Proper pushforward of algebraic cycles

Question

Let $f:X\to Y$ be a finite surjective morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0. Denote by $CH_i(W):=Z_i(W)/\sim$ the Chow group of $i$-cycles on $W$, where $Z_i(W)$ is the group of $i$-cycles on $W$ and $\sim$ means rational equivalence. The proper push-forward $f_\ast:Z_i(X)\to Z_i(Y)$ is a group homomorphism that is compatible with rational equivalence, meaning that it descends to a group homomorphism $f_\ast:CH_i(X)\to CH_i(Y)$. When $i=n−1$, under the above assumptions on $X$ and $Y$ we have that $CH_{n−1}(X)=Pic(X)$ and $CH_{n−1}(Y)=Pic(Y)$. Hence, $f_\ast$ is a group homomorphism $f_\ast:Pic(X)\to Pic(Y)$. Assuming that $Pic(Y)$ and $Pic(X)$ are representable by group schemes (which they are, under our assumptions), my question is:

Is $f_\ast:Pic(X) \to Pic(Y)$ a morphism of group schemes?

Answer 1

Per the request of the OP, I am posting my comment as an answer.

Yes, that morphism of group schemes is called the "norm". One place to read about norms is in Mumford's "Lectures on Curves on an Algebraic Surface", although there is also discussion in GIT, in Knudsen-Mumford, and in Fogarty