ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
- To ensure that students understand the unifying language of category theory and know how to use it in different contexts.
- To learn about derived functors and the theory of homological dimension.
- To know some motivating examples from algebraic topology and algebraic geometry, as well as some of the most important applications of homological algebra to the study of several algebraic structures.
1. Categorical language.
Categories, functors and natural transformations. Universal constructions. Limits and colimits. Adjoint functors. 2-categories. Abelian and semi-abelian categories.
Teaching:
Expository hours: 9
Hours study / work: 14 / 8
Tutorial: 0.9
2. Homology.
The category of modules. Free, injective, projective and flat modules. Complexes and homology. Derived functors. Ext and extensions, Tor and flatness. Universal coefficient theorems and the Künneht formula.
Teaching:
Expository hours: 10
Hours study / work: 14 / 8
Tutorial: 0.9
3. Some examples and applications.
Simplicial methods. Cotriple homology. Examples: Group homology and Lie algebra homology. André-Quillen homology and Hochschild homology.
Teaching:
Expository hours: 5
Hours study / work: 4 / 3
Tutorial: 0.2
Basic bibliography
- M. Farinati, Tópicos de álgebra homológica. Cursos y seminarios de matemática 14, Universidad de Buenos Aires (2021).
http://cms.dm.uba.ar/depto/public/Serie%20B/serieB14.pdf
- P. J. Hilton, U. Stammbach, A course in homological algebra. Graduate Texts in Math. 4, Springer-Verlag (1997).
- T. Leinster, Basic Category Theory. Cambridge University Press (2014).
https://arxiv.org/abs/1612.09375
- E. Lluis-Puebla, Álgebra Homológica, Cohomología de Grupos y K-Teoría Algebraica Clásica, Publicaciones Electrónicas
Sociedad Matemática Mexicana, Serie: Textos. Vol. 5, 2005.
http://www.pesmm.org.mx/Serie%20Textos_archivos/T5.pdf
- T. Pannila, An Introduction to Homological Algebra, Master’s thesis, University of Helsinki, 2016.
http://hdl.handle.net/10138/161100
- E. Riehl, Category theory in context, Dover Publications, Inc., 2016.
http://www.math.jhu.edu/~eriehl/context.pdf
Complementary bibliography
- J. Adamek, H. Herrlich, G. Strecker, Abstract and Concrete Categories, TAC reprints, 2006.
- S. Awodey, Category theory, The Clarendon Press, Oxford University Press, 2006.
- F. Borceux, Handbook of Categorical Algebra 1. Basic Category Theory, Encyclopedia of Mathematics and its applications 50, Cambridge University Press, 1994.
- S. I. Gelfand, Y. L. Manin, Methods of Homological Algebra, Springer, 2003.
- S. MacLane, Categories for the Working Mathematician, Springer-Verlag, Berlin, 1971.
- S. Roman, An Introduction to the Language of Category Theory, Birkhäuser, 2017.
- J. Rotman, An Introduction to Homological Algebra, 2nd Ed., Springer, 2009.
- H. Simmons, An Introduction to Category Theory, Cambridge University Press, 2011.
- C. A. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994.
GENERAL COMPETENCE:
- Acquisition of high-level mathematical tools for various applications, meeting the expectations of graduates in mathematics and other basic sciences (CG02).
- To know the great influence of categorical algebra on diverse fields of contemporary mathematics (CG03).
- To train for problem analysis and problem-solving in new or unfamiliar environments within broader contexts (CG04).
• To prepare for decision-making from abstract considerations in order to organize, plan and resolve complex issues (CG05).
TRANSVERSAL COMPETENCE:
- To make use of bibliography and bibliography search tools, including Internet usage (CT01).
- To manage time and other available resources in an optimal way and to maximize the ability to work in collaborative environments (CT02, CT03).
SPECIFIC COMPETENCE:
- To train for the study and research in developing mathematical theories (CE01).
- To apply mathematical tools in various fields of science, technology and social sciences (CE02).
- To develop the skills required for oral and written transmission of mathematical knowledge with formal correctness and communication efficiency (CE03).
- The program will run alternately through lectures and practical classes, encouraging student participation. Weekly expositions will be held so that the student can deepen in the development of the topics. Therefore, in addition to the presentations by the teacher of the different topics of the program, the student will have to develop some of the topics throughout the course (competence CG02, CG03, CE01, CE02, CE03).
- Exercise worksheets will be periodically given to students, some of which will be proposed to be presented at the end of the course while the rest will be solved in class under the supervision of the teacher. Students will be encouraged to attend the different seminars that may be held throughout the course on research topics related to the contents of the program (CG04, CG05, CT01, CT02, CT03, CE03).
1. Categorical language.
Teaching:
Expository hours: 9
Hours study / work: 14 / 8
Tutorial: 0.9
2. Homology.
Teaching:
Expository hours: 10
Hours study / work: 14 / 8
Tutorial: 0.9
3. Some examples and applications.
Teaching:
Expository hours: 5
Hours study / work: 4 / 3
Tutorial: 0.2
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.
The students will have to make presentations of some parts of the syllabus and will deliver several proposed exercises. The evaluation may be completed by means of a written examination or the completion of a work, in addition to considering active participation in the classes and the completion of the proposed exercises.
The qualification will be based on these expositions, as well as on the completion of the exercises. The levels of clarity of exposure and concision will be evaluated, as well as the answers of the student to questions that will be asked during the presentations.
In another case, the evaluation system contemplates, on the one hand, a mark of the final exam (E) and, on the other hand, a continuous evaluation (C), made throughout the course, based mainly on the participation of each student in the classroom, the realization of written controls, delivered works, tutorials and other means.
First Call (February):
With the exception of those not presented, for the calculation of the final mark (F) the continuous evaluation (C) and the final exam mark (E) will be taken into account and the following formula will be applied:
F= max (E, 0.4*C+0.6*E)
The evaluation of the final exam is done by means of a written examination.
Second Call (July):
The mark (C) obtained in the continuous evaluation of the first call in February is preserved and a new final writing exam (E) will be carried out.
With the exception of those not presented, for the calculation of the final mark (F) the continuous evaluation (C) and the final exam mark (E) will be taken into account and the following formula will be applied:
F= max (E, 0.25*C+0.75*E)
The evaluation of the final exam is done by means of a written examination.
According to article 5.2 of the Regulations on permanence in bachelor's and master's degrees of the University of Santiago de Compostela, students who do not carry out any assessable academic activity according to that established in the teaching programme will be included in the minutes as "not presented".
The competence {CG02, CG03, CG04, CG05, CT01, CT02, CT03, CE01, CE02, CE03} will be evaluated both in the processes of continuous evaluation and in the written examination.
Class meeting:
Expository lectures in large groups: 11
Interactive lessons in small groups: 11
Tutorships in a very small group: 2
Work hours total in person: 24
Independent work:
Individual self-study or in a group: 32
Writing exercises, conclusions or other derivative works of the subject: 19
Total hours working personal: 51
Math skills to bachelor level.
Manuel Eulogio Ladra Gonzalez
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813138
- manuel.ladra [at] usc.es
- Category
- Professor: University Professor
| Thursday | |||
|---|---|---|---|
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 10 |
| 13:00-14:00 | Grupo /CLIL_01 | Spanish | Classroom 10 |
| 01.20.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.18.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |