A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?

Question

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. Then the graph of $s$ could be considered both as a finite correspondence from $Y$ to $X$ and as a finite correspondence from $X$ to $Y$ (in the sense of Voevodsky, or in the sense of the 'classical' intersection theory). Now compose these correspondences to get a correspondence from $X$ to $X$ (using intersection theory); is this true that the result is $d\cdot id_{X}$? I would say yes, since the composite correspondce seems to have only the diagonal component as a cycle in $X\times X$, and it would be very strange if its multiplicity would be anything else but d. Yet I would be glad both for any explanations/references here as well as for pointing out any subtle points (since I don't know much about intersection theory). Does one require any extra conditions here? It would be ok for me to replace $X$ and $Y$ by some open subvarieties, but I don't want to demand $s$ to be generically etale.

Answer 1

Let $f:Y \to X\times Y$ be the graph of $s$ and $f^T:Y \to Y\times X$ the transpose of the graph. By definition you should take the fiber product of $(1\times f^T):X\times Y \to X\times Y\times X$ and $(f\times 1):Y\times X \to X\times Y\times X$. It is easy to see that the fiber product is $Y$. Then we should take the composition of the resulting map $Y \to X\times Y\times X$ with the projection $X\times Y\times X \to X\times X$. It is easy to see that this map factors as $Y \stackrel{s}\to X \stackrel{\Delta}\to X\times X$. This is (by definition) the composition of the correspondences. It remains to note that the pushforward of the fundamental cycle of $Y$ along $s$ is $d$ times the fundametal cycle of $X$, and the the diagonal gives the identity map.