use this directly. If we wish to stay in the world of algebraic geometry, we may instead "approximate" by projective spaces of larger and larger dimension. Thus we consider the map induced by the -bundle corresponding to This map induces an isomorphism in cohomology in degrees up to 2N. Thus the even (resp. odd) Betti numbers of are 1 (resp. 0), and the l-adic Galois representation on the (2n)th cohomology group is the nth power of the cyclotomic character. The second part is a consequence of the fact that the cohomology of is generated by algebraic cycle classes. This shows that Note that Multiplying by , one obtains the predicted equality. Notes References Theorems in algebraic geometry