EGA Chapter 1: The Language of Schemes | Hacker News Reader

EGA Chapter 1: The Language of Schemes

golem.ph.utexas.edu

51 pointschmaynard27 comments

Some background, for those unfamiliar with this area. Algebraic geometry grew out of the study of solutions of systems of polynomial equations. This is naturally a question of geometry, which can be studied analytically (using tools from calculus, for example). If you are willing to consider solutions over the complex numbers, and not just the real numbers, you can study the question algebraically, as well. Once you take that step, you can ask what happens when you look at solutions over exotic objects such as finite fields or rational functions.

Alexander Grothendieck pushed for a very general formulation of this algebraic approach, using the notion of "scheme". Together with Jean Dieudonne, he wrote an introduction to this topic, called EGA, which is French for "Elements of Algebraic Geometry". This is one of the foundational texts in algebraic geometry now.

At some point, Grothendieck became disillusioned with both mathematics and academia, quit both, and isolated himself from most people. Not long before his death, he expressed the desire that all of his work cease to be published. He died 5 years ago now, and I'm not actually sure of the legal status of republishing his work, though people are determined to go ahead because of its great historical significance. While I think EGA has largely been superseded by newer sources, some of his later work (known as SGA) has not.

I'd just like to point out that Grothendieck's notion of a scheme is simpler and more natural than other notions that were floating around at the same time and attempting to capture similar ideas.

Why did mathematicians seek a general formulation of geometry (in fact, schemes assign a "natural" geometric interpretation to every commutative ring)?

Well, the theory of schemes, has allowed number theoretic questions to be interpreted geometrically. An important result along these lines, that was proven only after the theory of schemes was made available, is an analogue of the Riemann hypothesis for finite fields. This result was famously proven by one of Grothendieck's students (Pierre Deligne).

>While I think EGA has largely been superseded by newer sources, some of his later work (known as SGA) has not

Can you elaborate what you mean by “superseded by”?

I just mean that the content can be found from later sources, such as Hartshorne or the Stacks Project. For some parts of SGA, it is my impression that literally the only reference is SGA.

A question to the experts: Of EGA I, there exist two editions:

1. the 1960 edition, which is available at http://www.numdam.org/item/?id=PMIHES_1960__4__5_0

2. the 1971 edition: mentioned under https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9o...

As I see it, this translation seems to be of the (older) 1960 edition. Does anybody know the reason why the translators chose the 1960 edition over the 1971 edition as the translation source?

Grothendieck left IHES in 1970, the second edition was after that. There was a lot of drama in the aftermath, essentially Grothendieck was not happy about how the more advanced material (SGA 4) he lectured on was kept private to the small group of students that attended the seminar and later reappeared in their publications without much acknowledgement. The eventual SGA 4.5 was a butchered version that only contained enough background to cover Deligne‘s proof.

Did Grothendieck himself not explicitly object to any re-publishing or translation of his manuscripts?

https://tqft.net/misc/Grothendieck's%20Declaration.pdf

``` Declaration of intent of non-publication

I do not intend to publish or republish any work or text of which I am the author, in any form whatsoever, printed or electronic, whether in full or in excerpts, texts of personal nature, of scientific character, or otherwise, or letters addressed to anybody, and any translation of texts of which I am the author. Any edition or dissemination of such texts which have been made in the past without my consent, or which will be made in the future and as long as I live, is against my will expressly specified here and is unlawful in my eyes. As I learn of these, I will ask the person responsible for such pirated editions, or of any other publication containing without permission texts from my hand (beyond possible citations of a few lines each), to remove from commerce these books; and librarians holding such books to remove these books from those libraries.

If my intentions, clearly expressed here, should go unheeded, then the shame of it falls on those responsible for the illegal editions, and those responsible for the libraries concerned (as soon as they have been informed of my intention).

Written at my home, January 3, 2010,

Alexandre Grothendieck. ```

They acknowledge this in the document:

"If it seems like we are over-eager to defend ourselves because we know that we are somehow in the wrong, it is because we are, at least partially. Working on this translation has meant going against Grothendieck’s explicit requests, and for that we are sorry. We only hope that the freedom of knowledge is an excusable defense."

´...or which will be made in the future and as long as I live,...´

There comes a point at which a book belongs more to its readers than to its author. EGA is well past that point. Try imagining the world we'd live in now if Euclid had managed to suppress all copies of his _Geometry_.

> There comes a point at which a book belongs more to its readers than to its author. EGA is well past that point. Try imagining the world we'd live in now if Euclid had managed to suppress all copies of his _Geometry_.

Tell this to the politicians who created the copyright laws.

> Tell this to the politicians who created the copyright laws.

I am sure Grothendieck would be on the front lines of those castigating such politicians. He made this request not to ensure personal gain, but to express his profound disappointment with and disengagement from society as he saw it.

This guy also hosts a large archive of Grothendieck's work, scans people have made and sent me over the years, etc.: https://wstein.org/sga/

It’s OK.

I have to just lightly wonder why he would have written anything, if he was so adamantly opposed to any copies in any form.

Unless I misread that and he only intended to restrict copies "without my consent". Hence as broad and varied distribution as he solely wished, and no others.

I think I read that initially as a fear of being misunderstood, since every new copy would have a non-zero chance of typos or changes. Or I have that entirely wrong, which is the more likely case.

Curious.

This was written well after he had done the work for which he is remembered. See https://en.wikipedia.org/wiki/Alexander_Grothendieck#Retirem... . That is, at the time the work was done, it was not his intention to limit copying.

Did we really just get an English translation now?

There's a lot of algebraic geometry literature that's in French, so many algebraic geometers just learn to read math in French (it's considerably easier than reading French literature).

The main problem has not been the language it was written in. It's hard going mathematically as well (for average graduate students at least). A few people have started translation projects, but gave up after realising that by the time they had translated more than a few pages, the French was actually pretty easy to follow. The maths, however, was not so easy.

It is really nice to see someone finally translated more than a whole chapter though.

Apparently Grothendiek stated he didn't want any translations as long as he lived. He died 5 years ago.

He didn't want his works republished at all; it wasn't an issue of translation.

Math texts in French are easy to understand, as long as you learn a few unfamiliar words, like ‘lorsque,’ par exemple.

The README of the Github repo details their progress in the translation:

https://github.com/ryankeleti/ega

It seems they're done with EGA 0_I, and EGA I, but they're still working on EGA 0_III, EGA 0_IV, EGA II, EGA III, EGA IV.

I wonder why there isn't an EGA 0_II mentioned...

What is EGA? (Enhanced Graphics Adapter? Extended Graphics Adapter?)

I clicked on the link assuming that there would be some kind of explanation, but the page itself doesn't make much sense unless you already know a rather significant amount of context.

> What is EGA?

Éléments de géométrie algébrique

https://en.wikipedia.org/w/index.php?title=%C3%89l%C3%A9ment...

Thank god it’s not GA (not again).

I really like how the current top two comment threads need to be glued together in order to get a coherent understanding of the timeline of this work.

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