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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.

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 $\mathop{\mathrm{CH}}\nolimits _*(Y \times _ X Z \times _ X W)$.

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

\[ \Delta _{12} : X \times _ S X \longrightarrow X \times _ S X \times _ S X, \quad (x, x') \mapsto (x, x, x') \]


\[ \Delta _{23} : X \times _ S X \longrightarrow X \times _ S X \times _ S X, \quad (x, x') \mapsto (x, x', x') \]

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

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

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

Proof. Denote $g : X \to S$ 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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