## 62.12 Action on chow groups

When $\alpha$ is a relative $r$-cycle, the operation $\alpha \cap -$ of Section 62.11 factors through rational equivalence and defines a bivariant class.

Lemma 62.12.1. Let $(S, \delta )$ be as in Section 62.11. Let $f : X' \to X$ be a proper morphism of schemes locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Chow Homology, Definition 42.29.1. Form the diagram

$\xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X }$

as in Chow Homology, Remark 42.29.7. If $\mathcal{L}|_ D \cong \mathcal{O}_ D$, then $i^*f_*\alpha ' = g_*(i')^*\alpha '$ in $Z_ k(D)$ for any $\alpha ' \in Z_{k + 1}(X')$.

Proof. The statement makes sense as all operations are defined on the level of cycles, see Chow Homology, Remark 42.29.6 for the gysin maps. Suppose $\alpha = [W']$ for some integral closed subscheme $W' \subset X'$. Let $W = f(W') \subset X$. In case $W' \not\subset D'$, then $W \not\subset D$ and we see that

$[W' \cap D']_ k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad \text{and}\quad [W \cap D]_ k = \text{div}_{\mathcal{L}|_ W}(s|_ W)$

and hence $f_*$ of the first cycle equals the second cycle by Chow Homology, Lemma 42.26.3. Hence the equality holds as cycles. In case $W' \subset D'$, then $W \subset D$ and both sides are zero by construction. $\square$

Lemma 62.12.2. Let $(S, \delta )$ be as in Section 62.11. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $r \geq 0$ and let $\alpha \in z(X/Y, r)$ be a relative $r$-cycle on $X/Y$. Let $(\mathcal{L}, s, i : D \to Y)$ be as in Chow Homology, Definition 42.29.1. Form the cartesian diagram

$\xymatrix{ E \ar[d] \ar[r]_ j & X \ar[d] \\ D \ar[r]^ i & Y }$

See Chow Homology, Remark 42.29.7. If $\mathcal{L}|_ D \cong \mathcal{O}_ D$, then for $e \in \mathbf{Z}$ the diagram

$\xymatrix{ Z_ e(D) \ar[rr]_{i^*\alpha \cap -} & & Z_{e + r}(E) \\ Z_{e + 1}(Y) \ar[u]^{i^*} \ar[rr]^{\alpha \cap -} & & Z_{r + e + 1}(X) \ar[u]_{j^*} }$

commutes where the vertical arrows $i^*$ and $j^*$ are the gysin maps on cycles as in Chow Homology, Remark 42.29.6.

Proof. Preliminary remark. Suppose that $g : Y' \to Y$ is an envelope (Chow Homology, Definition 42.22.1). Denote $D', i', E', j', X', \alpha '$ the base changes of $D, i, E, j, X, \alpha$ by $g$ and denote $f : X' \to X$ the projection. Assume the lemma holds for $D', i', E', j', X', Y', \alpha '$. Then, if $\beta ' \in Z_{e + 1}(Y')$, we have

\begin{align*} i^*\alpha \cap i^*g_*\beta ' & = i^*\alpha \cap f_*(i')^*\beta ' \\ & = f_*(f^*i^*\alpha \cap (i')^*\beta ') \\ & = f_*((i')^*\alpha ' \cap (i')^*\beta ') \\ & = f_*((j')^*(\alpha ' \cap \beta ')) \\ & = j^*(f_*(f^*\alpha \cap \beta ')) \\ & = j^*(\alpha \cap g_*\beta ') \end{align*}

Here the first equality is Lemma 62.12.1, the second equality is Lemma 62.11.6, the third equality is the definition of $\alpha '$, the fourth equality is the assumption that our lemma holds for $D', i', E', j', X', \alpha '$, the fifth equality is Lemma 62.12.1, and the sixth equality is Lemma 62.11.6. Thus we see that our lemma holds for the image of $g_* : Z_{e + 1}(Y') \to Z_ e(Y)$. However, since $g$ is completely decomposed this map is surjective and we conclude the lemma holds for $D, i, E, j, X, Y, \alpha$.

Let $\beta \in Z_{e + 1}(Y)$. We have to show that $(D \to Y)^*\alpha \cap i^*\beta = j^*(\alpha \cap \beta )$ as cycles on $E$. This question is local on $E$ hence we can replace $X$ and $Y$ by open subschemes. (This uses that formation of the operators $i^*$, $j^*$, $\alpha \cap -$ and $(D \to Y)^*\alpha \cap -$ commute with localization. This is obvious for the gysin maps and follows from Lemma 62.11.2 for the others.) Thus we may assume that $X$ and $Y$ are affine and we reduce to the case discussed in the next paragraph.

Assume $X$ and $Y$ are quasi-compact. By the first paragraph of the proof and Lemma 62.6.9 we may in addition assume that $\alpha$ is in the image of (62.6.8.1). By linearity of the operations in question, we may assume that $\alpha = [Z/X/Y]_ r$ for some closed subscheme $Z \subset X$ which is flat and of relative dimension $\leq r$ over $Y$. Also, as $Y$ is quasi-compact, the cycle $\beta$ is a finite linear combination of prime cycles. Since the operations in question are linear, it suffices to prove the equality when $\beta = [W]$ for some integral closed subscheme $W \subset Y$ of $\delta$-dimension $e + 1$.

If $W \subset D$, then on the one hand $i^*[W] = 0$ and on the other hand $\alpha \cap [W]$ is supported on $E$ so also $j^*(\alpha \cap [W]) = 0$. Thus the equality holds in this case.

Say $W \not\subset D$. Then $i^*[W] = [D \cap W]_ e$. Note that the pullback $i^*\alpha$ of $\alpha = [Z/X/Y]_ r$ by $i$ is $[(E \cap Z)/E/D]_ r$ and that $(E \cap Z) = E \times _ Y Z = D \times _ Y Z$ is flat over $D$. Hence by Lemma 62.11.5 used twice we have

$i^*\alpha \cap i^*[W] = [(E \cap Z) \times _ D (D \cap W)]_{r + e} = [E \cap (Z \times _ Y W)]_{r + e} = j^*(\alpha \cap [W])$

as desired. $\square$

Proposition 62.12.3. Let $(S, \delta )$ be as in Section 62.11. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $r \geq 0$ and let $\alpha \in z(X/Y, r)$ be a relative $r$-cycle on $X/Y$. The rule that to every morphism $g : Y' \to Y$ locally of finite type and every $e \in \mathbf{Z}$ associates the operation

$g^*\alpha \cap - : Z_ e(Y') \to Z_{r + e}(X')$

where $X' = Y' \times _ Y X$ factors through rational equivalence to define a bivariant class $c(\alpha ) \in A^{-r}(X \to Y)$.

Proof. The operation factors through rational equivalence by Lemma 62.12.2 and Chow Homology, Lemma 42.35.1. The resulting operation on chow groups is a bivariant class by Chow Homology, Lemma 42.35.2 and Lemmas 62.11.6, 62.11.3, and 62.12.2. $\square$

Remark 62.12.4. Let $(S, \delta )$ be as in Section 62.11. Let $X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $r \geq 0$. Let $c$ be a rule that to every morphism $g : Y' \to Y$ locally of finite type and every $e \in \mathbf{Z}$ associates an operation

$c \cap - : Z_ e(Y') \to Z_{r + e}(X')$

compatible with proper pushforward, flat pullback, and gysin maps as in Lemma 62.12.2. Then we claim there is a relative $r$-cycle $\alpha$ on $X/Y$ such that $c \cap = g^*\alpha \cap -$ for every $g$ as above. If we ever need this, we will carefully state and prove this here.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).