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COMMUTATIVE ALGEBRA - 8 CFU

Teacher

Remke Nanne Kloosterman

Scheduled Period

I Year - 1 Semester | 04/10/2021 - 15/01/2022

Hours: 64 (32 esercitazione, 32 lezione)

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Prerequisites

Basic notions of algebra (groups, rings, ideals, fields, quotients, etc.), as acquired in the "Algebra 1" course.

Target skills and knowledge

A good knowledge of the algebraic objects used in Algebraic Geometry and Number Theory:
- Modules;
- Tensor products;
- Prime spectrum;
- Localization;
- Integral extensions;
- Noetherian rings;
- Dedekind domains and discrete valuation rings;
- Basics on dimension theory.

Examination methods

Written exam

Assessment criteria

The student will be evaluated on his/her understanding of the topics, on the acquisition of concepts and methodologies proposed and on the ability to apply them in full independence and awareness.

Course contents

Commutative rings with unit, ideals, homomorphisms, quotient rings. Fields, integral domains, zero divisors, nilpotent elements. Prime ideals and maximal ideals. Local rings and their characterization. Operations on ideals (sum, intersection, product). Extension and contraction of ideals w.r.t. homomorphisms. Annihilator, radical ideal, nilradical and Jacobson radical of a ring. Direct product of rings.

Modules, submodules and their operations (sums, intersection). Annihilator of a module. Direct sums and direct products of modules. Exact sequences of modules, snake lemma. Projective and injective modules. Finitely generated and finitely presented modules, free modules. Cayley-Hamilton theorem and Nakayama's lemma.

Tensor product and its properties. Extension of scalars for modules. Algebras over a ring and their tensor product. Adjunction and exactness of the Hom and tensor product functors. Flat modules. Kahler differentials

Rings of fractions and localisation. Exactness of localisation. of rings and modules. Local properties.

Integral elements, integral extension of rings and integral closure. Going Up, Going Down and geometric translation. Valuation rings. Overview of completions.

Chain conditions, Artinian and Noetherian rings and modules. Hilbert's basis theorem. Normalization Lemma and Nullstellensatz.

Discrete valuation rings. Fractional ideals and invertible modules. Cartier and Weil divisors, Picard group, cycle map. Dedekind domains and their extensions. Decomposition of ideals, inertia, ramification.

Krull dimension, height of a prime ideal. Principal ideal theorem. Characterisation of factorial domains. Regular local rings. Finiteness of dimension for local noetherian rings.

Planned learning activities and teaching methods

Lectures. Recommended exercises.

Additional notes about suggested reading

Lecture notes available at https://www.mathematik.uni-kl.de/~gathmann/de/commalg.php

Textbooks (and optional supplementary readings)

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