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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