夜凉如水

“I have loved the stars too fondly to be fearful of the night.”

Let me recall one of my recent conversations with friends. So I was learning rigid geometry in the past month. One of fundamental things about rigid geometry was functional analysis over non-archimedean fields. In general, things would be nicer because of strong triangle inequality. However there is this weird thing that Huhn-Banach theorem may fail when working over some non-archimedean fields. One condition of ground field to make sure such a strange thing not happen was to ask the field to be spherically complete, which means a list of nested balls have nonempty intersection.

I was told that C_p is not spherically complete. If you think about it you will find it hard to believe. So suppose one was given nested balls B_i with radii r_i and (naturally?) suppose r_i goes to 0. Then if I pick an a_i from B_i, my claim is that a_i will have limit which is contained in the intersection of B_i (it’s easy to verify). So surprisingly (or maybe I’m the only one shocked by that?) the strange nested balls in C_p has radii bounded from below.

And indeed, let’s construct such a strange nested balls. One important lemma is the following:

Lemma: For every positive integer d, and every field K which is a finite extension of Q_p, there are only finitely many extension L/K of degree less than or equal to d. And let’s call the set of these fields to be F_d.

proof is easy, as every Galois extension of K is solvable. And cyclic extensions of K are understood in local class field theory…

Now let’s take B_1 to be the unit disc in C_p, and I want to construct the balls inductively. Suppose I have found B_i with the following property: r_i is bigger than 1/3+1/3i, and the intersection of B_i with O_L is empty where L runs over the set F_i. Then obviously one can always find a element a_{i+1} in B_i which is far away from all the integers of L in F_{i+1}, now let’s choose B_{i+1} to be the ball with center=a_{i+1} and radius 1/3+1/3(i+1). In this way we are given nested balls B_i and I claim the intersection is empty. Suppose there is an x inside all of B_i, then x is in O_{C_p} and will be at least 1/3 away from all the integers in finite extension of K which is a contradiction of the fact that union of such O_L’s is dense in O_{C_p}.

Anyhow, the amazing/strange fact to me is really that such a nested balls are forced to have a lower bound for the radii.

PS, I know that I promised to post once a month… Not to sure if I can do so, we’ll see.