I’m taking an introductory course on algebraic geometry. I’d like to show that $\varphi:C \rightarrow V$ (where $C$ is a smooth projective curve and $V$ an algebraic variety) is a closed immersion iff $\varphi$ is injective and $\varphi^*:O_{V,\varphi(P)} \rightarrow O_{C,P}$ is surjective.
Here $O_{C,P}$ denotes the local ring of the curve $C$ at the point $P$, which I know is equal to the localization of the ring of regular functions on $C$, $O(C)$, at the maximal ideal corresponding to $P$: $O_{C,P}=O(C)_{m_P}$.
Here’s the definition of an immersion I’m using:
Let $(X,O_X)$ and $(Z,O_Z)$ be ringed spaces. Then a morphism $\varphi: (Z,O_Z) \rightarrow (X,O_X)$ is an immersion if it induces an isomorphism of ringed spaces $(Z,O_Z) \cong (\varphi(Z),O_{X |\varphi(Z)})$.
This means that $\varphi$ induces an homeomorphism of $Z$ on $\varphi(Z)$ and that $\varphi^*$ induces an isomorphism between sheafs of functions $O_{X|\varphi(Z)} \cong O_Z$. Here $O_{X |\varphi(Z)}$ is the sheaf of functions that are locally restrictions of functions in $O_X$. To be more precise, if $Y \subset X$ is a subset of the topological space $X$,
$$O_{X|Y}(V) := \{ f:V \rightarrow k ~|~ \forall y \in V,~ \exists U \text{ open subset and } \exists g\in O_X(U) \text{ such that } y\in U \text{ and } f_{|V\cap U}=g_{|V\cap U} \}$$
Also, we say the immersion is closed if $\varphi(Z)$ is a closed subspace in $X$.
Apart from that, I just think I know so far that $\varphi$ being an immersion implies that it is injective and $\varphi^*:O_{V} \rightarrow O_{C}$ is surjective, but I don’t know how to conclude that the induced map on the local rings must be surjective too (should the universal property of localization be used here?). For the other direction, I don’t know how to proceed.
Any help would be welcome as I’m not yet familiar with all these concepts.