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This paper deals with a class
𝒦N of domains in Stein manifolds and with certain algebras of holomorphic functions
naturally associated with them.
The class 𝒦N consists of those relatively compact domains Δ in N-dimensional
Stein manifolds such that for some neighborhood Ω of Δ and some neighborhood W
of UN, the closure of UN = {(z1,⋯,zN) ∈ CN : |z1|,⋯,|zN| < 1}, the unit polydisc in
CN, there exists a proper holomorphic map Φ : Ω → W which is nonsingular
at every point of Φ−1(TN), TN the distinguished boundary of UN, and
which has, in addition, the property that Δ = Φ−1(UN). The collection of
all such maps Φ is denoted by ℳ(Δ,UN), and if Δ, Δ′∈𝒦N, ℳ(Δ,Δ′)
denotes the set of all maps Ψ: Δ → Δ′ such that if Φ ∈ℳ(Δ′,UN), then
Φ ∘ Ψ ∈ℳ(Δ,UN). For Δ ∈𝒦N let 𝒜(Δ) = {f ∈𝒞(Δ) : f is holomorphic in Δ},
and let H∞(Δ) = {f : f is holomorphic and bounded in Δ}. If Φ ∈ℳ(Δ,Δ′), then
𝒜(Δ) is a module over its subalgebra Φ∗𝒜(Δ′) = {f ∘ Φ : f ∈𝒜(Δ′)}, and
this paper treats the structure of 𝒜(Δ) as a Φ∗𝒜(Δ′)-module. The first
section of the paper presents an example to show that 𝒜(Δ) need not be free
over Φ∗𝒜(Δ′), and the second section shows that it is a finitely generated,
projective Φ∗𝒜(Δ′)-module. The final section establishes certain conditions
sufficient for the freeness of 𝒜(Δ). Parallel results obtain for H∞(Δ) as a
ΦH∞(Δ′)-module.
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