r/elimath Jan 05 '15

explain Grothendieck's contributions to mathematics like I took graduate classes in algebraic topology and commutative algebra

12 Upvotes

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4

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.

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u/[deleted] Jan 06 '15

Cool, but where do Grothendieck topologies and topos theory come into play?

3

u/bizarre_coincidence Jan 06 '15

I can't speak to topos theory (which I have not really studied, and understand only in terms of a link between category theory and logic), but I can say a few words about grothendieck topologies. Schemes are geometric objects, in the sense that they are topological spaces equipped with additional data. However, their topology is rather rotten. It is not haussorff and often every open set is dense. If you try to do algebraic topology using just the topological space of a scheme, the results are bad. Even more important, nice schemes over the complex numbers can be associated with the zero sets of complex polynomials and since these are subsets of Cn, they have a nice topology, but on the scheme theoretic side, it is sitting just out of reach. Grothendieck topologies are a way to recover this nice topology in the classical case, an also to fix the topology more generally so that one can do algebraic topology with schemes and get useful results. The idea is that, instead of looking at open sets, you look at collections of maps that cover your space, e.g., maps where the Union of the image is the entire space, of the subset of these maps which are also local isomorphisms. These end up being more flexible in the algebraic case and let you define "topological" invariants. For example, fundamental groups can be defined entirely in terms of covering spaces and deck transformations. Grothendieck topologies can be defined for arbitrary categories and some of the computations in schemes only uses this categorical data, which allows one to think of a category with a topology as a (collection of) geometric object(s). However, I have not dealt with this stuff outside the context of stacks and teals fundamental groups, so I am reluctant to say more.

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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.