Krull Dimensions of Rings of Holomorphic Functions

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least the cardinality of continuum if and only if it is > 0. Let R be a commutative ring. Recall that the Krull dimension dim(R) of R is the supremum of cardinalities lengths of chains of distinct proper prime ideals in R. Our main result is: Theorem 1. Let M be a connected complex manifold and H(M) be the ring of holomorphic functions on M . Then the Krull dimension of H(M) either equals 0 (if and only if H(M) = C) or is infinite, if and only if M admits a nonconstant holomorphic function M → C. More precisely, unless H(M) = C, dimH(M) ≥ c, i.e., the ring H(M) contains a chain of distinct prime ideals whose length has cardinality of continuum. Our proof of this theorem mostly follows the lines of the proof by Sasane [S], who proved that for each nonempty domain M ⊂ C the Krull dimension of H(M) is infinite (he did not prove that dimH(M) ≥ c). Remark 2. We note that Henricksen [H] was the first to prove that the Krull dimension of the ring of entire functions on C has cardinality at least continuum. In our proof we will use the Axiom of Choice in two ways: (a) to establish existence of certain maximal ideals and (b) to get existence of a nonprincipal ultrafilter ω on N and, hence of the ordered field ∗R of nonstandard real (or, hyperreal) numbers. The field ∗R contains ∗N, the nonstandard natural (or hypernatural) numbers. The field ∗R is a certain quotient of the countable direct product ∏ k∈N R; we will denote the equivalence class (in ∗R) of a sequence (xk) in R by [xk]. Accordingly, ∗N consists of equivalence classes [nk] of sequences of natural numbers. Roughly speaking, we will use ∗N and a certain order relation on it to compare rates of growth of sequences of natural numbers. 2010 Mathematics Subject Classification. Primary 32A10, 16P70.