Commutative Algebra

A.Y. 2021/2022
9
Max ECTS
73
Overall hours
SSD
MAT/02
Language
Italian
Learning objectives
The main task is to give an introduction to modern commutative algebra with a special regard to commutative ring theory, arithmetic, homological methods and algebraic geometry.
Expected learning outcomes
(first part) Theory and computations of primary decompositions, integral extensions, regular rings & a first step in dimension theory. (9 credits) The additional 3 credits course is providing the next key step in dimension theory.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
More specific information on the delivery modes of traininig activities for the academic year 20021/2022 will be provided in the next few months, based on the evolution of the public health situation.
Prerequisites for admission
The contents of teh courses Algebra 1, 2, 3 ,4.
Assessment methods and Criteria
Homeworks, written exam, oral exam.
Commutative Algebra (first part)
Course syllabus
First steps in dimension theory in the sense of Krull and via trascendence theory, examples in dimension 0 and 1, integral extensions, going up theorem, Noether normalization lemma, regular rings, Kahler differentials, finite ètale extensions, covers.
Teaching methods
Traditional blackboard lectures, exercise sessiones.
Teaching Resources
Notes of the course (available on Ariel).
"Galois Theory for schemes", Notes by H. Lenstra, available online.
Commutative Algebra mod/2
Course syllabus
Perfectoid algebras and applications, especially Hochster's direct summand conjecture
Teaching methods
Traditional blackboard lectures.
Teaching Resources
Yves André , "La conjecture du facteur direct"
Publications mathématiques de l'IHÉS n. 127, 71-93 (2018)
Commutative Algebra (first part)
MAT/02 - ALGEBRA - University credits: 6
Practicals: 24 hours
Lessons: 28 hours
Commutative Algebra mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours