# Homology of a semisimplicial scheme

This is a question about the homology of a complex made of algebraic varieties. Consider the following subgroups of $$\mathrm{SL}_3$$ (defined over $$\mathbb{Z}$$). $$P_{1,2} = \left\{\left(\begin{smallmatrix}*&*&*\\&*&*\\&&*\end{smallmatrix}\right)\right\} \quad P_{1} = \left\{\left(\begin{smallmatrix}*&*&*\\*&*&*\\&&*\end{smallmatrix}\right)\right\} \quad P_{2} = \left\{\left(\begin{smallmatrix}*&*&*\\&*&*\\&*&*\end{smallmatrix}\right)\right\} \quad P_{\emptyset} = \left\{\left(\begin{smallmatrix}*&*&*\\*&*&*\\*&*&*\end{smallmatrix}\right)\right\}$$

$$U = U_{1,2} = \left\{\left(\begin{smallmatrix}1&&\\*&1&\\*&*&1\end{smallmatrix}\right)\right\} \quad U_{1} = \left\{\left(\begin{smallmatrix}1&&\\&1&\\*&*&1\end{smallmatrix}\right)\right\} \quad U_{2} = \left\{\left(\begin{smallmatrix}1&&\\*&1&\\*&&1\end{smallmatrix}\right)\right\} \quad U_{\emptyset} = \left\{1\right\}$$ For each $$I \subseteq \{1,2\}$$ let $$V_I = UP_I/P_I$$. This can be identified with $$U_I$$ and is therefore a variety (a $$\mathbb{Z}$$-scheme, I suppose). For $$I \supseteq J$$ the map $$gP_I \mapsto gP_J$$ defines a morphism $$V_I \to V_J$$. The $$V_I$$ together with these maps form a semisimplicial object in the category of schemes over $$\mathbb{Z}$$.

So there should be an associated homology consisting of functors $$H_i \colon \mathcal{Fields} \to \mathbb{Z}\mathcal{-mod}$$, $$F \mapsto H_i(V_*(F);\mathbb{Z})$$.

My (vague) question is this: is there a way to work with this homology (e.g. compute it) that makes use of the variety structure rather than evaluate it field by field? (There is an obvious generalization to $$\mathrm{SL}_n$$ for other $$n$$ and a slightly less obvious one to other Chevalley groups; these are implicitly included in the question.)

I would be grateful for relevant references and clarifications about imprecisions in the exposition.

Context: For any field $$F$$ the associated complex is the augmentation of the complex opposite a chamber in a spherical building and Abramenko shows that its homology vanishes in all degrees except for the top degree provided $$F$$ is (explicitly) large compared to $$n$$. My motivation for the question is twofold:

1. Could it be possible to recover Abramenko's result (without the explicit bound) for abstract model-theoretic reasons? (Like: it is true for infinite fields so it has to be true for large enough finite fields.)
2. Abramenko's bound on the size of $$F$$ is exponential in $$n$$, yet no counterexample is known for $$F \ne \mathbb{F}_2,\mathbb{F}_3$$. So conceivably there is a bad prime phenomenon that is uniform to all complexes. Can one guess where in the above description such a bad prime phenomenon arises?
• I guess $P_I$ should be $B_I$? – ThiKu Jan 29 at 17:42