Skip to content →

Tag: Dieudonne

From Weil’s foundations to schemes

Last time, we’ve seen that the first time ‘schemes’ were introduced was in ‘La Tribu’ (the internal Bourbaki-account of their congresses) of the May-June 1955 congress in Chicago.

Here, we will focus on the events leading up to that event. If you always thought Grothendieck invented the word ‘schemes’, here’s what Colin McLarty wrote:

“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)

What were Weil’s foundations of algebraic geometry?

Well, let’s see how Weil defined an affine variety over a field $k$. First you consider a ‘universal field’ $K$ containing $k$, that is, $K$ is an algebraically closed field of infinite transcendence degree over $k$. A point of $n$-dimensional affine space is an $n$-tuple $x=(x_1,\dots,x_n) \in K^n$. For such a point $x$ you consider the field $k(x)$ which is the subfield of $K$ generated by $k$ and the coordinates $x_i$ of $x$.

Alternatively, the field $k(x)$ is the field of fractions of the affine domain $R=k[z_1,\dots,z_n]/I$ where $I$ is the prime ideal of all polynomials $f \in k[z_1,\dots,z_n]$ such that $f(x) = f(x_1,\dots,x_n)=0$.

An affine $k$-variety $V$ is associated to a ‘generic point’ $x=(x_1,\dots,x_n)$, meaning that the field $k(x)$ is a ‘regular extension’ of $k$ (that is, for all field-extensions $k’$ of $k$, the tensor product $k(x) \otimes_k k’$ does not contain zero-divisors.

The points of $V$ are the ‘specialisations’ of $x$, that is, all points $y=(y_1,\dots,y_n)$ such that $f(y_1,\dots,y_n)=0$ for all $f \in I$.

Perhaps an example? Let $k = \mathbb{Q}$ and $K=\mathbb{C}$ and take $x=(i,\pi)$ in the affine plane $\mathbb{C}^2$. What is the corresponding prime ideal $I$ of $\mathbb{Q}[z_1,z_2]$? Well, $i$ is a solution to $z_1^2+1=0$ whereas $\pi$ is transcendental over $\mathbb{Q}$, so $I=(z_1^2+1)$ and $R=\mathbb{Q}[z_1,z_2]/I= \mathbb{Q}(i)[z_2]$.

Is $x=(i,\pi)$ a generic point? Well, suppose it were, then the points of the corresponding affine variety $V$ would be all couples $(\pm i, \lambda)$ with $\lambda \in \mathbb{C}$ which is the union of two lines in $\mathbb{C}^2$. But then $i \otimes 1 + 1 \otimes i$ is a zero-divisor in $\mathbb{Q}(x) \otimes_{\mathbb{Q}} \mathbb{Q}(i)$. So no, it is not a generic point over $\mathbb{Q}$ and does not define an affine $\mathbb{Q}$-variety.

If we would have started with $k=\mathbb{Q}(i)$, then $x=(i,\pi)$ is generic and the corresponding affine variety $V$ consists of all points $(i,\lambda) \in \mathbb{C}^2$.

If this is new to you, consider yourself lucky to be young enough to have learned AG from Fulton’s Algebraic curves, or Hartshorne’s chapter 1 if you were that ambitious.

By 1955, Serre had written his FAC, and Bourbaki had developed enough commutative algebra to turn His attention to algebraic geometry.

La Ciotat congress (February 27th – March 6th, 1955)

With a splendid view on the mediterranean, a small group of Bourbaki members (Henri Cartan (then 51), with two of his former Ph.D. students: Jean-Louis Koszul (then 34), and Jean-Pierre Serre (then 29, and fresh Fields medaillist), Jacques Dixmier (then 31), and Pierre Samuel (then 34), a former student of Zariski’s) discussed a previous ‘Rapport de Geometrie Algebrique'(no. 206) and arrived at some unanimous decisions:

1. Algebraic varieties must be sets of points, which will not change at every moment.
2. One should include ‘abstract’ varieties, obtained by gluing (fibres, etc.).
3. All necessary algebra must have been previously proved.
4. The main application of purely algebraic methods being characteristic p, we will hide nothing of the unpleasant phenomena that occur there.



(Henri Cartan and Jean-Pierre Serre, photo by Paul Halmos)

The approach the propose is clearly based on Serre’s FAC. The points of an affine variety are the maximal ideals of an affine $k$-algebra, this set is equipped with the Zariski topology such that the local rings form a structure sheaf. Abstract varieties are then constructed by gluing these topological spaces and sheaves.

At the insistence of the ‘specialistes’ (Serre, and Samuel who had just written his book ‘Méthodes d’algèbre abstraite en géométrie algébrique’) two additional points are adopted, but with some hesitation. The first being a jibe at Weil:
1. …The congress, being a little disgusted by the artificiality of the generic point, does not want $K$ to be always of infinite transcendent degree over $k$. It admits that generic points are convenient in certain circumstances, but refuses to see them put to all the sauces: one could speak of a coordinate ring or of a functionfield without stuffing it by force into $K$.
2. Trying to include the arithmetic case.

The last point was problematic as all their algebras were supposed to be affine over a field $k$, and they wouldn’t go further than to allow the overfield $K$ to be its algebraic closure. Further, (and this caused a lot of heavy discussions at coming congresses) they allowed their varieties to be reducible.

The Chicago congress (May 30th – June 2nd 1955)

Apart from Samuel, a different group of Bourbakis gathered for the ‘second Caucus des Illinois’ at Eckhart Hall, including three founding members Weil (then 49), Dixmier (then 49) and Chevalley (then 46), and two youngsters, Armand Borel (then 32) and Serge Lang (then 28).

Their reaction to the La Ciotat meeting (the ‘congress of the public bench’) was swift:

(page 1) : “The caucus discovered a public bench near Eckhart Hall, but didn’t do much with it.”
(page 2) : “The caucus did not judge La Ciotat’s plan beyond reproach, and proposed a completely different plan.”

They wanted to include the arithmetic case by defining as affine scheme the set of all prime ideals (or rather, the localisations at these prime ideals) of a finitely generated domain over a Dedekind domain. They continue:

(page 4) : “The notion of a scheme covers the arithmetic case, and is extracted from the illustrious works of Nagata, themselves inspired by the scholarly cogitations of Chevalley. This means that the latter managed to sell all his ideas to the caucus. The Pope of Chicago, very happy to be able to reject very far projective varieties and Chow coordinates, willingly rallied to the suggestions of his illustrious colleague. However, we have not attempted to define varieties in the arithmetic case. Weil’s principle is that it is unclear what will come out of Nagata’s tricks, and that the only stable thing in arithmetic theory is reduction modulo $p$ a la Shimura.”

“Contrary to the decisions of La Ciotat, we do not want to glue reducible stuff, nor call them varieties. … We even decide to limit ourselves to absolutely irreducible varieties, which alone will have the right to the name of varieties.”

The insistence on absolutely irreducibility is understandable from Weil’s perspective as only they will have a generic point. But why does he go along with Chevalley’s proposal of an affine scheme?

In Weil’s approach, a point of the affine variety $V$ determined by a generic point $x=(x_1,\dots,x_n)$ determines a prime ideal $Q$ of the domain $R=k[x_1,\dots,x_n]$, so Chevalley’s proposal to consider all prime ideals (rather than only the maximal ideals of an affine algebra) seems right to Weil.

However in Weil’s approach there are usually several points corresponding to the same prime ideal $Q$ of $R$, namely all possible embeddings of the ring $R/Q$ in that huge field $K$, so whenever $R/Q$ is not algebraic over $k$, there are infinitely Weil-points of $V$ corresponding to $Q$ (whence the La Ciotat criticism that points of a variety were not supposed to change at every moment).

According to Ralf Krömer in his book Tool and Object – a history and philosophy of category theory this shift from Weil-points to prime ideals of $R$ may explain Chevalley’s use of the word ‘scheme’:

(page 164) : “The ‘scheme of the variety’ denotes ‘what is invariant in a variety’.”

Another time we will see how internal discussion influenced the further Bourbaki congresses until Grothendieck came up with his ‘hyperplan’.

Leave a Comment

The birthplace of schemes

Wikipedia claims:

“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.

Leave a Comment

Bourbaki and the miracle of silence

The last pre-war Bourbaki congress, held in september 1938 in Dieulefit, is surrounded by mystery. Compared to previous meetings, fewer documents are preserved in the Bourbaki archives and some sentences in the surviving notules have been made illegible. We will have to determine the exact location of the Dieulefit-meeting before we can understand why this had to be done. It’s Bourbaki’s own tiny contribution to ‘le miracle de silence’…

First, the few facts we know about this Bourbaki congress, mostly from Andre Weil‘s autobiography ‘The Apprenticeship of a Mathematician’.

The meeting was held in Dieulefit in the Drome-Provencale region, sometime in september 1938 prior to the Munich Agreement (more on this next time). We know that Elie Cartan did accept Bourbaki’s invitation to join them and there is this one famous photograph of the meeting. From left to right : Simone Weil (accompanying Andre), Charles Pison, Andre Weil (hidden), Jean Dieudonne (sitting), Claude Chabauty, Charles Ehresmann, and Jean Delsarte.

Failing further written documentation, ‘all’ we have to do in order to pinpoint the exact location of the meeting is to find a match between this photograph and some building in Dieulefit…

The crucial clue is provided by the couple of sentences, on the final page of the Bourbaki-archive document deldi_001 Engagements de Dieulefit, someone (Jean Delsarte?) has tried to make illegible (probably early on).



Blowing the picture up, it isn’t too hard to guess that the header should read ‘Décision du 22 septembre 1938’ and that the first sentence is ‘Le Bourbaki de 2e classe WEIL fera pour le 15 octobre’. The document is signed

Camp de Beauvallon, le 22.IX.38.
L’adjudant de jour
DIEUDONNE

Now we are getting somewhere. Beauvallon is the name of an hamlet of Dieulefit, situated approximately 2.5km to the east of the center.

Beauvallon is rather famous for its School, founded in 1929 by Marguerite Soubeyran and Catherine Krafft, which was the first ‘modern’ boarding school in France for both boys and girls having behavioral problems. From 1936 on the school’s director was Simone Monnier.

These three women were politically active and frequented several circles. Already in 1938 (at about the time of the Bourbaki congress) they knew the reality of the Nazi persecutions and planned to prepare their school to welcome, care for and protect refugees and Jewish children.

From 1936 on about 20 Spanish republican refugees found a home here and in the ‘pension’ next to the school. When the war started, about 1500 people were hidden from the German occupation in Dieulefit (having a total population of 3500) : Jewish children, intellectuals, artists, trade union leaders, etc. etc. many in the Ecole and the Pension.

Because of the towns solidarity with the refugees, none were betrayed to the Germans, Le miracle de silence à Dieulefit.
It earned the three Ecole-women the title of “Juste” after the war. More on this period can be read here.

But what does this have to do with Bourbaki? Well, we claim that the venue of the 1938 Bourbaki congress was the Ecole de Beauvallon and they probably used Le Pension for their lodgings.

We have photographic evidence comparing the Bourbaki picture with a picture taken in 1943 at the Ecole (the woman in the middle is Marguerite Soubeyran). Compare the distance between door and window, the division of the windows and the ivy on the wall.

Below two photographs of the entire school building : on the left, the school with ‘Le Pension’ next to it around 1938 (the ivy clad wall with the Bourbaki-door is to the right) and on the right, the present Ecole de Beauvallon (this site also contains a lot of historical material). The ivy has gone, but the main features of the building are still intact, only the shape of the small roof above the Bourbaki-door has changed.

During their stay, it is likely the Bourbakis became aware of the plans the school had would war break out. Probably, Jean Delsarte removed all explicit mention to the Ecole de Beauvallon from the archives upon their return. Bourbaki’s own small contribution to Dieulefit’s miracle of silence.

Leave a Comment