The Stacks project

Lemma 42.52.2. In Situation 42.50.1 let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then

\[ P_ p(Z \to X, E) \circ c = c \circ P_ p(Z \to X, E), \]


\[ c_ p(Z \to X, E) \circ c = c \circ c_ p(Z \to X, E) \]

in $A^*(Y \times _ X Z \to X)$.

Proof. This follows from Lemma 42.49.5. More precisely, let

\[ b : W \to \mathbf{P}^1_ X \quad \text{and}\quad Q \quad \text{and}\quad T' \subset T \subset W_\infty \]

be as in the proof of Lemma 42.50.2. By definition $c_ p(Z \to X, E) = c'_ p(Q)$ as bivariant operations where the right hand side is the bivariant class constructed in Lemma 42.49.1 using $W, b, Q, T'$. By Lemma 42.49.5 we have $P'_ p(Q) \circ c = c \circ P'_ p(Q)$, resp. $c'_ p(Q) \circ c = c \circ c'_ p(Q)$ in $A^*(Y \times _ X Z \to X)$ and we conclude. $\square$

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