Over the complex numbers, the projective space \(\mathbb{P}^d\) has no cohomology in odd dimensions, and a rank-1 cohomology group in dimensions $$ 1,\; 2, \; 4,\; \dots,\; 2d $$ because it's made of chunks called "Schubert varieties" that are copies of $$ \mathbb{C}^0,\; \mathbb{C}^1,\; \mathbb{C}^2,\; \dots,\; \mathbb{C}^d $$ Over the field with q elements, \(\mathbb{P}^d\) is made of Schubert varieties that are copies of $$ \mathbb{F}_q^0, \; \mathbb{F}_q^1, \; \mathbb{F}_q^2, \; \dots, \; \mathbb{F}_q^d $$ so its number of points is $$ 1 + q + q^2 + \cdots + q^d $$

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