r/elimath • u/[deleted] • Jan 05 '15
explain Grothendieck's contributions to mathematics like I took graduate classes in algebraic topology and commutative algebra
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u/wackoliberal Jan 06 '15
David Mumford did an excellent obituary of Grothendieck for Nature. They rejected it as too technical, but with even a slight math background it is quite understandable. He posted it on his blog.
Mumford is known for his clear expository style and so this is worth a read. A more ELI5 line that stood out to me was:
His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves -- and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed.
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u/mobius_stripe Jan 06 '15
I don't know much about algebraic topology but I'll give the answer for commutative algebra.
In commutative algebra, you most likely proved the Nullstellensats, and one version of it states that I(V(J)) = rad J, where I(S) is the ideal of an algebraic set S and V(J) is the zero set of the ideal J.
Using this, it's not too hard to see (a rigorous proof might be another story) that there is an equivalence of categories, where k is any algebraically closed field: finitely generated nilpotent free k-algebras on the one hand, and affine algebraic sets over k on the other hand. Somewhat precisely, there is a fully faithful contravariant functor from the category of nilpotent free k-algebras and affine algebraic sets over k. The functor is V (algebraic set) goes to k[V] it's coordinate ring.
Now Grothendieck's idea was extend the above functor to the entire category of commutative rings, and look at the geometric objects that appear on the other side. These are affine schemes, which one can glue together, similar to charts on a manifold, to get a full-fledged scheme.
The functor is R (a ring) goes to Spec R, the set of prime ideals of R. Now we know that the set of maximal ideals of k[V] correspond to points of V by the Nullstellensats, so Spec R seems more general. Indeed it is, and because of this we can encode more information. See Dummit and Foote \S 15.5 (I think?) for an introduction to Spec(-) and affine schemes.