My Books in Progress

I am writing two books: First, a self-contained introductory book in algebraic geometry and invitation to equivariant geometry aimed at both graduate students and researchers, and second a book introducing basic category theory.

The goal for the first book is to introduce algebraic geometry from a categorical perspective and embrace the categories and sheaves as the geometric objects they are. The big benefit to students and researchers here is that the book not only emphasizes descent-theoretic techniques, but also is completely self-contained (so while everyone should go check out a tonne of different introductory books, having to only check one resource is really quite nice).  A particular focus in the writing and way the material is developed lies in the perspective that the geometry of a scheme or of an object is really recorded by the sheaves defined on/over the objects at hand. So far there are six or so main chapters with a few appendices roughly organized as follows:

1. This is a motivational chapter and nothing more. It motivates why we may want to study equivariant algebraic geometry and why the question of ``what is the equivariant derived category'' is actually quite interesting and subtle.
2. This is a chapter introducing sheaves on a topological space, ringed spaces, and locally ringed spaces. A subsection is devoted also to stalks and skyscraper sheaves.
3. This long chapter introduces schemes and varieties. The construction is done ``from the ground up'' and uses the locally ringed space perspective so that readers can go from reading this to reading other references in algebraic geometry immediately. Special attention is paid to the various category-theoretic properties one can put on schemes and the relationships between the various categories of schemes that arise in practice.
4. This chapter introduces group objects and equivariant objects within a category as well as what it means for a sheaf to be equivariant. While still under construction, eventually there will also be a discussion of the issue regarding the existence of quotients of schemes by group schemes.
5.  This chapter introduces quasi-coherent sheaves on a scheme and some of the subtleties that arise in their study.
6. This chapter introduces the etale topology on a scheme as well as etale sheaves on a scheme. Eventually I will write about the category of l-adic sheaves here, but that remains to be done.
7. This chapter introduces the category of smooth varieties and some of the basics regarding smooth morphisms and algebraic groups. So far there is essentially nothing in this chapter
8. An introduction to equivariant algebraic geometry. Needs to be redone based on some ongoing research work I have in progress regarding pseudocones and instead turned into a chapter on two-dimensional category theory.
9. An introduction to equivariant categories on varieties.
10. An introduction to equivariant functors between equivariant categories.
11. An introduction to equivariant triangulated categories.
12. Appendix A: An introduction/recollection on the basics of sites and sheaves.
13. Appendix B: An introduction/recollection on the basics of localizations of categories and left/right calculi of fractions.
14. Appendix C: A universal (no set theory!) introduction to additive and Abelian categories.
15. Appendix D: A universal introduction to cohomology and derived functors.
16. Appendix E: A discussion of internal categories, internal groupoids, and stackification via torsors.

This is a definite work in progress, but I have placed a copy of the book as it is at the moment here so that people can take a look and let me know what they think as they take a glance. I have tried to include many exercises and stuff like that. As of the moment, Chapters 1 -- 3 (modulo adding exercises to Chapter 3), most of Chapter 4, the first half of Chapter 5, the first chunk of Chapter 6, Chapters 9 -- 11 (modulo exercises), as well the appendices (modulo some of Appendix D.3) are complete.

• The Algebraic Geometry Book, last updated on: 2023 July 21, 23:17 Atlantic Daylight Time.

The goal for the second book is simply to be an introductory book on category theory. So far it serves as precisely that save for the fact that the chapter on monads is incomplete and some of the specialized topics (sheaves and two-dimensional category theory) are missing. Note that some of the chapter titles and the like are inaccurate, as this was originally going to be an algebraic geometry book (the algebraic geometry book became the project above while the category theory notes remain here). There are over 200 exercises in this book so far.

• The Category Theory Book, last updated on: 2023 September 19, 18:22 Atlantic Daylight Time