What I Learned Today

. . (3/20/17) Today I learned that every if R is a ring and if S ⊂ R is a multiplicative subset of R not containing 0, then the prime ideals of R whose intersection with S is empty are in one to one correspondence with the prime ideals of RS , the ring localizing S. The way I look at this is to think, well, if I want a prime ideal in RS from an ideal in R, say, I, if I ∩ S = ∅, then you really didn’t change the ideal. I’m not 100 percent sure how right that is-we’ll see. (3/21/17) Today on a homework problem I learned about the Tubular Neighborhood Theorem, which says that if one is given manifolds Z ⊂ Y , then there is a neighborhood in the normal bundle of Z in Y ,written N(Z;Y ), containing Z, that is diffeomorphic to a neighborhood of Z. The picture I drew of this was a central circle for Z and a sphere for Y , which makes it seem super obvious, but I’m sure it isn’t in general. (3/22/17) So fun fact. The fact about prime ideals and localization in (3/20/17) was in the book I was reading for fun at night called Algebraic Number Fields. But today, I learned a very similar fact about prime ideals, but this time relating to quotienting, in Vakil’s notes. In particular, there’s a natural bijective correspondence between prime ideals in a ring R containing an ideal I and prime ideals in the quotient ring R/I. This in particular gives some good visualization of Spec(R/I) as a subset of Spec(R). This sort of explains how we can relate ideals–for example, we can view any element of Spec(C[x, y]/(x2−y2)) as the set of ideals I ⊂ C[x, y] containing (x2−y2). So for example, the point (x − 1, y − 2) isn’t considered. This will probably have a similar story with the above localization and prime ideals. Also today I learned a fact in DiffTop which will probably serve well in the future, well, maybe. A submanifold Z ⊂ Y is globally cut out by the zeroes of independent coordinate functions if and only if its normal bundle in Y N(Z;Y ) is trivial. I feel like now that we’re getting into intersection theory stuff that might come in handy? I should think about why this applies to the example I learned in class about projecting a donut down to its height–why one circle by its lonesome can’t be cut out in this case. (3/23/17) The first thing I can think of that I learned today is the Van Kampen theorem for groupoids. In particular, if X is a topological space, then Pi(X) is a groupoid whose objects are the points of x and whose morphisms between objects are homotopy classes of paths between the two points. In particular, if S = {U} is an open cover of X that is closed under finite intersection such that all of those open sets are path connected*, we obtain an S shaped diagram if we view S as a category whose morphisms are ”inclusion”. Then the Van Kampen theorem for groupoids says that Π(X) ∼= colimS(Π(U)). *Although to be honest, I’m not sure if this condition was necessary. I combed through the proof a few times to see, and it didn’t seem to. Then I saw this Stackexchange post. http://math.stackexchange.com/questions/198348/does-mays-version-of-groupoidseifert-van-kampen-need-path-connectivity-as-a-hy Also, the ”closed under finite intersection” thing might be a similar thing to being a ”filtered category.”