Rational cohomology ring of complex Grassmannian.

Question

The rational cohomology ring of complex projective space $\mathbb{CP}^{m}$ is $\frac{\mathbb{Q}[X]}{(X^{m+1})}.$ This is given in the book of Allen Hatcher. I want to understand the rational cohomology ring of complex Grassmannian. The simplest case after complex projective spaces is $Gr(m,2).$ What is the rational cohomology ring of $Gr(m,2)?$ Moreover what is the Betti numbers of $Gr(m,2).$ If some one know good reference about general case please share with me. In the paper ''ENDOMORPHISMS OF THE COHOMOLOGY OF COMPLEX GRASSMANNIANS'' MICHAEL HOFFMAN cite a paper of A. Borel in which the presentation is given but this paper is not in English.

Answer 1

There is a presentation of the (integral) cohomology ring given in Section 5.8 (in the published version) of the book 3264 And All That by Eisenbud and Harris. (They talk in terms of the Chow ring, but in this case the two things are the same.)