p-adic Orlik-Solomon algebras

In this blog post I’m going to finish up discussing the classical Orlik-Solomon algebra and then move on to discuss section 2.1 of de Shalit. Note that I’ve made the title of this post up on a whim, and I have no idea if it’s at all close to standard terminology. 1. The Orlik-Solomon algebra [...]

Hyperplane arrangements

Let be a finite extension of . We are on our way towards understanding the cohomology of Drinfeld’s -adic symmetric domain of dimension over , and its connection with the building of . Recall that Drinfeld’s domain is with all -rational hyperplanes removed. When these spaces are a little mysterious to me, so a more [...]

An announcement

Dear reader(s): We proudly announce the first release of our code for computing with arithmetic quotients of the Bruhat-Tits of . We hope that it will make it into Sage some day, but for now it is available on a private space in Assembla. If you want to try it out, here is what you [...]

The Bruhat-Tits building of PGL(n+1) (III)

It is now time to introduce the action of on our building. Of course, acts on homothethy classes of flags on the left, by acting on the vector space (remember that this is the definition of ). Here is where types start to become relevant: the action of can’t be transitive in general: it will [...]

The Bruhat-Tits tree

In this post I’ll specialize Marc’s discussion to the building of . It turns out that the building is an infinite tree in this case. 1. Definition Recall that the vertices of are lattices in taken up to rescaling. If is a lattice, then we’ll write for the homothety class of -multiples of . Two [...]

The Bruhat-Tits building of PGL(n+1) (II)

As promised in the previous post, we will start this one with distances on the BT building . First, if are two vertices, then one can find bases of adapted to them: that is, so that is represented by the standard -lattice , and is represented by some sublattice of the form This is just [...]

The Bruhat-Tits building of PGL(n+1)

We will be following [dS] for a while. Let’s fix some notation: take a finite extension of , with a choice of uniformizer . Let denote the size of and normalize the norm on (and on for that matter) so that . Fix also a vector space over of dimension , and denote by the [...]

The name of the game

The first goal of this blog is to understand the paper “Residues on buildings, and de Rham cohomology of -adic symmetric domains” of Ehud de Shalit. C and i have been playing around trees for a while now, and decided that if a computer can deal with trees, then it should also be able to [...]