Completions of Countable Excellent Domains and Countable Noncatenary Domains

Abstract

We find necessary and sufficient conditions for a complete local ring containing the rationals to be the completion of a countable excellent local (Noetherian) domain. Furthermore, we find necessary and sufficient conditions for a complete local ring to be the completion of a countable noncatenary local domain, as well as necessary and sufficient conditions for it to be the completion of a countable noncatenary local unique factorization domain.

Completions of Uncountable Local Rings with Countable Spectra

“…We then show that the uncountable domain also has property X. The countable local domain with property X with which we begin comes from previous results in the literature (for example, [6,Theorem 8] and results from [11]).…”

“…We characterize completions of uncountable excellent local domains with countable spectra, assuming that the completion contains the rationals. We make use of results from [11] to accomplish this. The following lemma identifies sufficient conditions on a local ring to be excellent.…”

“…Note that dim A = 1 and so A has a countable spectrum. Since the completion of A is T , the map A → T /M 2 is surjective (see, for example, Proposition 2.4 of [11]). By Proposition 2.7, we have that…”

“…We now use similar arguments as those in Propositions 4.23 and 4.28 to identify sufficient conditions for a complete local ring to be the completion of an uncountable noncatenary local UFD with a countable spectrum. The following results from [1] and [11] characterize completions of noncatenary local UFDs and countable noncatenary local UFDs. (1) no integer of T is a zero divisor, (2) depth (T ) > 1, and (3) there exists P ∈ Min(T ) such that 2 < dim(T /P ) < dim T .…”

“…First, we provide results from [1] and [11], which identify necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local domain and necessary and sufficient conditions for a complete local ring to be the completion of a countable noncatenary local domain.…”