generated modules over a noetherian ring. Primary decomposition, existence and uniqueness theorems.

6. Integral extensions. (5 sessions)
Integral dependence. The going up theorem. Integrally closed domains. The going down theorem. Krull dimension of a ring Dimension of integral extensions.

7. Finitely generated algebras over a field. (6 sessions)
Algebraically independent subsets in an algebra, transcendence basis of a field extension, transcendence degree. The Hilbert Nullstellensatz. The Noether normalization lemma. Dimension of Finitely generated algebras over a field.

8. Dimension of noetherian rings. (6 sessions)
The Hilbert-Samuel polynomial. The dimension theorem. Krull's principal ideal theorem. Regular local rings.

9. Dedekind domains. (3 sessions)
Discrete valuation rings. Dedekind domains. Fractional ideals and invertible ideals. Unique decomposition of ideals into product of prime ideals. Class groups of ideals.

BASIC

1. Atiyah, M.F., Macdonald, I.G., Introduction to commutative algebra, Addisson-Wesley (1969)
2. Bourbaki, N.: Commutative algebra, Chap I-VII. Springer Verlag (1989)
3. Matsumura, H.: Commutative Algebra (2ª ed). Benjamin (1980)


COMPLEMENTARY

1. Altman, A., Kleiman, S.: A term of commutative algebra. Worldwide Center of Mathematics (2013)
2. Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Springer GTM 150 (1995)
3. Kunz, E.: Introduction to commutative algebra and algebraic geometry. Birkhäuser (1985)
4. Matsumura, H.: Commutative ring theory. Cambridge University Press (2004)
5. Raghavan Balwant Singh, S., Sridharan, R.: Homological methods in commutative algebra. Tata Institute of Fundamental
Research, Bombay (1975)
6. Sharp, R.Y.: Steps in commutative algebra (2ª ed). London Mathematical Society Student Texts 51, Cambridge University Press, (2000)
7. Zariski, O., Samuel, P.: Commutative algebra I, II. Springer, GTM 28, 29 (1975)

BASIC AND GENERAL COMPETENCES
GENERAL
• CG02 - Acquisition of high level mathematical tools for diverse applications covering the expectations of graduates in mathematics and other basic sciences.
• CG03 - Know the broad panorama of current mathematics, both in its lines of research, as well as in methodologies, resources and problems it addresses in various fields
BASICS
• CB6 - Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context
• CB7 - That students know how to apply the knowledge acquired and their ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their area of study.
• CB9 - That students know how to communicate their conclusions and the knowledge and ultimate reasons that sustain them to specialized and non-specialized audiences in a clear and unambiguous way
• CB10 - That students have the learning skills that allow them to continue studying in a way that will be largely self-directed or autonomous.
3.2 TRANSVERSAL COMPETENCES
• CT01 - Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access
• CT02 - Optimally manage work time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making
• CT03 - Enhance capacity for work in cooperative and multidisciplinary environments.
3.3 SPECIFIC COMPETENCES
• CE01 - Train for the study and research in mathematical theories in development.
• CE03 - Develop the necessary skills for the transmission of mathematics, oral and written, both in regard to formal correction, as well as in terms of communicative effectiveness, emphasizing the use of appropriate ICT

- Theory classes with lectures by the professor.
- Presentations of topics of the program by the students.

It will be taken into account for the student's grade:
- Class participation (10%)
- Presentation of topics of the program by the student (20%).
- Final written exam (70%)
For the second opportunity, the grades for class participation and exposition of topics of the program by the student will be maintained with the same relative weight, and a new final written exam will be carried out.

In cases of fraudulent performance of exercises or tests, the provisions of the Regulations on the Evaluation of Students' Academic Performance and the Review of Grades will apply.

In addition to the 3 or 4 weekly sessions, the student should aim to spend 10 hours per week working on the course.

There will be an e-learning platform to support the teaching of this subject at USC.