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 (Xan, Oan). 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, …, an} ⊂ 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.