Homology of projective space over finite fields (Q1356046)

Let \(PG_n (q)\) denote the \(n\)-dimensional projective space over the finite field \(\mathbb{F}_q\). The author fixes a field \(F\) whose characteristic divides \(q+1\), and defines the chain group \(C_k\) as all sums of \(k\)-flats in \(PG_n (q)\) with coefficients in \(F\). There is a natural boundary map \(\partial: C_k\to C_{k-1}\), and so he can define homology groups \(H_i (PG_n(q))\). The main theorem is that \[ H_i \bigl(PG_n(q) \bigr)= 0\text{ if } n\text{ is even, or } n\text{ is odd and } i\neq (n-1)/2, \] and that \[ \text{rank} H_i \bigl(PG_n (q)\bigr) =(q-1) (q^3-1) \dots (q^n-1) \text{ if }n \text{ is odd and } i=(n-1)/2. \]