The great reformulation of algebraic geometry

“Tate helped shape the great reformulation of arithmetic and geometry which has taken place since the 1950’s.” — Andrew Wiles

At the Heidelberg Laureate Forum I has a chance to interview John Tate. In his remarks below, Tate briefly comments on his early work on number theory and cohomology. Most of the post consists of his comments on the work of Alexander Grothendieck.

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JT: My first significant work after my thesis was to determine the cohomology groups of class field theory. The creators of the theory, including my thesis advisor Emil Artin, didn’t think in terms of cohomology, but their work could be interpreted as finding the cohomology groups H0, H1, and H2.

I was invited to give a series of three talks at MIT on class field theory. I’d been at a party, and I came home and thought about what I’d talk about. And I got this great idea: I realized I could say what all the higher groups are. In a sense it was a disappointing answer, though it didn’t occur to me then, that there’s nothing new in them; they were determined by the great work that had already been done. For that I got the Cole prize in number theory.

Later when I gave a talk on this work people would say “This is number theory?!” because it was all about cohomology groups.

JC: Can you explain what the great reformulation was that Andrew Wiles spoke of? Was it this greater emphasis on cohomology?

JT: Well, in the class field theory situation it would have been. And there I played a relatively minor part. The big reformulation of algebraic geometry was done by Grothendieck, the theory of schemes. That was really such a great thing, that unified number theory and algebraic geometry. Before Grothendieck, going between characteristic 0, finite characteristic 2, 3, etc. was a mess.

Grothendieck’s system just gave the right framework. We now speak of arithmetic algebraic geometry, which means studying problems in number theory by using your geometric intuition. The perfect background for that is the theory of schemes. ….

Grothendieck ideas [about sheaves] were so simple. People had looked at such things in particular cases: Dedekind rings, Noetherian rings, Krull rings, …. Grothendieck said take any ring. … He just had an instinct for the right degree of generality. Some people make things too general, and they’re not of any use. But he just had an instinct to put whatever theory he thought about in the most general setting that was still useful. Not generalization for generalization’s sake but the right generalization. He was unbelievable.

He started schemes about the time I got serious about algebraic geometry, as opposed to number theory. But the algebraic geometers classically had affine varieties, projective varieties, … It seemed kinda weird to me. But with schemes you had a category, and that immediately appealed to me. In the classical algebraic geometry there are all these birational maps, or rational maps, and they’re not defined everywhere because they have singularities. All of that was cleared up immediately from the outset with schemes. ….

There’s a classical algebraic geometer at Harvard, Joe Harris, who works mostly over the complex numbers. I asked him whether Grothendieck made much of a difference in the classical case — I knew for number theorists he had made a tremendous difference — and Joe Harris said yes indeed. It was a revolution for classical algebraic geometry too.

Related: Applied number theory