# Cohomology ring of Grassmannians

I’m reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing):

Let $w=1+w_1+ \ldots + w_m$ be the total Stiefel-Whitney class of the canonical $m$-plane bundle over $G_m(\mathbb{R}^{m+n})$ and let $\bar{w}=1+\bar{w_1}+\ldots+ \bar{w_n}$ be its dual. Then $H^\ast G_m (\mathbb{R}^{m+n})$ is the quotient of the polynomial algebra $\mathbb{Z}_2[1,w_1,\ldots,w_m]$ by the ideal $(\bar w_{m+1},\cdots,\bar {w}_{m+n})$ generated by the relation $w\bar{w}=1$.

The reference provided is to Borel’s La cohomolgie mod 2 de espaces homogenes. As the title suggests, this paper is in French, a language with which I am not familiar.

I’m familiar with the fact that the cohomology ring of the infinite Grassmannian $G_m(\mathbb{R}^\infty)$ is freely generated by $w_1,\ldots,w_m$ over $\mathbb{Z}_2$ (as proved in Hatcher’s Vector Bundles), but I can’t see how to prove this variant. Any help would be much appreciated. Perhaps someone can even translate the proof given in Borel’s paper.