Gr1(R^(2,1))

The cohomology of $\text{Gr}_1(\mathbb{R}^{2,1})$

Also known as $S^{1,1}$ or as $\mathbb{RP}^1_{tw}$.
Poincaré polynomial: $$x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{2,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2$$