# INSTITUTIONS OF HIGHER ALGEBRAModule MODULO I

Academic Year 2023/2024 - Teacher: Marco D'ANNA

## Expected Learning Outcomes

The aim of this course is to deepen the study of coomutative ring theory, taking particular attention to polynomial rings and their quotients, with a view towards applications to allgebraic geometry and number theory.

The student will also improve his skill of making abrstact argomentations and will learn that a deep theoretical knowledge allows to develop signifinant applicative tools.

## Course Structure

In the course the will be lectures and exercises, given at the blackboard by the lecturer, and class exercises. Ussually the lecturer alternates exercises and theoretical parts in the same day. As for class exercises, the lecturer gives some exercises to the students, that have to try to solve them working in small groups; the lecturer helps the students to find the proper way to appoach the exercises.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

## Detailed Course Content

I. Rings and ideals. Definitions and first properites. Prime and maximal ideals. Local rings. Nilradical and Jacobson radical. Ideals operations.Homomorphisms.

II. Modules. Definitions and first properties. DIrect product and direct sum; free modules. Finite modules and Nakayama's lemma. Module homomorphisms. Algebras.

III. Frctions rings and modules. Definition and properties. Localization and local properties. Ideals in fraction rings.

IV. Noetherian rings. Affine varieties, affine K-algebras, correspondance betwwen algebra and algebraic-geometry concepts. Krull dimension. Noetherian rings and modules: definitions and first properties. Hilbert's basis theorem. Conditions for a sub-algebra to be finitely generated.

V. Artinian rings. Artinian rings and modules. Composition series. Length. A ring is artinian if and only if it is noetherian and zero-dimensional.

VI. Primary decomposition. Primary ideals and primary decomposition. Associated primes and their characterization. Zero-divisors. Unicity of isolated components. The noetherian case.

VII. Hilbert Nullstellensatz: weak and strong formulations.

VIII. Integral dependance. Definitions and first properties. Going Up theorem. Normal domains and Going Down theorem. Noether's normalization lemma.

IX. First steps in dimension theory. Chain of primes, height, dimension. Krull's principal ideal theorem.Krull's height theorem.Dimension for polynomial rings with coefficient in a field. Local rings. System of parameters. Embedding dimension. Regular local rings (only definition and geometric relevance).

## Textbook Information

1. M.F. Atiyah, I.G. Macdonald , Introduzione to commutative algebra.

2. E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkauser 1985

## Course Planning

SubjectsText References
1Anelli e ideali: prime proprietà degli anelli commutativi con unità. Ideali primi e ideali massimali. Anelli locali. Nilradicale e radicale di Jacobson.1
2Operazioni con gli ideali; radicale di un ideale. Omomorfismi. Ideali estesi e ideali contratti.1
3Moduli: definizione e prime proprietà. Prodotto diretto e somma diretta: moduli liberi. Moduli finitamente generati e lemma di Nakayama. Omomorfismi tra moduli. Algebre.1
4Anelli e moduli di frazioni: definizione e proprietà. Localizzazione e proprietà locali. Ideali negli anelli di frazioni.1
5Varietà affini, K-algebre affini e dizionario di base algebra-geometria algebrica. Dimensione di Krull. 2
6Anelli e moduli noetheriani: definizioni e prime proprietà. Il teorema della base di Hilbert. Condizioni perché una sottoalgebra sia finitamente generata.1
7Anelli e moduli artiniani. Serie di composizione. Lunghezza. Un anello è artiniano se e soltanto se è noetheriano e ha dimensione zero.1
8Ideali primari; decomposizione primaria. Primi associati e loro caratterizzazione. Divisori dello zero. Unicità delle componenti isolate. Il caso noetheriano.1
9Teorema degli zeri di Hilbert: forma debole e forma forte.1 oppure 2
10Dipendenza integrale: definizioni e prime proprietà. Teorema del Going Up. Domini normali e Teorema del Going Down.1
11Lemma di normalizzazione di Noether.2
12Catene di primi, altezza, dimensione. Teorema dell'ideale principale di Krull. Teorema dell'altezza di Krull. 2
13Dimensione degli anelli di polinomi a coefficienti in un campo. Anelli locali. Sistema di parametri. Dimensione di immersione. Anelli locali regolari (solo definizione e importanza geometrica).2