The Ramification Tree and Almost Dedekind Domains of Prescribed SP-Rank

Given a valuation [Formula Presented] and a chain [Formula Presented] of finite extensions of [Formula Presented], we construct a weighted tree [Formula Presented] encoding information about the ramification of [Formula Presented] in the extensions [Formula Presented]; conversely, when [Formula Presented] is a discrete valuation, we show that a weighted tree [Formula Presented] can be expressed as [Formula Presented] under some mild hypotheses on [Formula Presented] or on [Formula Presented]. We use this correspondence to construct, for every countable successor ordinal number [Formula Presented] and every discrete valuation ring V, an almost Dedekind domain D integral over V whose SP-rank is [Formula Presented]. Subsequently, we extend this result to countable limit ordinal numbers by considering integral extensions of Dedekind domains with countably many maximal ideals.