Algebra 4

A.Y. 2020/2021
Overall hours
Learning objectives
The course aims to give an introduction to category theory, module theory, commutative and multilinear algebra.
Expected learning outcomes
Basic knowledge of commutative and multilinear algebra. Familiarity with the language of categories and functors.
Course syllabus and organization

Single session

Lesson period
Second semester
If it will not be possible to have face-to-face classes, they will be held remotely, using Zoom or a similar platform.
Similarly, if in-persons exams are not possible, they will also be held remotely, using Zoom or a similar platform.
Course syllabus
Extension and contraction of ideals. Categories, functors, natural transformations. Adjoint functors, limits and colimits. Representable functors and Yoneda lemma. Hom and Ext. Free, projective and injective modules. Tensor product and flat modules. Localizations and local properties. Nakayama lemma. Noetherian rings and modules: finiteness conditions, finitely presented modules, Hilbert base theorem. 1-dimensional Noetherian rings and Artinian rings. Multilinear algebra: tensor algebra, symmetric and external algebra.
Prerequisites for admission
Basic knowledge of ring and module theory: principal ideal domains, prime and maximal ideals, definition of a module over a commutative ring with unit.
Teaching methods
Traditional lectures.
Teaching Resources
Atiyah e MacDonald, "Introduction to Commutative Algebra"
Conrad, "Tensor products I e II",
Rotman, "Advanced modern algebra"
Leinster, "Basic category theory"
Mac Lane, "Categories for the working mathematician"
Assessment methods and Criteria
The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems related to the topics treated during the lectures. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). The outcomes of the written exams will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/02 - ALGEBRA - University credits: 6
Practicals: 24 hours
Lessons: 36 hours