The Stacks project

Proof. Consider $n$ and $m$ cycles $\alpha = \sum _{i \in I} n_ i[X_ i]$ and $\beta = \sum _{j \in J} m_ j[Y_ j]$ with $X_ i \subset X$ and $Y_ j \subset Y$ locally finite families of integral closed subschemes of $\delta$-dimensions $n$ and $m$. Let $K \subset I \times J$ be the set of pairs $(i, j) \in I \times J$ such that $X_ i$ or $Y_ j$ dominates an irreducible component of $S$. Then $\{ X_ i \times _ S Y_ j\} _{(i, j) \in K}$ is a locally finite collection of closed subschemes of $X \times _ S Y$ of $\delta$-dimension $n + m - 1$. This means we can indeed consider

as a $(n + m - 1)$-cycle on $X \times _ S Y$. In this way we obtain an additive map $\times : Z_ n(X) \otimes _{\mathbf{Z}} Z_ m(Y) \to Z_{n + m}(X \times _ S Y)$. The problem is to show that this procedure is compatible with rational equivalence.

Let $i : X' \to X$ be the inclusion morphism of an integral closed subscheme of $\delta$-dimension $n$ which dominates an irreducible component of $S$. Then $p' : X' \to S$ is flat of relative dimension $n - 1$, see More on Algebra, Lemma 15.22.11. Hence flat pullback along $p'$ is an element $(p')^* \in A^{-n + 1}(X' \to S)$ by Lemma 42.33.2 and hence $c' = i_* \circ (p')^* \in A^{-n + 1}(X \to S)$ by Lemma 42.33.4. This produces maps

which sends $[Y']$ to $[X' \times _ S Y']_{n + m - 1}$ for any integral closed subscheme $Y' \subset Y$ of $\delta$-dimension $m$.

Let $i : X' \to X$ be the inclusion morphism of an integral closed subscheme of $\delta$-dimension $n$ such that the composition $X' \to X \to S$ factors through a closed point $s \in S$. Since $s$ is a closed point of the spectrum of a Dedekind domain, we see that $s$ is an effective Cartier divisor on $S$ whose normal bundle is trivial. Denote $c \in A^1(s \to S)$ the gysin homomorphism, see Lemma 42.33.3. The morphism $p' : X' \to s$ is flat of relative dimension $n$. Hence flat pullback along $p'$ is an element $(p')^* \in A^{-n}(X' \to S)$ by Lemma 42.33.2. Thus

by Lemma 42.33.4. This produces maps

which for any integral closed subscheme $Y' \subset Y$ of $\delta$-dimension $m$ sends $[Y']$ to either $[X' \times _ S Y']_{n + m - 1}$ if $Y'$ dominates an irreducible component of $S$ or to $0$ if not.

From the previous two paragraphs we conclude the construction $([X'], [Y']) \mapsto [X' \times _ S Y']_{n + m - 1}$ factors through rational equivalence in the second variable, i.e., gives a well defined map $Z_ n(X) \otimes _{\mathbf{Z}} \mathop{\mathrm{CH}}\nolimits _ m(Y) \to \mathop{\mathrm{CH}}\nolimits _{n + m - 1}(X \times _ S Y)$. By symmetry the same is true for the other variable and we conclude. $\square$