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This question refers to the Definition in p. 137 from Hartshorne. Let $f: X \rightarrow Y$ be a finite morphism of nonsingular curves. Let $Q$ be a closed point of $Y$ and $P \in X$ such that $f(P)=Q$. Let $t \in \mathcal{O}_{Y,Q}$. Then $\mathcal{O}_{Y,Q}$ is a discrete valuation ring and we can talk about the quantity $v_Q(t)$, where $v_Q$ is the corresponding valuation. Now, $\mathcal{O}_{X,P}$ is also a discrete valuation ring with valuation $v_P$, but Hartshorne is talking about the quantity $v_P(t)$. Does $v_P(t)$ really mean $v_P\left(f_Q(t)\right)$, where $f_Q : \mathcal{O}_{Y,Q} \rightarrow \mathcal{O}_{X,P}$ is the local ring homomorphism induced by the morphism $f$?

The above notation $v_P(t)$ appears in the definition of the group homomorphism $f^*:Div(Y) \rightarrow Div(X)$ by $f^*Q = \sum_{f(P)=Q} v_P(t) \cdot P$

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Dear Manos, you should change your notation for the induced local morphism to $f^*_P : \mathcal{O}_{Y,Q} \rightarrow \mathcal{O}_{X,P}$, because that morphism depends on $P$, not on $Q$. In other words if several $P_i$'s are sent by $f$ to the same $Q$ the induced local morphisms $f^*_{P_i} : \mathcal{O}_{Y,Q} \rightarrow \mathcal{O}_{X,P_i}$ are different and your notation would then be ambiguous.Anyway, you should adopt this modified notation because it is the one universally used since EGA was published :-) –  Georges Elencwajg Oct 23 '12 at 20:04

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Yes. And I believe he's taking $t$ to be a uniformizing parameter of the DVR $\mathcal{O}_{Y,Q}$ in the definition of that homomorphism.

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