Archive for February, 2011

This is cool (Freyd’s initial object theorem)

February 7, 2011

I blogged about it here, and added it to the CRing project here. Basically, the point is that complete categories are prone to having initial objects (of course, cocomplete categories are!) if they have a “weak” initial object: one that homs into every other object, but not necessarily uniquely.

I don’t think we’ll ever use it in the project, but it is neat. Covering a bit of basic category theory for its own sake seems like a reasonable thing to do, moreover.

Changes

February 5, 2011

We have received a new set of donations, again from a live-TeXed course. One thing that I hadn’t seen before is a theorem of McCoy that gives a criterion (in the “density” of certain ideals generated by minors) of when a matrix mapping of free modules is injective.

In addition, various edits keep being made, slowly. There have been several articles in the Monthly that give short semi-expository pieces of mathematics; a brief account of some of them have appeared in the book. Here is one, giving a short proof of the following lemma: a field extension which is finitely generated as an algebra is algebraic.

To see this, one can use the following argument of McCabe. Let $k'/k$ be an extension finitely generated as an algebra. There is a subring $k[x_1, \dots, x_n]$ over which $k’$ is algebraic. The finite generation hypothesis implies that we can find a localization $k[x_1, \dots, x_n]_y$ over which $k’$ is even integral. But if one has an integral extension and the top ring is a field, then so is the bottom one; this implies $k[x_1, \dots,x_n]_y$ is a field. So $y \in k[x_1, \dots, x_n]$ must lie in every maximal ideal, meaning it is zero, contradiction.