Lemma 62.11.6. Let $(S, \delta )$ be as above. Let
be a cartesian diagram of schemes locally of finite type over $S$ with $g$ proper. Let $r, e \geq 0$. Let $\alpha $ be a family of $r$-cycles on the fibres of $X/Y$. Let $\beta ' \in Z_ e(Y')$. Then we have $f_*(g^*\alpha \cap \beta ') = \alpha \cap g_*\beta '$.
Proof. Since we are proving an equality of cycles on $X$, we may work locally on $Y$, see Lemma 62.11.2. Thus we may assume $Y$ is affine. Thus $Y'$ is quasi-compact. In particular $\beta '$ is a finite linear combination of prime cycles. Since $- \cap -$ is linear in the second variable (Lemma 62.11.1), it suffices to prove the equality when $\beta ' = [Z']$ for some integral closed subscheme $Z' \subset Y'$ of $\delta $-dimension $e$. Set $Z = g(Z')$. This is an integral closed subscheme of $Y$ of $\delta $-dimension $\leq e$. For simplicity we are going to assume $Z$ has $\delta $-dimension equal to $e$ and leave the other case (which is easier) to the reader. Let $y \in Z$ and $y' \in Z'$ be the generic points. Write $\alpha _ y = \sum m_ j[V_ j]$ with $V_ j \subset X_ y$ integral closed subschemes of dimension $r$.
Assume first $g$ is a closed immersion. Then $g_*\beta ' = [Z]$ and $(g^*\alpha )_{y'} = \sum n_ j[V_ j]$; this makes sense because $V_ j$ is contained in the closed subscheme $X'_{y'}$ of $X_ y$. Thus in this case the equality is obvious: in both cases we obtain $\sum m_ j[\overline{V}_ j]$ where $\overline{V}_ j$ is the closure of $V_ j$ in the closed subscheme $X' \subset X$.
Back to the general case with $\beta ' = [Z']$ as above. Set $W = Z \times _ X Y$ and $W' = Z' \times _{X'} Y'$. Consider the cartesian squares
Since we know the result for the first two squares with by the previous paragraph, a formal argument shows that it suffices to prove the result for the last square and the element $\beta ' = [Z'] \in Z_ e(Z')$. This reduces us to the case discussed in the next paragraph.
Assume $Y' \to Y$ is a generically finite morphism of integral schemes of $\delta $-dimension $e$ and $\beta ' = [Y']$. In this case both $f_*(g^*\alpha \cap \beta ')$ and $\alpha \cap g_*\beta '$ are cycles which can be written as a sum of prime cycles dominant over $Y$. Thus we may replace $Y$ by a nonempty open subscheme in order to check the equality. After such a replacement we may assume $g$ is finite and flat, say of degree $d \geq 1$. Of course, this means that $g_*\beta ' = g_*[Y'] = d[Y]$. Also $\beta ' = [Y'] = g^*[Y]$. Hence
as desired. The second equality is Lemma 62.11.3 and the third equality is Chow Homology, Lemma 42.15.2. $\square$
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)