Lemma 42.51.2. In the situation of Definition 42.49.4 assume $P_ p(Z \to X, E)$, resp. $c_ p(Z \to X, E)$ is defined. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then

respectively

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

Lemma 42.51.2. In the situation of Definition 42.49.4 assume $P_ p(Z \to X, E)$, resp. $c_ p(Z \to X, E)$ is defined. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then

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

respectively

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

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

**Proof.**
This follows from Lemma 42.48.4. Namely, our assumptions say $E$ is represented to a bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ X$-modules. Let

\[ b : W \to \mathbf{P}^1_ X \quad \text{and}\quad \mathcal{Q}^\bullet \]

be the blowing up and complex of $\mathcal{O}_ W$-modules constructed in More on Flatness, Section 38.44. Let $T \subset W_\infty $ be the closed subscheme whose existence is averted in More on Flatness, Lemma 38.44.1. Let $T' \subset T$ be the open and closed subscheme such that $\mathcal{Q}^\bullet |_{T'}$ is zero, resp. isomorphic to a finite locally free sheaf of rank $< p$ placed in degree $0$. By definition

\[ c_ p(Z \to X, E) = c'_ p(\mathcal{Q}^\bullet ) \]

as bivariant operations (and not just on cycles over $X$) where the right hand side is the bivariant class constructed in Lemma 42.48.1 using $W, b, \mathcal{Q}^\bullet , T'$. By Lemma 42.48.4 we have

\[ P'_ p(\mathcal{Q}^\bullet ) \circ c = c \circ P'_ p(\mathcal{Q}^\bullet ) \quad \text{resp.}\quad c'_ p(\mathcal{Q}^\bullet ) \circ c = c \circ c'_ p(\mathcal{Q}^\bullet ) \]

in $A^*(Y \times _ X Z \to X)$ and we conclude. $\square$

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