# Oldish and newish algebraic geometry texts

Since I’m not sure these links are well-known (i.e., I didn’t know them myself until yesterday in one case, and until a few weeks in the other), it may be useful to point them out:

* The work-in-progress new edition of SGA 3 (the part of Grothendieck’s algebraic geometry seminar dedicated to algebraic groups), which is edited by Philippe Gille and Patrick Polo; aficionados of the functorial point of view will enjoy it for remarks like:

Pour obtenir un langage qui “colle” sans effort à l’intuition géométrique, et éviter des circonlocutions insupportables à la longue, nous identifions toujours un préschéma X sur un autre S au foncteur
$(Sch)_S \rightarrow (Ens)$
qu’il représente, et il est nécessaire de donner de nombreuses définitions de telle façon qu’elles s’appliquent à des foncteurs quelconques, représentables ou non.

which I translate as follows:

To obtain a language which “sticks” effortlessly to geometric intuition, and avoid circumlocutions which quickly become unbearable, we always identify a prescheme X over another S with the functor
$(Sch)_S \rightarrow (Ens)$
it represents, and it is necessary to give many definitions in such a way that they apply to arbitrary functors, whether representable or not.

(A footnote says this point of view was first really considered by P. Cartier in the theory of formal groups).

* The Algebraic Stacks Project, which is a book-in-constant-progress about algebraic stacks. It is (as far as I know) the mathematical writing project which is closest in style to Open Source software. Indeed, although one can download a single PDF file with the current version, it also has distributed version control access (using Git), a maintainer (in the sense that Linus Torvalds is maintainer of the Linux kernel), who is A. J. de Jong, a free license (GNU Free Documentation License), complete history, and so on. Looking at the log entries shows that the project is fast moving: the currently 1298 pages long PDF version is just a convenience local in time (the project page indicates that no published version is projected to appear).

New contributors are encouraged to participate, and one can see that it would be quite easy to start by correcting a few typos, then looking at a part where one has particular expertise, and so on.

Thanks to the availability of source files, one can even (say) decide to “fork” the project, and create a new/derivative version where the set-theoretic foundations are done differently; or one where wimpy 2-categories are throughout upgraded to n-categories. More realistically, the license allows (I think: I haven’t read it in full detail) someone to produce a printed edition of a given — possibly edited — version (just like Linux distributions used to be available as physical boxes representing, in effect, a particular state of many independent software projects, before such distribution became almost non-existent — except maybe as DVDs included in books).

I haven’t yet looked much at the actual content of the book, but I must say I find quite appealing the idea of writing such a text this way. Maybe I’ll try to read/browse through a sufficient part to write some kind of review of its style.

### Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

## 4 thoughts on “Oldish and newish algebraic geometry texts”

1. Kevin Lin says:

Ens = Ensemble = the category of sets? Then the translation should be “Set” :)

2. Anonymous says:

On the other hand, I remember Arthur Ogus using “Ens” to denote the category of Sets because, as he said, it took fewer letters than “Sets.”

3. Anonymous2 says:

Well, in French they use R and L for right and left derived functors, so I don’t see anything wrong with us using Ens. Also, when you translate math, you generally don’t translate symbolic notations.