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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?
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    $\begingroup$ I guess $P_I$ should be $B_I$? $\endgroup$ – ThiKu Jan 29 at 17:42

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