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This is lemma 3.6 page 20 of Hartshorne's Algebraic Geometry book. Let $X$ be a variety and let $Y$ be an affine variety. A map $f: X \rightarrow Y$ is a morphism if and only if $x_{i} \circ f$ is a regular function on $X$ for each $i$ where $x_{i}$ are the coordinate functions.

The first part says: if $f$ is a morphism then $x_{i} \circ f$ must be regular by definition of morphism. I'm assuming he's considering the pullback, however in order to do this we need to know that each $x_{i}$ is a regular function on $X$, why is this?

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    @Dylan Moreland: never mind, I see my mistake now, thanks.2012-05-11

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