In an article about the life of Grothendieck, available here:


Allyn Jackson writes about how Mumford was profoundly impressed:

Mumford found the leaps into abstraction to be breathtaking. Once he asked Grothendieck how to prove a certain lemma and got in reply a highly abstract argument. Mumford did not at first believe that such an abstract argument could prove so concrete a lemma. “Then I went away and thought about it for a couple of days, and I realized it was exactly right,” Mumford recalled.

What were the lemma and proof that so impressed Mumford?

(I have tried asking algebraic geometers and category theorists; the tags attached to this question are speculative.)

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    $\begingroup$ This post could be better at home on History of Science and Mathematics, though it is probably not off-topic here, either. $\endgroup$ – Danu Jul 16 '16 at 15:08
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    $\begingroup$ Based on things I've heard people say, I would guess something close to the Theorem of the Cube from Abelian Varieties. $\endgroup$ – Tabes Bridges Jul 16 '16 at 15:24
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    $\begingroup$ Why not ask Mumford? $\endgroup$ – Timothy Chow Jul 17 '16 at 16:43
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    $\begingroup$ Asking Mumford directly is almost surely the best idea. If you want a guess, then I suspect it is Zariski’s theorem that "the inverse image of every normal point under a proper birational morphism from one variety onto another is connected" as described here (p. 4). $\endgroup$ – Benjamin Dickman Jul 23 '16 at 3:47
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    $\begingroup$ If you get an answer tell us! $\endgroup$ – richarddedekind Jul 23 '16 at 14:29

The concrete lemma was "The Theorem of The Cube", (Mumford, Abelian Varieties [AV], Section 6) as indicated by Tabes Bridges. The abstract argument was the second theorem in Section 5 of AV (p.46).

In Mumford's collected works, Volume 2, p. 689 there is a letter of Grothendieck to Mumford in which the abstract argument is discussed, and footnote 2 indicates that it was published by Mumford in AV as the second theorem in Chapter 5.

All of this was confirmed in an email correspondence with Mumford.

I am obliged Mumford (of course), to Tabes Bridges for giving the right answer and to Benjamin Dickman for a reminder to post. I should look at MO more frequently!


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