Sunday, January 23, 2005

Zombie Conjecture

Maybe it lives after all. Maybe I can't decide whether or not all the hypotheses of the conjecture apply in the would-be counter-example. Maybe I'm just really, really confused. (By my count there are about, oh, 2 people reading this who might understand the technical details, so I'll be uselessly brief: A is an algebra over R which is, in turn, an algebra over k. And so it depends if you're thinking about A as an R-algebra or as a k-algebra. And I think that I was thinking about the R-algebra structure when he was working with a as a k-algebra.)

Of course this changes all my problems.

Before, when it was dead, I was pretty sure that I had exhausted everything that I could do to think about the problem that I was working on. The generalizations seemed to be all settled as to true/untrue, and strengthening the results was way above me. And so my task was to find something new to work on. I already had a few meta-ideas: I was going to specialize in something with well-written, easy-to-read papers. Since it takes me longer than everyone else to read a math paper, this is important to me. So far I've noticed two consistently good writers among the papers that I've read. And I've found that people who've recently earned their PhDs tend to leave in all the details (and this goes extra for papers based on their dissertations). Also, work done by people new to the field tends not to require so much background. So I was planning on thinking about something related to work done by either of these two people and/or by new mathematicians -- most ideal would be to find recent students of the good writers.

But now I'm back to asking myself: Do the primitive ideal of A intersect down to maximal ideals of R? And, if so, are the endomorphism rings of the simple A/P-modules algebraic over R/p? And, if so, is the radical of each factor ring of A nilpotent?

Which is a shame (in some ways) as I was going to use the excuse "everything has already been solved by someone else" to justify spending much of the weekend sewing or knitting. I even bought an invisible zipper foot!