## September 4th, 2008

### (no subject)

's post linked to the current homework for Luis von Ahn's 15-396 "Science of the Web" which mentioned signed graphs, which apparently have something to do with root systems (something John Baez talks about a lot in connection with the classification of semisimple lie groups). There is a generalization of a root system called a root datum, introduced in Grothendieck's Séminaire de géométrie algébrique (volume III) which was kind of a sequel to his earlier Éléments de Géométrie Algébrique, which importantly introduced the notion of (affine) scheme, which have to do with local rings. I thought at first a "locally ringed space" was just, you know, a topological space that was ringed, er, locally. Like, you require a function that maps each open set to a ring, and each open-set-subset-relation to a ring homomorphism, maybe. Just a functor from the poset of open sets to the category Rng. But no! Not only are you supposed to have a sheaf of rings, but every stalk of the sheaf is supposed to be a local ring, which means quite unexpectedly that the ring has a unique maximal ideal.

Why this is called being "local" is because it's a typical property of rings of germs of functions. An equivalent property (one whose equivalence I can't prove to myself yet - I'm not very fluent thinking about ideals) is that the sum of two non-units in the ring is again a non-unit. For germs of, say, functions RR at zero, this latter property is pretty easy to see: a non-unit is precisely a germ whose value at zero is zero.