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I've been reading Hartshorne on my own for the past month and stumbled up on something that seems important, but I can't prove... It is proposition II.2.16.

The part I've been unable to prove is basically the last part of exercise II.2.15. After showing part a) and b) it's trivial to prove that a morphism of schemes t(V)->t(W) induces a map f:V->W and the injectivity part is also trivial. But how do you continue proving that this is a morphism of varieties?

So how do you prove that:

1. the map f:V->W is a morphism?

2. the morphism of sheaves of regular functions induced by f is the same sheaf morphism as the sheaf morphisms in the morphism of schemes?

The exercises and theorems are quite long, so I won't repeat them here. I assume that every algebraic geometer has a copy of Hartshorne. I've been banging my head into this for quite some time already, so I'm hoping that someone could help.

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# About a theorem in Hartshorne's AG

Can you offer guidance or do you also need help?

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