The Stacks project

43.26 Chow rings

Let $X$ be a nonsingular projective variety. We define the intersection product

\[ \mathop{\mathrm{CH}}\nolimits _ r(X) \times \mathop{\mathrm{CH}}\nolimits _ s(X) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{r + s - \dim (X)}(X),\quad (\alpha , \beta ) \longmapsto \alpha \cdot \beta \]

as follows. Let $\alpha \in Z_ r(X)$ and $\beta \in Z_ s(X)$. If $\alpha $ and $\beta $ intersect properly, we use the definition given in Section 43.17. If not, then we choose $\alpha \sim _{rat} \alpha '$ as in Lemma 43.24.3 and we set

\[ \alpha \cdot \beta = \text{class of }\alpha ' \cdot \beta \in \mathop{\mathrm{CH}}\nolimits _{r + s - \dim (X)}(X) \]

This is well defined and passes through rational equivalence by Theorem 43.25.2. The intersection product on $\mathop{\mathrm{CH}}\nolimits _*(X)$ is commutative (this is clear), associative (Lemma 43.20.1) and has a unit $[X] \in \mathop{\mathrm{CH}}\nolimits _{\dim (X)}(X)$.

We often use $\mathop{\mathrm{CH}}\nolimits ^ c(X) = \mathop{\mathrm{CH}}\nolimits _{\dim X - c}(X)$ to denote the Chow group of cycles of codimension $c$, see Chow Homology, Section 42.42. The intersection product defines a product

\[ \mathop{\mathrm{CH}}\nolimits ^ k(X) \times \mathop{\mathrm{CH}}\nolimits ^ l(X) \longrightarrow \mathop{\mathrm{CH}}\nolimits ^{k + l}(X) \]

which is commutative, associative, and has a unit $1 = [X] \in \mathop{\mathrm{CH}}\nolimits ^0(X)$.

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