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Mathematics

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SUPERIOR ALGEBRA

Academic Year 2024/2025 - Teacher: CARMELO ANTONIO FINOCCHIARO

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

The course will consist of lectures and exercises, given at the blackboard by the lecturer, and class exercises. Usually the lecturer alternates exercises and theoretical parts in the same lecture. As for class exercises, the lecturer gives some questions to the students, that have to try to solve them by working in small groups; the lecturer helps the students to find the proper way to approach the exercises, by providing hints and useful observations.


Detailed Course Content

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

II. Topological rings. Topologies on a ring. Completions. Hensel's Lemma. Examples and exercises.

III. The prime spectrum of a ring. Zariski topology, constructible topology, inverse topology. Topological properties of the prime spectrum of a ring. Examples and exercises.

IV. Spectral spaces. Topological characterization for the topological spaces that are homeomorphic to the prime spectrum of a ring. Examples and exercises.

V. Introduction to Multiplicative Ideal Theory . Invertible ideals. Dedekind domains. Prufer domains. Ideal class group. Elements of algebraic number theory. Examples and exercises.

VI. Riemann-Zariski spaces. Zariski topology on spaces of valuation domains. Riemann-Zariski spaces are spectral. Examples and exercises.

Textbook Information

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. Lecture notes.