Random Math

Here is a semi-random collection of math stuff 🙂

Pi = 0, or, how I learned to stop worrying and let delta depend on x.

How to learn what an étale topos is

Self-assigned student numbers. How can N students assign their own unique student numbers without any student knowing anything but his own number? If you want the answer, email me! I don’t think you’ll find it anywhere else 😉

Compact convergence does not imply local uniform convergence. A friend asked me if uniform convergence on compact sets implies local uniform convergence. On a locally compact space, these notions are equivalent. The space I made up for a counterexample to the general claim was interesting enough to me that I decided to write it up.

Divisors and invertible sheaves (an arrowful tale). This is a one-page annotated commutative diagram summarizing the mappings that relate Weil and Cartier divisors and invertible sheaves, with references. (The green arrow is the one from my talk at the MAGIC seminar.)

The long exact sequence of Tor (a worked example). Computing Tor modules can be done using any choice of projective resolution, and this is usually enough for most applications. However, finding the maps involved in the long exact sequence connecting them (which arises from tensoring a short exact sequence of modules with some other module) requires the “Snake Lemma” applied to an exact sequence of complexes constructed using the “Horseshoe Lemma.” This is an example we worked out for our homological algebra seminar.

Stalk-local detection of irreducibility. If a locally Noetherian scheme is connected and stalk-locally irreducible, then it is irreducible. This is sort of geometrically intuitive when you think of how, for example, reducibility of the variety {xy=0} in the plane can be detected in an “infinitely small” neighborhood of the origin; what I find neat is how this can be seen directly by using elementary algebra and Noetherianity instead of higher theorems (see link), and so maybe this would make a good introductory textbook exercise. Thanks to my officemate Adam Boocher for sharing an interest in this problem!(As an applicaiton of this, the “integral” hypothesis can be removed from Hartshorne’s AG II.6.11.)