Lemma 41.51.2. Let $k$ be a field. Let $X$ be a smooth scheme over $k$, equidimensional of dimension $d$. The map

is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above.

Lemma 41.51.2. Let $k$ be a field. Let $X$ be a smooth scheme over $k$, equidimensional of dimension $d$. The map

\[ A^ p(X) \longrightarrow A_{d - p}(X),\quad c \longmapsto c \cap [X]_ d \]

is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above.

**Proof.**
Denote $g : X \to \mathop{\mathrm{Spec}}(k)$ the structure morphism. The map is the composition of the isomorphisms

\[ A^ p(X) \to A^{p - d}(X \to \mathop{\mathrm{Spec}}(k)) \to A_{d - p}(X) \]

The first is the isomorphism $c \mapsto c \circ g^*$ of Proposition 41.49.2 and the second is the isomorphism $c \mapsto c \cap [\mathop{\mathrm{Spec}}(k)]$ of Lemma 41.50.2. From the proof of Lemma 41.50.2 we see that the inverse to the second arrow sends $\alpha \in A_{d - p}(X)$ to the bivariant class $c_\alpha $ which sends $\beta \in A_*(Y)$ for $Y$ locally of finite type over $k$ to $\alpha \times \beta $ in $A_*(X \times _ k Y)$. From the proof of Proposition 41.49.2 we see the inverse to the first arrow in turn sends $c_\alpha $ to the bivariant class which sends $\beta \in A_*(Y)$ for $Y \to X$ locally of finite type to $\Delta ^!(\alpha \times \beta ) = \alpha \cdot \beta $. From this the final result of the lemma follows. $\square$

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