I continue to be bowled over daily by the high quality of the discussion at MathOverflow, and the prominence of many of the frequent participants. But this one was special:
A newbie poster asked for a pointer to a proof of the “de Rham-Weil” theorem. There’s a bit of ambiguity about what theorem this might refer to, but I had a pretty good of what the poster meant, so I responded that the earliest reference I know of is in Grothendieck‘s 1957 Tohoku paper — which led another poster to ask if this meant de Rham and Weil had had nothing to do with it.
This triggered an appearance from the legendary Roger Godement (had he been lurking all this time?), now aged 91 and one of the last survivors of the extraordinary circle of French mathematicians who rewrote the foundations of topology and geometry in the mid-20th century and changed the look, feel and content of mathematics forever. I tend to think of them as gods and demigods. Godement’s indispensable Theorie des Faisceaux was my constant companion in late graduate school. And now he has emerged from retirement for the express purpose of chastising me:
There was mathematics before Grothendieck …
Of course Weil did it…and I even lectured on it in Paris 40 years ago….
Before Grothendieck’s Tohoku, there was a Cartan Seminar on sheaves theory (ca. 1948-49) in which everybody, at any rate in France, learned the theory, including Groth and myself. I even wrote a book on the subject which was still on sale (and found customers) two or three years ago, and possibly still is.
And indeed on rechecking the Tohoku paper, I find this from Grothendieck’s introduction:
Conversations with Roger Godement and Henri Cartan were very valuable for perfecting the theory. In particular, Godement’s introduction of flabby sheaves and soft sheaves, which can usefully be substituted for injective sheaves in many situations, has turned out to be extremely convenient. A more complete description, to which we will turn for a variety of details, will be given in a book by Godement in preparation.
In my own limited defense, I was pretty sure these ideas had come out of Cartan’s seminar, and almost said so in my posting, but since I wasn’t sure I went with the factually correct statement that “the earliest reference I know of is Grothendieck…”.
Had I known Godement was watching, I’d have been more thorough. Now I wonder who else is out there.