Tag: Chevalley

The birthplace of schemes

Published August 11, 2022 by lievenlb

“The word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski’s ideas.”

and refers to the lecture by Chevalley ‘Les schemas’, given on December 12th, 1955 at the ENS-based ‘Seminaire Henri Cartan’ (in fact, that year it was called the Cartan-Chevalley seminar, and the next year Chevalley set up his own seminar at the ENS).

Items recently added to the online Bourbaki Archive give us new information on time and place of the birth of the concept of schemes.

From May 30th till June 2nd 1955 the ‘second caucus des Illinois’ Bourbaki-congress was held in ‘le grand salon d’Eckhart Hall’ at the University of Chicago (Weil’s place at that time).

Only six of the Bourbaki members were present:

Jean Dieudonne (then 49), the scribe of the Bourbaki-gang.
Andre Weil (then 49), called ‘Le Pape de Chicago’ in La Tribu, and responsible for his ‘Foundations of Algebraic Geometry’.
Claude Chevalley (then 46), who wanted a better, more workable version of algebraic geometry. He was just nominated professor at the Sorbonne, and was prepping for his seminar on algebraic geometry (with Cartan) in the fall.
Pierre Samuel (then 34), who studied in France but got his Ph.D. in 1949 from Princeton under the supervision of Oscar Zariski. He was a Bourbaki-guinea pig in 1945, and from 1947 attended most Bourbaki congresses. He just got his book Methodes d’algebre abstraite en geometrie algebrique published.
Armand Borel (then 32), a Swiss mathematician who was in Paris from 1949 and obtained his Ph.D. under Jean Leray before moving on to the IAS in 1957. He was present at 9 of the Bourbaki congresses between 1955 and 1960.
Serge Lang (then 28), a French-American mathematician who got his Ph.D. in 1951 from Princeton under Emil Artin. In 1955, he just got a position at the University of Chicago, which he held until 1971. He attended 7 Bourbaki congresses between 1955 and 1960.

The issue of La Tribu of the Eckhart-Hall congress is entirely devoted to algebraic geometry, and starts off with a bang:

“The Caucus did not judge the plan of La Ciotat above all reproaches, and proposed a completely different plan.

I – Schemes
II – Theory of multiplicities for schemes
III – Varieties
IV – Calculation of cycles
V – Divisors
VI – Projective geometry
etc.”

In the spring of that year (February 27th – March 6th, 1955) a Bourbaki congress was held ‘Chez Patrice’ at La Ciotat, hosting a different group of Bourbaki members (Samuel was the singleton intersection) : Henri Cartan (then 51), Jacques Dixmier (then 31), Jean-Louis Koszul (then 34), and Jean-Pierre Serre (then 29, and fresh Fields medaillist).

In the La Ciotat-Tribu,nr. 35 there are also a great number of pages (page 14 – 25) used to explain a general plan to deal with algebraic geometry. Their summary (page 3-4):

“Algebraic Geometry : She has a very nice face.

Chap I : Algebraic varieties
Chap II : The rest of Chap. I
Chap III : Divisors
Chap IV : Intersections”

There’s much more to say comparing these two plans, but that’ll be for another day.

We’ve just read the word ‘schemes’ for the first (?) time. That unnumbered La Tribu continues on page 3 with “where one explains what a scheme is”:

So, what was their first idea of a scheme?

Well, you had your favourite Dedekind domain $D$, and you considered all rings of finite type over $D$. Sorry, not all rings, just all domains because such a ring $R$ had to have a field of fractions $K$ which was of finite type over $k$ the field of fractions of your Dedekind domain $D$.

They say that Dedekind domains are the algebraic geometrical equivalent of fields. Yeah well, as they only consider $D$-rings the geometric object associated to $D$ is the terminal object, much like a point if $D$ is an algebraically closed field.

But then, what is this geometric object associated to a domain $R$?

In this stage, still under the influence of Weil’s focus on valuations and their specialisations, they (Chevalley?) take as the geometric object $\mathbf{Spec}(R)$, the set of all ‘spots’ (taches), that is, local rings in $K$ which are the localisations of $R$ at prime ideals. So, instead of taking the set of all prime ideals, they prefer to take the set of all stalks of the (coming) structure sheaf.

But then, speaking about sheaves is rather futile as there is no trace of any topology on this set, then. Also, they make a big fuss about not wanting to define a general schema by gluing together these ‘affine’ schemes, but then they introduce a notion of ‘apparentement’ of spots which basically means the same thing.

It is still very early days, and there’s a lot more to say on this, but if no further documents come to light, I’d say that the birthplace of ‘schemes’, that is , the place where the first time there was a documented consensus on the notion, is Eckhart Hall in Chicago.

The (somewhat less) Secret Bourbaki Archive

Published July 14, 2022 by lievenlb

It has been many, many years since I’ve last visited the Bourbaki Archives.

The underground repository of the Bourbaki Secret Archives is a storage facility built beneath the cave of the former Capoulade Cafe. Given its sporadic use by staff and scholars, the entire space – including the Gallery of all intermediate versions of every damned Bourbaki book, the section reserved to Bourbaki’s internal notes, such as his Diktats, and all numbers of La Tribu, and the Miscellania, containing personal notes and other prullaria once belonging to its members – is illuminated by amber lighting activated only when movement is detected by strategically placed sensors, and is guarded by a private security firm, hired by the ACNB.

This description (based on that of the Vatican Secret Archives in the book The Magdalene Reliquary by Gary McAvoy) is far from the actual situation. The Bourbaki Archive has been pieced together from legates donated by some of its former members (including Delsarte, Weil, de Possel, Cartan, Samuel, and others), and consist of well over a hundredth labeled carton and plastic cases, fitting easily in a few standard white Billy Ikea bookcases.

The publicly available Bourbaki Archive is even much smaller. The Association des collaborateurs de Nicolas Bourbaki has strong opinions on which items can be put online. For years the available issues of La Tribu were restricted to those before 1953. I was once told that one of the second generation Bourbaki-members vetoed further releases.

As a result, we only had the fading (and often coloured) memories of Bourbaki-members to rely on if we wanted to reconstruct key events, for example, Bourbaki’s reluctance to include category theory in its works. Rather than to work on source material, we had to content ourselves with interviews, such as this one, the relevant part starts at 51.40 into the clip. See here for some other interesting time-slots.

On a recent visit to the Bourbaki Archives I was happy to see that all volumes of “La Tribu” (the internal newsletter of Bourbaki) are now online from 1940 until 1960.

Okay, it’s not the entire story yet but, for all you Grothendieck aficionados out there, it should be enough as G resigned from Bourbaki in 1960 with this letter (see here for a translation).

Grothendieck was present at just twelve Bourbaki congresses in the period between 1955 and 1960 (he was also present as a ‘cobaye’ at a 1951 congress in Nancy).

The period 1955-60 was crucial in the modern development of algebraic geometry. Serre’s ‘FAC’ was published, as was Grothendieck’s ‘Tohoku-paper’, there was the influential Chevalley seminar, and the internal Bourbaki-fight about categories and the functorial view.

Perhaps the definite paper on the later issue is Ralf Kromer’s La ‘Machine de Grothendieck’ se fonde-t-elle seulement sur les vocables metamathematiques? Bourbaki et les categories au cours des annees cinquante.

Kromer had access to most issues of La Tribu until 1962 (from the Delsarte archive in Nancy), but still felt the need to justify his use of these sources to the ACNB (footnote 9 of his paper):

“L’autorisation que j’ai obtenue par le Comité scientifique des Archives de la création des mathématiques, unité du CNRS qui fut chargée jusqu’en 2003 de la mise à disposition de ces archives, me donne également le droit d’utiliser les sources datant des années postérieures à l’année 1953, que j’avais consultées auparavant aux Archives Jean Delsarte, soit avant que l’ACNB (Association des Collaborateurs de Nicolas Bourbaki) ne rende publique sa décision d’ouvrir ses archives et ne décide des parties qui seraient consultables.

J’ai ainsi bénéficié d’une occasion qui ne se présenterait sans doute plus aujourd’hui, mais c’est en toute légitimité que je puis m’appuyer sur cette riche documentation. Toutefois, la collection des Archives Jean Delsarte étant à son tour limitée aux années antérieures à 1963, je n’ai pu étudier la discussion ultérieure.”

The Association des Collaborateurs de Nicolas Bourbaki made retirement from active B-membership mandatory at the age of 50. One might expect of it to open up all documents in its archives which are older than fifty years.

Meanwhile, we’ll have a go at the 1940-1960 issues of La Tribu.

Grothendieck’s Café

Published July 1, 2014 by lievenlb

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”. At the time only an undefined idea of “schéma” was current in Paris, meaning more or less whatever would improve on Weil’s foundations.” (McLarty in The Rising Sea)

Finding that particular café in Paris, presumably in the 5th arrondissement, seemed like looking for a needle in a haystack.

Until now.

In trying to solve the next riddle in Bourbaki’s death announcement:

A reception will be held at the Bar ‘The Direct Products’, at the crossroads of the Projective Resolutions (formerly Koszul square)

I’ve been reading Mathematics, a novel by Jacques Roubaud (the guy responsible for the announcement) on Parisian math-life in the 50ties and 60ties.

It turns out that the poor Bourbakistas had very little choice if they wanted to have a beer (or coffee) after attending a seminar at the IHP.

On page 114, Roubaud writes:

“Père Plantin presided over his bar, which presided over the Lhomond/Ulm street corner. It is an obvious choice. rue Pierre-et-Marie-Curie had no bars; rue d’Ulm had no bars in eyeshot either. If we emerged, as we did, on this side of the Institut Henri Poincaré (for doing so on the other side would have meant fraternizing with the Spanish and Geography students in the cafés on rue Saint-Jacques, which was out of the question), we had no choice. Café Plantin had a hegemony.”

It is unclear to me whether Plantin was once actually the name of the café, or that it’s just Roubaud’s code-word for it. At other places in the book, e.g. on pages 82 and 113, he consistently writes “Plantin”, between quotes.

Today, the café on the crossroads of rue d’Ulm (where the Ecole Normal Superieure is located) and de rue Lhomond is the Interlude Café

and here’s what Roubaud has to say about it, or rather about the situation in 1997, when the French version of his book was published:

“the thing that would currently be found at the very same corner of rues Lhomond/Ulm would not be what I am here terming “Plantin”.”

So, we can only hope that the Café ‘Aux Produits Directs’ was a lot cosier, way back then.

But let us return to Grothendieck’s “What is a scheme?” story.

Now that we have a fair idea of location, what about a possible date? Here’s a suggestion: this happened on monday december 12th, 1955, and, one of the friends present must have been Cartier.

Here’s why.

The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry.

The lecturer was Claude Chevalley, and the date was december 12th 1955.

I’m fairly certain Grothendieck and Cartier attended and that Cartier was either briefed before or understood the stuff at once (btw. he gave another talk on schemes, a year later at the Chevalley seminar).

A couple of days later, on december 15th, Grothendieck sends a letter to his pal Serre (who must have been out of Paris for otherwise they’d phone each other) ending with:

Note the phrase: I am exploiting him most profitably. Yes, by asking him daft questions over a pint at Café “Plantin”…