I was wondering how to prove the two definitions of pullbacks are compatible:
Let $f: X \to Y$ be a morphism between nonsingular curves. Take a point $Q$ on Y, its pullback divisor is $\sum_{P=f(Q)}v_{p}(t)P$. On the other hand, we can look at $Cl(Y) \to Pic(Y)$ by $Q \mapsto \mathscr{L}(Q)$, and the pullback invertible sheaf is $f^{-1}\mathscr{L}(Q) \otimes \mathcal{O}_{x}$. Now how to show that $\mathscr{L}(\sum_{P=f(Q)}v_{p}(t)P)$ is indeed $f^{-1}\mathscr{L}(Q) \otimes \mathcal{O}_{x}$ (i.e. $f^{*}(\mathscr{L}(Q))=\mathscr{L}(f^{*}(Q))$)?
Thank you!