The Stacks project

Lemma 62.11.6. Let $(S, \delta )$ be as above. Let

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

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

\[ \xymatrix{ W \ar[r] \ar[d] & X \ar[d] \\ Z \ar[r] & Y } \quad \xymatrix{ W' \ar[r] \ar[d] & X' \ar[d] \\ Z' \ar[r] & Y' } \quad \xymatrix{ W' \ar[r] \ar[d] & W \ar[d] \\ Z' \ar[r] & Z } \]

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

\[ f_*(g^*\alpha \cap \beta ') = f_*(g^*\alpha \cap g^*[Y]) = f_*f^*(\alpha \cap [Y]) = d (\alpha \cap [Y]) = \alpha \cap g_*\beta ') \]

as desired. The second equality is Lemma 62.11.3 and the third equality is Chow Homology, Lemma 42.15.2. $\square$

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