Algebraic Geometry Codes

Spring 2018

Other notes

Below are my handwritten notes in case you find them useful. These notes are meant for my own use and so they are not complete and don't have a "soul" (unlike, hopefully, the lectures themselves). I'm mostly following Lorenzini's truly marvellous book An Invitation to Arithmetic Geometry and so you can see more details there.

Chapter 0 - Why bother?

  1. Goppa codes

  2. What we'll learn (and what will have to wait for the followup course)

Chapter 1 - Introduction to algebraic curves

This chapter closely follows Lorenzini 2.1-2.5. 

  1. Introduction

  2. Rings of polynomial functions

  3. Points and maximal ideals (aka Hilbert's Nullstellensatz)

  4. Singular points

Chapter 2 - Commutative algebra and a tiny tiny bit of Galois theory

But for (1), (3), this chapter is closely follows Lorenzini, Chapter 1 and 2.6 

  1. Modules

  2. Integral elements

  3. Field embeddings and separable extensions

  4. Noetherian rings

  5. Dedekind domains

  6. Localization

Chapter 3 - Algebraic curves continued

This chapter closely follows Lorenzini 2.7-2.10 

  1. Localization and plane curves

  2. Singularity and local PID

  3. Localization take 2 - localization of modules

  4. Hilbert's basis theorem

Chapter 4 - Factorization of ideals

This chapter closely follows Lorenzini 3.1-3.8

  1. A worked out example

  2. Unique factorization of ideals and Dedekind domains

  3. Ramification index, residual degree, and the fundamental equality

  4. Explicit factorization in Dedekind domains obtained via a monic polynomial

  5. Ramification

  6. Simple sufficient conditions for Dedekind-ness

  7. Factorization in Artin-Schreier extensions and Kummer extensions

  8. A tiny bit more on Galois theory

  9. Factorization in Galois extensions

Chapter 5 - Valuations

This chapter closely follows Lorenzini 4.2,4.6,5.2,5.3,5.6,5.8.

  1. Norm, trace and the discriminant of a polynomial

  2. The ideal norm map

  3. The ideal class group

  4. Rings with finite quotients

  5. Absolute values and valuations

  6. Discrete valuation rings

Chapter 6 - Projective curves and Nonsingular complete curves

  1. Nonsingular complete curves

  2. Affine curves and complete curves

  3. The projective plane and projective curves

  4. Functions on the projective curve

  5. Projective curves and valuations