Preliminary program alggeo24
Table of contents
Welcome to a small preliminary program concerning a reading course in algebraic geometry. Here you will find all information on when, how and if we meet, as well as some more material, such as a small captains log as well as literature. One of course may ask why I made a website for this. The answer is simple: I was bored.
Organization
We will hold meetings in persona as well as by Zoom.
The first meeting will be on 13.02.2024 at 16:00 via
| In persona | Zoom |
|---|---|
| Hubland Nord (31.00.017) | link gone |
Afterwards we meet twice a week on Tuesdays and Thursdays again at 16:00, same place/link as above. This is of course not fix. We can change this later. Maybe we also go to 18:00.
Zoom links, telegram links and other more private stuff will be found in link gone.
Schedule
This is a preliminary schedule which will be extended with time:
| Part | Description |
|---|---|
| 1 | Organizational stuff |
| 2 | p 1-26 in Görtz, Wedhorn, Algebraic geometry 1, Affine varieties as spaces with functions. Some basic properties of the Zariski topology, Ideals of polynomial rings, localization in rings, Hilbert Nullstellen and Basis theorem as well as Noether normalization Lemma |
| 3 | p 27-46 in Görtz, Wedhorn, Algebraic geometry 1. Definition of a projective Variety and the Spectrum of a ring as a topological space. Motivation for the use of sheaves. |
| 4 | p 68-76 in Vakil, The rising sea. Definitions of sheaves and presheaves. |
| 5 | p 77-87 in Vakil, The rising sea. Morphisms of sheaves and presheaves, all properties can be checked on stalks, Sheafification, Sheaf on a base |
| 6 | p 88-94 in Vakil, The rising sea to finish up part 1 with some sheaves of abelian groups and inverse image sheaves. |
| 7 | p 95-148 in Vakil, The rising sea, Chapter 3 deals with the definition of the base topological space of an affine scheme and some topological properties. Chapter 4 deals with the definition of affine, projective and general schemes. |
| 8 | p 106-155 Chapter 3 in MacLane, Moerdijk, Sheaves in Geometry and Logic. Here one first introduces the notion of a topology on a small category. For this one considers sieves on one hand and coverings on the other, ultimately both yield the same definition of a sheaf on a site. This then leads to the definition of a Grothendieck-Topos. |
Literature
Our main reference for a classical approach to algebraic geometry is Görtz, Wedhorn, Algebraic geometry 1. Here we will look at chapters 1 and 2 concerning prevarieties, spectra of rings and in general zero-sets of polynomials in $n$-variables.
For the rest of the course I plan to look at Vakil, The rising sea which favours the opposite approach. Here we begin by looking at sheaves in an algebraic way and use this knowledge to define schemes.
An other reference, which I do not plan on reading, is Hartshorne, Algebraic Geometry. This is a very densely written compendium and not much of a textbook for novices in the field (at least in my opinion).
One of my favourite references for commutative algebra is Atiyah, MacDonald, Introduction to commutative algebra. Here we can find all that we need concerning basic algebraic vocabulary dealing with rings and their ideals.
For diagram lemmata and abelian category theory my main reference is the great book Freyd, Abelian categories. I do not think we will need this often but it is also a nice read in of itself.
For the recent stuff of our endeavour will refer to MacLane, Moerdijk, Sheaves in Geometry and Logic which is our main reference on sheaves, sites and topoi.
For other topics in category theory my main reference are the three books by Francis Borceux called “Handbook of categorical algebra”. I do not think we will need this.
This will hopefully be sufficient, but we can add more resources if needed.
Goals
My personal goal is to understand schemes in sufficient detail to continue on reading about quasicoherent schemes. But officially my goal for the reading course is Chapter 8 in Vakil, The rising sea. At this point we will have learned about sheaves, schemes and especially their morphisms (in any sensible mathematics one considers morphisms of the objects).
We then turn to some more categorical generalizations of sheaves: namely topoi. For this our main reference is MacLane, Moerdijk, Sheaves in Geometry and Logic. Here we talk about some basic definitions and continue to look at classifying topoi and other interesting stuff and I ultimately hope to understand stacks in some detail.
Material
I plan on collecting some exercises for a given chapter and their solutions. We can hopefully solve these exercises together at one of our meetings. I think it will be very important to actually do these exercises in detail, as category theory is very unwieldy otherwise. If a proof is very confusing and unclear I will also try to write it down in more detail.
For a collection of all notes see Notes
For a collection on some topoi stuff and maybe some stacks later, see Stacks and stuff
For a collection of all exercises see Exercises