OVERVIEW

The aim of this course is to introduce basic notions of commutative algebra, a discipline that flourished after the work of Hilbert and Dedekind at the end of the 19-th century, and developed during the 20-th century. Commutative algebra is largely concerned with the study of certain rings, called Noetherian, in which every ideal is finitely generated. The development of commutative algebra is often motivated by natural and simple-looking questions. Among them: what kind of factorization is available for Noetherian rings? What is the "correct" notion of dimension for a ring?

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to introduce some basic notions in commutative algebra such as localization, tensor product, Noetherian and Artinian modules, Krull dimension and integral dependence.

AIMS AND LEARNING OUTCOMES

The aims of the course are:

1. To introduce the student to some techniques which are typical of of commutative algebra such as localization, tensor product and extension/contraction of ideals along morphisms. 
2. To introduce Notherian and Artinian modules, and to develop the theory of associated primes of a module in relation to primary decomposition (a.k.a. the "substitute" of unique factorization) for ideals in Noetherian rings.
3. To characterize Artinian rings by introducing the length of a module and the Krull dimension of a ring; moreover, using such a characterization, the goal is to prove Krull's Height Theorem. 
4. To introduce the concept of integral dependence in order to compute the Krull dimension of algebras which are finitely generated over a field. 

The expected learning outcomes are:

1. A strengthening in the knowledge of basic algebraic notions thanks to the theoretical aspects developed in class and thanks to the discussion of assigned exercises; a good handling of standard techniques in commutative algebra.
2. By the end of Algebra Commutativa 1, students have learned the features of Noetherian rings, know the main features of primary decompositions of ideals, and know how to compute them in some notable cases (e.g. monomial ideals).
3. By the end of Algebra Commutativa 1 ], students have learned what Krull dimension is, and know ways to compute it in the case of local rings and in the case of algebras which are finitely generated over a field.

PREREQUISITES

Students are expected to be familiar with groups, rings, ideals, modules, algebras, quotients and homomorphisms.

TEACHING METHODS

Most of the lectures will be devoted to cover the theoretical part of the course; exercise sheets will be made available during the semester, and will be discussed collectively in the remaining hours. 

Attendance, even if not mandatory, is highly recommended. In fact, lectures are often a key tool to apprehend how the subject developed throughout the years, as well as the historic reasons that led to such a development.

SYLLABUS/CONTENT

The main topics are:

1. Rings, Ideal, modules and algebras. Rings and modules of fractions.
2. Chain conditions, Noetherian and Artinian rings.
3. Primary decomposition, associated primes.
4. Tensor products of modules, tensor algebra, symmetric algebra and exteriro algebra.
5. Rings of invariants, Reynolds operator, pure subrings, direct summands and algebra retracts.
6. Dimension theory. Krullhauptidealssatz and its variations. System of parameters and dimension of local rings.
7. Integral dependence, Noether normalization and dimension of finitely generated K-algebras.

RECOMMENDED READING/BIBLIOGRAPHY

• Notes (in Italian), Authors: E. De Negri and A. De Stefani (available on AulaWeb).
• Introduction to Commutative Algebra, Authors: Atiyah and MacDonald, Addison-Weysley 1969.
• Commutative Algebra with a View toward algebraic geometry, Author: D. Eisenbud, Springer 1994

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The dates of the academic calendar are available on the webpages of the Corso di Studi. Further information regarding the schedule is available here.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of an oral examination.

ASSESSMENT METHODS

The student will be evaluated on the theoretical aspects developed during the lectures and on the resolution of some exercises, typically along the lines of those discussed in class.

The assessment will be based both on the knowledge of the topics and on the ability to present them in a formal, coincise and correct way. 

The grade is based on the performance at the oral exam, and participation during the activities (in class lectures, exercise sessions) of the semester.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it