Holomorphic Extensions of Δ-Domains

Abstract

In this chapter we take up the study of extensions of a Δ-domain relative to algebras of germ-valued rather than complex-valued functions. If (Ф, p) and (Ψ, q) are two A-domains and ρ: (Φ, p) → (Ψ, q) is a morphism of these domains then since p is an open local homeomorphism (Lemma 48.1) we always have [0]_Ψ∘ρ [0]_Φ just as for complex-valued functions. Moreover if Ψ is connected then by Proposition 68.3 the mapping of [0]_Ψ into [0]_Φ is an isomorphism. Also, since p = q∘ρ the isomorphism obviously preserves derivatives. As in the case of complex-valued functions, the morphism is called an extension of (Φ, p) relative to [0]_Ψ provided [0] _Ψ ∘ρ = [0]_Φ, and we write

$$ \rho :(\Phi, p) \Rightarrow (\Psi, q) $$

using the “closed” arrow “=>“in place of the “open” arrow “=>“in order to distinguish the germ-valued from the complex-valued case. Maximal extensions and maximal A-domains for germ-valued functions may be defined exactly as in the case of complex-valued functions.