# About a theorem in Hartshorne's AG

1. Apr 14, 2008

### eof

Hi,

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.

Can you offer guidance or do you also need help?
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