| Center |
| Faculty of Mathematics |
| Departament |
| Algebra |
| Lecturers in charge |
| Sin datos cargados |
| Met. Docent |
| In the theoretical sessions, the active part of its development will correspond basically to the professor. The participation and work of the students in the classroom will be promoted, specially in the practical sessions. |
| Met. Avaluació |
| A theoretic and practical examination will be made. In the final qualification, may also be valued the resolution of questions, problems and exposition of subjects that colud possibly be proposed. |
| Bibliografia |
| - Blyth, T.S.: Module Theory - An approach to linear algebra. Clarendon Press - Oxford - Hilton, P.J. - Stammbach, U.: A course in homological algebra. Springer-Verlag - Hungerford, T.W.: Algebra. Springer-Verlag - Lafon, J.P.: Les formalismes fondamentaux d'àlgèbre commutative. Hermann - MacLane, S.: Categories for the working mathematician. Springer-Verlag - Mitchell, B.: Theory of categories. Academic Press - Northcott, D.G.: A first course of homological algebra. Cambridge University Press - Raghavan, S. - Singh, B. - Sridharan, R.: Homological Methods in Commutative Algebra Oxford Univ. Press - Rotman, J.J.: An introduction to homological algebra. Academic Press - Stenström, B.: Rings of quotients. Springer-Verlag |
| Continguts |
| The first part of the subject is dedicated to introduce to the categorical language and its basic concepts: monic arrow, epic arrow, isomorphism, initial final and zero objects, as well as to give an extensive list of examples, presented from the axiomatic theory of sets. We study the concepts of functor and natural transformation as well as the important concept of pair of adjoint functors The second part of the subject is dedicated to introduce the concept module over a unitary ring. The general theory is developed: submodule, quotient module, homomorphisms and the isomorphism theorems, dedicating an special attention to the concept of free module and related concepts as the basis of a module, showing the differences in relation to which it’s know in the particular case of vectorial spaces We also study the problem of exactness of sequences and additive functors in categories of modules giving an special attention to the two important exemples of the additive functors Hom y tensor product, proving in particular that form a pair of adjoint functors.. In the last part, which constitutes tha main part of the subject, we study the two infinite sequences associated to the functors Ext and Tor. Finally as an application of the theory of the functors Ext and Tor we characterize the different concepts of dimension over a ring . |
| Objetius |
| - To introduce the language of categories and to prove that module categories are abelian categories. - To define the functors Hom and tensor product and to study properties of exactness of both functors. - Considering projetive resolutions of modules, to define the functors Ext and Tor as derived functors of certain functors Hom and tensor product. - To study properties related to projective dimension of modules, in order to define left and right global dimension and homological dimension of a ring. |
| URL de Fitxa |