Lemma 45.14.12. Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A7). Given integers 0 < l < n and a nonempty equidimensional smooth projective scheme X over k consider the projection morphism p : X \times \mathbf{G}(l, n) \to X. Then \gamma commutes with pushforward along p.
Proof. If l = 1 or l = n - 1 then p is a projective bundle and the result follows from Lemma 45.14.9. In general there exists a morphism
h : Y \to X \times \mathbf{G}(l, n)
such that both h and p \circ h are compositions of projective space bundles. Namely, denote \mathbf{G}(1, 2, \ldots , l; n) the partial flag variety. Then the morphism
\mathbf{G}(1, 2, \ldots , l; n) \to \mathbf{G}(l, n)
is a compostion of projective space bundles and similarly the structure morphism \mathbf{G}(1, 2, \ldots , l; n) \to \mathop{\mathrm{Spec}}(k) is of this form. Thus we may set Y = X \times \mathbf{G}(1, 2, \ldots , l; n). Since every cycle on X \times \mathbf{G}(l, n) is the pushforward of a cycle on Y, the result for Y \to X and the result for Y \to X \times \mathbf{G}(l, n) imply the result for p. \square
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