Teaching plan for the course unit

 General information

Course unit name: Local Algebra

Course unit code: 568184

Coordinator: Santiago Zarzuela Armengou

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 Estimated learning time Total number of hours 150

 Face-to-face and/or online activities 70
 -  Lecture Face-to-face 30 -  Problem-solving class Face-to-face 30 -  Seminar Face-to-face 10
 Supervised project 20
 Independent learning 60

 Recommendations

 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. It also includes an introduction to the necessary concepts and techniques of category theory and homological algebra.

 Learning objectives
 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.

 Teaching blocks

1. Introduction

1.1. Elements of commutative rings, ideals and modules

2. Tor and Ext functors

2.1. Module categories; Limits and colimits

2.2. Tensor product and Hom functors

2.3. Projective and injective resolutions

2.4. Tor and Ext functors

2.5. Applications for commutative ring theory

3. Homological dimension; Syzygy theorem

3.1. Global dimension of a ring

3.2. Regular local rings; Characterization: The Auslander-Buchsbaum and Serre theorem

3.3. Syzygy theorem

3.4. Factoriality of regular local rings: Auslander-Buchsbaum theorem

4. Completion

4.1. Completion of abelian topological groups

4.2. Adic completion of a ring

4.3. Noetherian completion

4.4. Hensel’s lemma and some applications of complete rings

5. Koszul complex

5.1. Koszul complex associated to a linear form

5.2. Koszul complex associated to a family of elements

5.3. Regular sequences and the Koszul complex

6.1. Module dimensions

6.2. The grade of a module

6.3. Ideals generated by regular sequences

6.4. Complete intersection ideals

7. Cohen-Macaulay rings and modules

7.1. Cohen-Macaulay modules

7.2. Cohen-Macaulay rings

7.3. Transfer of Cohen-Macaulay properties for morphisms

8. Computation and applications of the Hilbert function

8.1. Hilbert functions

8.2. Hilbert-Samuel functions

8.3. Multiplicity of local rings

8.4. 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; • Synchronous non-contact teaching (theory and 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. There will be one hour face-to-face class of theory that will be used to comment on the theoretical resuts. The proofs and some examples will be given by written notes or other audiovisual media. Their study will be part of the independent learning. Another hour of face-to-face class will be used to discuss the solutions to the exercises submitted by the students individually. There will be two more hours of synchronous non-contact teaching to complete the face-to face classes, both of theory or practical according to how the program be develping.   In the case of partial or full virtual teaching required by the health situation, the schedules and distribution of the face-to-face classes will be mantained and changed to synchronous non-contact teaching accordingly

 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.   Examination-based assessment Students who wish to opt out of continuous assessment must notify the teacher in charge within fifteen days from the commencement of the teaching activities for the subject.