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How can one interprete Grassmanian in a projective space? Why it is true that $Gr(V,r)=Gr(P(V),r-1)$? How to compute the cohomology of Grassmanian?

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    Dear Jacob, your last question is too general: computing the cohomology of grassmannians is a whole sophisticated chapter of algebraic topology or algebraic geometry, for which you should consult a book like that of Bott-Tu. Users should ask specific questions on this site, to which we all are happy to welcome you.2012-06-26

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