# Commutative AlgebraSS 2023

Dietrich Burde

Lectures: Monday and Thursday 11:30 - 13:00 in Seminarraum 12

• Commutative algebra studies commutative rings, their ideals, and modules over such rings. It has a long and fascinating history, and it is also a fundamental basis for algebraic geometry, invariant theory and algebraic number theory. In the second half of the 19th century, two concrete classes of commutative rings (and their ideal theory) marked the beginning of commutative algebra: rings of integers of algebraic number fields, on the one hand, and polynomial rings occurring in classical algebraic geometry and invariant theory, on the other hand. In the first half of the 20th century, after the basics of abstract algebra had been established, commutative algebra was developed further by E. Noether, E. Artin, W. Krull, B. L. van der Waerden, and others. This was applied in the 1940's to classical algebraic geometry by C. Chevalley, O. Zariski, and A. Weil, creating a revolution in this field. The 1950's and 1960's saw the development of the structural theory of local rings, the foundations of algebraic multiplicity theory, Nagata's counter-examples to Hilbert's 14th problem, the introduction of homological methods into commutative algebra, and other pioneering achievements. However, the most important mark of this period was A. Grothendieck's creation of the theory of schemes, the (till now) ultimate revolution of algebraic geometry. His foundational work lead to a far-reaching alliance of commutative algebra and algebraic geometry.

Nowadays also many computational methods have been developed, and the field of computer algebra is of growing interest. In particular we mention Groebner bases and its applications.

• Here is a syllabus and a bibliography available.

• Exercises: This lecture has no exercise class, but I have collected some exercises below, which might be interesting.

Exam 2023: Planned 29th of June 2023, 11:30 - 12:45.

## pdf-files:

 No. Topic Date pdf-file 1 Lecture notes 2023 commutative_algebra.pdf 2 Exercises 2023 exercises_cca.pdf

Topics for the exam:

1. Basic notions and examples of commutative rings.
2. Localizations
3. Noetherian rings
4. Affine algebraic sets and Zariski-Topology
5. Gröbner Bases
6. Module Theory
7. Integral ring extensions
8. Dedekind rings and DVRs (discrete valuation rings)

Dietrich Burde