Welcome to Algebraic Geometry and Number Theory Essentials, a reading course offered as part of the Decoding Dimensions program on Algebraic Geometry and Number Theory.
The goal of this course is to develop the algebraic and categorical background needed for further study in Algebraic Geometry and Number Theory, in particular for later courses on scheme theory and class field theory.
The course is not intended to give a comprehensive treatment of abstract algebra, commutative algebra, category theory, or Galois theory. Instead, we will focus on the ideas and constructions that will be used repeatedly later on. Some of the material will already be familiar to participants from previous algebra courses. In such cases, the aim will be less to repeat the theory from scratch and more to develop fluency with the relevant constructions and techniques.
The class is scheduled for Wednesday, from 4 pm to 6:00 pm (Tehran time).
I will be sharing the weekly reading material and exercises on the News page. All class-related communication and updates will also be posted there. So, please stay connected…
Main references
We will not follow a single textbook from beginning to end. Instead, selected portions of the following books and notes will form the main reading material:
- T. Leinster, Basic Category Theory
for the category-theoretic foundations. - A. Gathmann, Commutative Algebra
for the algebra and commutative algebra portions of the course. - P. Morandi, Field and Galois Theory
for fields and Galois theory.
The precise selection of topics and chapters will be announced as the course progresses.
Additional references
The following sources provide alternative treatments, additional examples and exercises, and material for further study. We may occasionally refer to some of them or use selected exercises from them:
Category Theory:
- E. Riehl, Category Theory in Cotext – highly recommended.
- H. Simmons, An Introduction to Category Theory
- A. Agore, A First Course in Category Theory
- S. MacLane, Categories for the Working Mathematician – classic; more advanced.
- Stacks Project, {0011} Categories – reference.
Algebra and Commutative Algebra:
- A. Knapp, Basic Algebra
- R. Y. Sharp, Steps in Commutative Algebra – especially useful for a more detailed treatment and additional exercises.
- A. Altman and S. Kleiman, A Term of Commutative Algebra
- M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra
- P. Aluffi, Algebra: Chapter 0 — particularly useful for its categorical viewpoint on algebra.
- S. Bosch, Algebraic Geometry and Commutative Algebra – more advanced.
- D. Eisenbud, Commutative Algebra, with a view Toward Algebraic Geometry – more advanced.
- E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry
- H. Matsumura, Commutative Ring Theory – more advanced.
- Stacks Project, {00AO} Commutative Algebra – reference.
In previous years, we studied J. Milne‘s lecture notes A Primer on Commutative Algebra and The CRing Project.