# Commutative Algebra

## Introduction

Commutative algebra studies commutative rings and their ideals. Its development has two sources: algebraic geometry and algebraic number theory, where the prototypes are systems of polynomial equations and ring of algebraic integers.

The study of commutative algebra dates back to the work of Kummer on Fermat Last Theorem, developed by Kronecker, Dedekind, Hilbert, Noether, Artin, Krull , … , and is highlighted later by the Grothendieck school. The importance of commutative algebra to algebraic geometry is like calculus to differential geometry.

This lecture serves as an introduction to commutative algebra, emphasising on its application to algebraic geometry. We plan to cover the following topics: Hilbert Nullstellensatz and algebraic sets; dimension theory; rings of small dimensions: discrete valuation rings (points), Dedekind rings (curves); binomial and monomial ideals; Gröbner basis. Tools like localisation, normalisation, primary decomposition will be developed along the way.

## ILIAS

The lecture (Vorlesung) and the exercise (Übung) will be on ILIAS:

## Informations

• Change of room: the Thursday lecture will be 14h-15h30 in Seminarraum 2.
• The exam will be on 10.07.2019 from 9.45 to 11.45, place to be announced.
• The lectures will be held on Wednesday 10h-11h30 in Cohen-Vossen Raum (3.13) and Thursday 14h-15h30 in Seminarraum 2. Office hour will be on Wednesday 14h-15h or by appointment.

## Contents

• Lecture 1 (03.04.2019): Introduction, rings.
• Lecture 2 (04.04.2019): Algebras, ideals.
• Lecture 3 (10.04.2019): Nullstellensatz, integral extension.
• Lecture 4 (11.04.2019): Modules.
• Lecture 5 (17.04.2019): Integral extension (cont.).
• Lecture 6 (18.04.2019): Normal ring, normalisation of cusp curve.
• Lecture 7 (24.04.2019): Noether normalisation, parameter space.
• Lecture 8 (25.04.2019): Proof of Nullstellensätze.
• Lecture 9 (02.05.2019): Prime spectrum.
• Lecture 10 (08.05.2019): Prime spectrum (cont.), arithmetic surface.
• Lecture 11 (09.05.2019): Zariski topology.
• Lecture 12 (15.05.2019): Noetherian rings and modules.
• Lecture 13 (16.05.2019): Hilbertscher Basissatz, Noetherian space.
• Lecture 14 (22.05.2019): Decomposition of ideals, localisation.
• Lecture 15 (23.05.2019):
• Lecture 16 (29.05.2019):
• Lecture 17 (05.06.2019):
• Lecture 18 (06.06.2019):
• Lecture 19 (19.06.2019):
• Lecture 20 (26.06.2019):
• Lecture 21 (27.06.2019):
• Lecture 22 (03.07.2019):
• Lecture 23 (04.07.2019): Revision.

## References

• [1]. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp.
• [2]. M. Atiyah; I. Macdonald, Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.
• [3]. J. Herzog; T. Hibi, Monomial ideals. Graduate Texts in Mathematics, 260. Springer-Verlag London, Ltd., London, 2011. xvi+305 pp.
• [4]. J. Herzog; T. Hibi; H. Ohsugi, Binomial ideals. Graduate Texts in Mathematics, 279. Springer-Verlag, 2018. xix+321 pp.

• Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln