$f : X\longrightarrow Y$ isomorphism $ \Rightarrow (?)\, \, f^*:Div(Y)\longrightarrow Div(X)$ isomorphism.

Question

Let $X$ be a complex manifold. Denote by Div$(X)$ the Weil divisors group of $X$.

We have to:

Let $f : X \longrightarrow Y$ be a holomorphic map of connected complex manifolds and suppose that $f$ is dominant, i.e. $f(X)$ is dense in $Y$. Then the pull-back defines a group homomorphism $$ f^* :Div(Y) \longrightarrow Div(X). $$

Question: If $f : X \longrightarrow Y$ is an analytic isomorphism ( bi-holomorphic map), then is it true that $f^*:Div(Y)\longrightarrow Div(X)$ is a group isomorphism?

Thanks

Answer 1

The inverse map gives rise to a map on Div's in the opposite direction. Since $\text{Div}(\cdot)$ is functorial, and composition of $f$ with its inverse gives the identity map, this means that the map on Div's is inverse to the original one.