The version of GAGA we plan to prove is about projective space, and it is high time that we disclose to the reader what projective space is. The differential geometer would think of projective space ℂℙn as the quotient of ℂn+1 − {0} by the action of ℂ*. Let us remind the reader.

The group ℂ* is the group of non-zero complex numbers under multiplication. The group ℂ* acts on ℂn+1 by the following rule: the action sends the pair (λ, x), where λ ∈ ℂ* and x ∈ ℂn+1, to λ−1x ∈ ℂn+1. The point 0 ∈ ℂn+1 is a fixed point for this action, and therefore ℂ* acts on the complement ℂn+1 − {0} of 0 ∈ ℂn+1. The orbit space is the space of lines through the origin in ℂn+1, and this is what a differential geometer would understand by projective space ℂℙn.

We have become more sophisticated by now. We would expect to define a ringed space (ℙn, O), which if we are lucky should be a scheme of finite type over ℂ, and so that {ℙ}an = ℂℙn. The object of the next two chapters is to give the construction. The idea is simple enough. The topological space ℂn+1 can most definitely be viewed as Xan for a scheme (X, OX) of finite type over ℂ; nothing could be easier.