SUPERIOR ALGEBRA

Academic Year 2022/2023 - Teacher: Marco D'ANNA

Expected Learning Outcomes

The aim of this course is to deepen the study of commutative rings and their modules, with applications of topological methods to Multiplicative Ideal Theory. One of the goals of the course is to make the students improve their skill of provinding abstract arguments and learn that a deep theoretical knowledge allows to develop relevant applicative tools.

Course Structure

Lectures and exercises, given at the blackboard by the lecturer, and class exercises.

Required Prerequisites

Basic Commutative Algebra and General Topology.

Attendance of Lessons

Strongly recommended

Detailed Course Content

I. Modules. Free modules, flat modules, injective modules, projective modules. Examples and exercises.

II. Introduction to Multiplicative Ideal Theory. Valuation domains. Invertible ideals. Dedekind domains. Prufer domains. Krull domains. Examples and exercises.

III. Noetherian local algebra. Regular sequences, depth, Cohen Macaulay rings, ideal generated by system of parameters, type, Gorenstein rings.

Textbook Information

0. A. Geramita, C. Small, Introduction to homological methods in commutative rings, Queen's papers in pure and applied mathematics - n. 43

1. R. Gilmer, Multiplicative Ideal Theory. M. Dekker (1972).

2. A. Grothendiek, Éléments de géométrie algébrique I. Le langage des schémas. Publications Mathématiques de l'IHÉS, Volume 4 (1960).

3. I. Kaplansky, Commutative Rings. Allyn and Bacon, Inc. (1970).

4. L. Salce, L. Fuchs, Modules over Non-Noetherian Domains. Mathematical Surveys and Monographs AMS (2000).

5. O. Zariski, P. Samuel, Commutative Algebra, Volume II. Graduate Texts in Mathematics (1976).

6. Lecturer's notes.

Course Planning

	Subjects	Text References
1	Free, flat, injective and projective modules	4
2	Valuation domains, invertible ideals, Dedekind domains, Prufer domains.	1,3
3	Regular sequences, depth, Cohen Macaulay rings	0
4	Ideal generated by a system of parameters, type, Gorenstein rings	0

Learning Assessment

Learning Assessment Procedures

The students will receive some homework or exercises to solve directly in the classroom. At the end of the course there will be a final examination (written and/or oral).

Degree Course in

Mathematics