# Push-forward of divisors and intersections

Let $$f:X\rightarrow Y$$ be a surjective finite morphism of varieties, with $$X$$ normal and $$Y$$ smooth. Let $$D\subset X$$ be a divisor and $$C\subset Y$$ a curve. Does the equality $$C\cdot f_{*}D = f^{*}C\cdot D$$ always hold under these hypothesis?

Thank you very much.

• Is $D$ a Cartier divisor? – Mohan Nov 17 '20 at 17:47
• Yes, $D$ is a Cartier divisor on $X$. – SmY Nov 17 '20 at 17:57

Let $$W = C \times_Y X$$. I imagine $$f^*C$$ is suitably interpreted as the chow class on $$W$$ given by $$C \cdot X$$. Write $$i : C \subseteq Y$$, $$f': W \to C$$, $$i' : W \to X$$. (Suppose $$i$$ is l.c.i. so $$i^!$$ makes sense, or use obstruction theories. This is automatic if $$C, Y$$ are smooth). Then Gysin pullback and pushforward commute, so $$i^! f_* [D] = f'_* i'^![D] \in A_*(C)$$. This is one interpretation of your formula, but beware that $$i'^!$$ and $$i^!$$ may differ by the "excess intersection formula" if $$f$$ isn't flat.
• Thank you for the answer. Just a question. Where did you use that $f:X\rightarrow Y$ is finite in your argument? – SmY Nov 19 '20 at 8:50