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I'm stuck on the following exercise, which seems rather simple:

Let $f: (X,\mathcal{O}_X)\longrightarrow(Y,\mathcal{O}_Y)$ be a morphism of varieties and assume that the corresponding morphism of $k$-algebras $f^{\ast}:\ \mathcal{O}_Y(Y)\longrightarrow\mathcal{O}_X(X)$ is injective. Show that $fX$ is dense in $Y$.

A hint would be very welcome.

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    Consider making the title more descriptive.2012-06-10

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Else, if $y \in Y \setminus \overline{f(X)},$ try to find $\phi \in \mathcal O_Y(Y)$ zero on $\overline{f(X)}$ and such that $\phi(y)=1$. Then...

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    Congratulations: you are a quick thinker!2012-06-10