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Tag: Weil

Weil photos used in Dema-lore

On April 20th of 2018, twenty one pilots updated their store page to include a video with a hidden message at the end of it.

and with a bit of sleuthing it led to a page on the dmaorg.info site containing:



This was immediately identified as part of the photo on the right, which is on the French Wikipedia page for Andre Weil.

The photo is clipped in such a way one cannot be certain whether the child is a boy or girl, so a logical explanation is that this is supposed to be the nine year old Clancy, shielding his eyes from the violence (vialism) he just discovered in Dema.

The full picture suggests that Clancy’s struggles might mirror some in Andre Weil’s life.

Andre Weil was born May 6th, 1906, so ‘in his ninth year’ World War 1 breaks out in 1914.

Last time we’ve seen that Bourbaki’s Dema = Ecole Normal Superieure in Paris during WW1, Vialism = militant patriotism sending ENS-graduates as trained reserve second lieutenants in the infantry to the trenches, and there getting killed ‘pour la patrie’ and the glory of the ENS and its director Ernest Lavisse, “L’instituteur national”.

Here’s a G-translation of his letter to the young French, published September 23rd 1914:

Dear children of France, You will be old one day, and, like the old, you will like to remember times past. There will come evenings when your little children, seeing you dreaming, will say to you: Tell us, grandfather. And you will tell. It will be a few episodes of the war, a long march, an alert, a bayonet assault, a cavalry charge, the feat of a battery of 75, the strewn enemy dead on the plain, or else, in the streets of a city, the serried ranks of corpses left standing for lack of room to fall; and then the death of comrades, the terrible losses of your company and your regiment, your wounds received in Belgium, in Champagne, on the banks of the Rhine, beyond the Rhine; but the joy of victories, the poles knocked down on too narrow frontiers, triumphal entries.

On those evenings, after the amazed children have gone to sleep, you will open a drawer where you will have collected precious objects, a bullet extracted from a wound, a piece of shell, a cloth where your blood will have turned pale, a cross of honour, I hope, or a military medal, at the very least a medal from the 1914 war, on the ribbon of which the silver clasps will bear the names of immortal battles. And whatever your life, happy or unhappy, you will be able to say: I lived great days such as the history of men had not yet seen. And you will be right to be proud of your youth, because you are sublime young people!

I have read your letters; I have spoken with the wounded. Through you, I know what heroism is. I had heard a lot about it, being a historian by profession, but now I see it, I touch it, and how beautiful your heroism is, embellished with grace and smiling in the French way! Young soldiers if you were given one chevron per battle, your march would not be enough to accommodate them, because at the end of the war you would count more chevrons than years;

Young soldiers you are glorious old warriors.

Oh! Thanks thanks! Thank you for the beautiful end of life that you give to the elderly who, for forty-four years have suffered so much from the abasement of the fatherland.

The 44 years refers to the Franco-Prussian war of 1870 in which Bourbaki (the general) played a dramatic role.

The next cycle of militant patriotism occurred in the years leading up to the second world war. Here, Andre Weil’s experiences mirror those of Clancy. He tried several times to escape, first from military action (although he too was a reserve officer in the French army), then from France itself. He was captured in Finland, brought back to France to face trial and imprisonment, was released on the condition that he did active military duty, escaped with the French army to England, there demobilised he refused to join de Gaulle’s troops, left England on a boat to Marseille, from where he escaped to the US.

All this, and much more, you can read in his autobiography The Apprenticeship of a Mathematician, especially Chapter VI, The War and I: A Comic Opera in Six Acts.



(for TØP-ers, note the Bishop-red cover…)

Comic or not, the book tries to ‘explain’ his actions in those years, but failed to convince the French from offering him a professorship at a French university after the war.

Perhaps it may be worth looking into a comparison between Weil’s autobiography and the collected Clancy letters.

I guess that’s the best I can do to explain the use of that Weil photo by TØP. Surely they didn’t search any deeper as to where and when this picture was taken, or who the girl was next to Weil.

In case anyone might be interested, I’ll be happy to explain my own theory about this in another post.

I’m sure the full photograph ended up in the ‘Trench-bible’, given to the director of their clip-movies. The scenery is used at the end of the Jumpsuit video when ‘Clancy’ takes out a jumpsuit from the burning car and walks away along a road very similar to that in the photo.



The boy/girl shielding his/her eyes for the violence, should have been used at about minute one into the Outside video



Now, there’s another Weil (or rather Bourbaki) photograph we know did inspire Twenty One Pilots, the classic picture at the Dieulefit/Beauvallon 1938 Bourbaki-congress



which was photoshopped in order to get Szolem Mandelbrojt in from the Chancay (quite similar to Clancy now that i type this) 1937 Bourbaki congress



Now, these were the only two Bourbaki-meetings Simone Weil (Andre’s sister) attended, and she features prominently in both pictures.

Probably this brother/sister thing struct a chord with Twenty One Pilots. But then, you quickly end up with this iconic picture of both of them, taken in the summer of 1922, just before Andre entered the Ecole Normale (he entered the ENS at age 16…)



I’d love to be send a copy of the ‘Trench bible’ because I’m fairly certain also this photograph is in it. At the end of the Nico and the niners-video you see this boy and girl (who may be around age 9 and discover the truth about Dema) finding a jumpsuit with the Bishops approaching



and they reappear a bit older at the end of the Outside-video, with a burning Dema in the background.



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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’.

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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.

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