Compositions of Consistent Systems of Rank One Discrete Valuation Rings

Abstract

Let V be a rank one discrete valuation ring (DVR) on a field F, and let L/F be a finite separable algebraic field extension with [L: F] = m. The integral closure of V in L is a Dedekind domain that encodes the following invariants: (i) the number s of extensions of V to a valuation ring W _i on L, (ii) the residue degree f _i of W _i over V, and (iii) the ramification degree e _i of W _i over V. These invariants are related by the classical formula . Given a finite set V of DVRs on the field F, an m-consistent system for V is a family of sets enumerating what is theoretically possible for the above invariants of each V ∈ V. The m-consistent system is said to be realizable for V if there exists a finite separable extension field L/F that gives for each V ∈ V the listed invariants. We investigate the realizability of m-consistent systems for V for various positive integers m. Our general technique is to “compose” several realizable consistent systems to obtain new consistent systems that are realizable for V. We apply the new results to the set of Rees valuation rings of a nonzero proper ideal I in a Noetherian domain R of altitude one.

Key Words:

• Dedekind domain
• DVR
• Radical power ideal
• Ramification degree
• Rees valuation ring
• Projectively equivalent
• Projectively full

2000 Mathematics Subject Classification:

• Primary 13A15, 13B02
• Secondary 13G05

Notes

¹Example 5.1 of [2] demonstrates that there exist integrally closed local domains (R, M) for which M is not projectively full. Remark 4.10 and Example 4.14 of [1] show that a sufficient, but not necessary, condition for I to be projectively full is that the gcd of the Rees integers of I is equal to one.

² D may have a residue field D/M _i that has no extension field K _i ⁽¹⁾ with [K _i ⁽¹⁾: (D/M _i )] = e _i ; for example, D/M _i may be algebraically closed, see also Example 3 in [15].

Communicated by I. Swanson.