Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are inclusions. Let $\mathcal{N}(\mathcal{C})$ be the nerve of $\mathcal{C}$, and $X:=|\mathcal{N}(\mathcal{C})|$ its geometric realization. What is the cohomology ring $H^*(X,k)$?
I'm having trouble working this out even in the case $n=2$. Any help or references would be appreciated. Thank you.
The simplicial complex you describe is called the Tits building for $GL_n(k)$. When $k=\mathbb{F}_p$ it is homotopy equivalent to a wedge of $p^{n(n-1)/2}$ spheres of dimension $n-2$. I think that theorem is due to Quillen, and am pretty sure that volume II of Benson's Representations and Cohomology contains a proof. I don't know what happens for other fields.