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Second Level Degree in Mathematics

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RINGS AND MODULES - 6 CFU

Teacher

Silvana Bazzoni

Scheduled Period

I Year - 2 Semester | 02/03/2020 - 12/06/2020

Hours: 48 (16 esercitazione, 32 lezione)

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Prerequisites

Notions from the Algebra courses of the first two years of the degree in Mathematics and basic notions on module theory over arbitrary rings.

Target skills and knowledge

The aim of the course is to learn the basic notions in catgeory theory and the related main constructions. To introduce the techniques and the tools of homological algebra and their applications to dimension theory.

Examination methods

Written exam consisting in answering to questions from the theory and in solving exercises.
Discussion of the composition and possible oral exam.

Assessment criteria

Check of the learning of the taught notions and on the ability of their application.

Course contents

Additive and Abelian categories. Functor categories. Freyd-Mitchell embedding theorem. Pull-back and push-out. Limits and colimits. Adjoint functors. Categories of chain complexes and the homotopy category. Foundamental Theorem in homology. Left and right derived functors. The functors Tor, flatness and purity. The funtors Ext and Yoneda extensions. Flat, projective and injective dimensions of modules and their characterization in terms of derived functors. Applications to the global dimension of rings and Hilbert's syzygies Theorem.

Planned learning activities and teaching methods

Lists of exercises to solve will be distributed to check and to deepen the learning of the taught notions.
The notes of the lectures will be distributed daily.

Additional notes about suggested reading

Notes of the lectures, solving of the proposed exercises. Reading of the books in the bibliography.

Textbooks (and optional supplementary readings)

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