06689 - Commutative Algebra

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.

Prerequisites: Algebra 1 and Algebra 2

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. Commutative Algebra through Exercises, Springer, 2024

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.

Office hours

See the website of Jacopo Gandini