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06689 - Commutative Algebra

Academic Year 2024/2025

                
                        Docente:
                        Alessandro D'Andrea
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/02
                    
                        Language:
                        Italian
                    
                        Moduli:
                        
                            Alessandro D'Andrea
                            (Modulo 1)
                        
                            Roberto Pagaria
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    Traditional lectures (Modulo 1)
                                
                                    Traditional lectures (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 5827)

                            Online Lessons
                        
                            Teaching resources on Virtuale
                        
                                        Course Timetable
                                    
from Sep 18, 2024 to Dec 19, 2024

Learning outcomes

Successful students will understand the basic properties of modules (finitely generated, noetherian, artinian, injective, projective) over commutative rings and will know about localization of rings and modules.

They will be able to apply theoretical knowledge towards solving problems and managing proofs.

Course contents

Commutative rings: homomorphisms and homomorphism theorems; (principal, prime, maximal) ideals; radical ideals (nilradical, Jacobson radical); zero divisors, nipotent and invertible elements.

Chain conditions (noetherianity, artinianity, composition series).

Localization of rings and modules.

Lying over, going up, going down theorems.

Basics of dimension theory.

Readings/Bibliography

M. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Reading Massachusetts, 1969.

Altman, Allen & Kleiman, Steven. (2013). A term of Commutative Algebra (https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf ).

A. Gathmann (2013). Commutative algebra (https://agag-gathmann.math.rptu.de/class/commalg-2013/commalg-2013.pdf ).

A. Bandini, P. Gianni, E. Sbarra. Esercizi di Algebra commutativa, Pisa University Press, 2022.

Inoltre,

D.G.Northcott: Ideal Theory Cambridge University Press, Cambridge 1953

D.Cox - J.Little - D.O'Shea: Ideals, Varieties and Algorithms. 4th Ed. Undergraduate Texts in Mathematics. Springer Verlag, New York 2007

Teaching methods

Chalk and blackboard. Groupwork. Office hours.

Assessment methods

Take home exam at the end of term.

Written and oral exam.

Teaching tools

Blackboards, printed handouts. Possible use of algebraic symbolic computation software.

Office hours

See the website of Alessandro D'Andrea

See the website of Roberto Pagaria