Vol. 29, No. 2, 1969

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Additional results on modules over polydisc algebras

Edgar Lee Stout

Vol. 29 (1969), No. 2, 427–437

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 Φ1(TN), TN the distinguished boundary of UN, and which has, in addition, the property that Δ = Φ1(UN). The collection of all such maps Φ is denoted by ,UN), and if Δ, Δ′∈𝒦N, ,Δ) denotes the set of all maps Ψ: Δ Δsuch that if Φ ∈ℳ,UN), then Φ Ψ ∈ℳ,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), then 𝒜(Δ) is a module over its subalgebra Φ𝒜) = {f Φ : f ∈𝒜)}, and this paper treats the structure of 𝒜(Δ) as a Φ𝒜)-module. The first section of the paper presents an example to show that 𝒜(Δ) need not be free over Φ𝒜), and the second section shows that it is a finitely generated, projective Φ𝒜)-module. The final section establishes certain conditions sufficient for the freeness of 𝒜(Δ). Parallel results obtain for H(Δ) as a ΦH)-module.

Mathematical Subject Classification
Primary: 32.49
Received: 13 June 1968
Published: 1 May 1969
Edgar Lee Stout