6 - The high road to analytification

Summary

Starting with a scheme (X, O) locally of finite type over ℂ we learned, in Chapters 4 and 5, how to construct an analytic space (X^an, O^an). The constructions we gave were local, and as always with local constructions one needs to check that the local data glue well.

There is a high road, which mentions local information as little as possible. In this chapter I sketch this for the interested reader. None of this chapter is essential to what follows. It is most sensible to begin with the high road description of polydiscs.

A coordinate-free approach to polydiscs

If S is a finitely generated ℂ–algebra, and if {a₁, a₂, …, a_n} ⊂ S is a set of generators, we can embed {Spec(S)}^an in ℂⁿ. The embedding allows us to form the open subsets V = Δ(g; w; r) ∩ {Spec(S)}^an of {Spec(S)}^an, which are the intersections of {Spec(S)}^an with polydiscs Δ(g; w; r) ⊂ ℂⁿ. Proposition 5.6.4(i) told us that the subsets of {Spec(S)}^an obtained this way are independent of the choice of generators. But it would still be nice to have a definition, of these open sets in {Spec(S)}^an, which does not mention generators anywhere. In this section we give such a definition.