Lemma 42.64.1. The product defined above is associative. More precisely, with $(S, \delta )$ as above, let $X$ be smooth over $S$, let $Y, Z, W$ be schemes locally of finite type over $X$, let $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(Y)$, $\beta \in \mathop{\mathrm{CH}}\nolimits _*(Z)$, $\gamma \in \mathop{\mathrm{CH}}\nolimits _*(W)$. Then $(\alpha \cdot \beta ) \cdot \gamma = \alpha \cdot (\beta \cdot \gamma )$ in $\mathop{\mathrm{CH}}\nolimits _*(Y \times _ X Z \times _ X W)$.
42.64 Intersection products over Dedekind domains
Let $S$ be a locally Noetherian scheme which has an open covering by spectra of Dedekind domains. Set $\delta (s) = 0$ for $s \in S$ closed and $\delta (s) = 1$ otherwise. Then $(S, \delta )$ is a special case of our general Situation 42.7.1; see Example 42.7.3 and discussion in Section 42.63.
Let $X$ be a smooth scheme over $S$. The bivariant class $\Delta ^!$ of Section 42.60 allows us to define a kind of intersection product on chow groups of schemes locally of finite type over $X$. Namely, suppose that $Y \to X$ and $Z \to X$ are morphisms of schemes which are locally of finite type. Then observe that
\[ Y \times _ X Z = (Y \times _ S Z) \times _{X \times _ S X, \Delta } X \]
Hence we can consider the following sequence of maps
\[ \mathop{\mathrm{CH}}\nolimits _ n(Y) \otimes _\mathbf {Z} \mathop{\mathrm{CH}}\nolimits _ m(Y) \xrightarrow {\times } \mathop{\mathrm{CH}}\nolimits _{n + m - 1}(Y \times _ S Z) \xrightarrow {\Delta ^!} \mathop{\mathrm{CH}}\nolimits _{n + m - *}(Y \times _ X Z) \]
Here the first arrow is the exterior product constructed in Section 42.63 and the second arrow is the gysin map for the diagonal studied in Section 42.60. If $X$ is equidimensional of dimension $d$, then $X \to S$ is smooth of relative dimension $d - 1$ and hence we end up in $\mathop{\mathrm{CH}}\nolimits _{n + m - d}(Y \times _ X Z)$. In general we can decompose into the parts lying over the open and closed subschemes of $X$ where $X$ has a given dimension. Given $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(Y)$ and $\beta \in \mathop{\mathrm{CH}}\nolimits _*(Z)$ we will denote
\[ \alpha \cdot \beta = \Delta ^!(\alpha \times \beta ) \in \mathop{\mathrm{CH}}\nolimits _*(Y \times _ X Z) \]
In the special case where $X = Y = Z$ we obtain a multiplication
\[ \mathop{\mathrm{CH}}\nolimits _*(X) \times \mathop{\mathrm{CH}}\nolimits _*(X) \to \mathop{\mathrm{CH}}\nolimits _*(X),\quad (\alpha , \beta ) \mapsto \alpha \cdot \beta \]
which is called the intersection product. We observe that this product is clearly symmetric. Associativity follows from the next lemma.
Proof. By Lemma 42.63.5 we have $(\alpha \times \beta ) \times \gamma = \alpha \times (\beta \times \gamma )$ in $\mathop{\mathrm{CH}}\nolimits _*(Y \times _ S Z \times _ S W)$. Consider the closed immersions
\[ \Delta _{12} : X \times _ S X \longrightarrow X \times _ S X \times _ S X, \quad (x, x') \mapsto (x, x, x') \]
and
\[ \Delta _{23} : X \times _ S X \longrightarrow X \times _ S X \times _ S X, \quad (x, x') \mapsto (x, x', x') \]
Denote $\Delta _{12}^!$ and $\Delta _{23}^!$ the corresponding bivariant classes; observe that $\Delta _{12}^!$ is the restriction (Remark 42.33.5) of $\Delta ^!$ to $X \times _ S X \times _ S X$ by the map $\text{pr}_{12}$ and that $\Delta _{23}^!$ is the restriction of $\Delta ^!$ to $X \times _ S X \times _ S X$ by the map $\text{pr}_{23}$. Thus clearly the restriction of $\Delta _{12}^!$ by $\Delta _{23}$ is $\Delta ^!$ and the restriction of $\Delta _{23}^!$ by $\Delta _{12}$ is $\Delta ^!$ too. Thus by Lemma 42.54.8 we have
\[ \Delta ^! \circ \Delta _{12}^! = \Delta ^! \circ \Delta _{23}^! \]
Now we can prove the lemma by the following sequence of equalities:
\begin{align*} (\alpha \cdot \beta ) \cdot \gamma & = \Delta ^!(\Delta ^!(\alpha \times \beta ) \times \gamma ) \\ & = \Delta ^!(\Delta _{12}^!((\alpha \times \beta ) \times \gamma )) \\ & = \Delta ^!(\Delta _{23}^!((\alpha \times \beta ) \times \gamma )) \\ & = \Delta ^!(\Delta _{23}^!(\alpha \times (\beta \times \gamma )) \\ & = \Delta ^!(\alpha \times \Delta ^!(\beta \times \gamma )) \\ & = \alpha \cdot (\beta \cdot \gamma ) \end{align*}
All equalities are clear from the above except perhaps for the second and penultimate one. The equation $\Delta _{23}^!(\alpha \times (\beta \times \gamma )) = \alpha \times \Delta ^!(\beta \times \gamma )$ holds by Remark 42.61.4. Similarly for the second equation. $\square$
Lemma 42.64.2. Let $(S, \delta )$ be as above. Let $X$ be a smooth scheme over $S$, equidimensional of dimension $d$. The map is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above.
\[ A^ p(X) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{d - p}(X),\quad c \longmapsto c \cap [X]_ d \]
Proof. Denote $g : X \to S$ the structure morphism. The map is the composition of the isomorphisms
\[ A^ p(X) \to A^{p - d + 1}(X \to S) \to \mathop{\mathrm{CH}}\nolimits _{d - p}(X) \]
The first is the isomorphism $c \mapsto c \circ g^*$ of Proposition 42.60.2 and the second is the isomorphism $c \mapsto c \cap [S]_1$ of Lemma 42.63.2. From the proof of Lemma 42.63.2 we see that the inverse to the second arrow sends $\alpha \in \mathop{\mathrm{CH}}\nolimits _{d - p}(X)$ to the bivariant class $c_\alpha $ which sends $\beta \in \mathop{\mathrm{CH}}\nolimits _*(Y)$ for $Y$ locally of finite type over $k$ to $\alpha \times \beta $ in $\mathop{\mathrm{CH}}\nolimits _*(X \times _ k Y)$. From the proof of Proposition 42.60.2 we see the inverse to the first arrow in turn sends $c_\alpha $ to the bivariant class which sends $\beta \in \mathop{\mathrm{CH}}\nolimits _*(Y)$ 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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