Valuation domains with a maximal immediate extension of finite rank

If R is a valuation domain of maximal ideal P with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals P = L0 ⊃ L1 ⊃ · · · ⊃ Lm ⊇ 0 such that RLj /Lj+1 is almost maximal for each j, 0 ≤ j ≤ m − 1 and RLm is maximal if Lm 6= 0. Then we suppose that there is an integer n ≥ 1 such that each torsion-free R-module of finite rank is a direct sum of modules of rank at most n. By adapting Lady’s methods, it is shown that n ≤ 3 if R is almost maximal, and the converse holds if R has a maximal immediate extension of rank ≤ 2. Let R be a valuation domain of maximal ideal P , R̂ a maximal immediate extension of R, R̃ the completion of R in the R-topology, and Q, Q̂, Q̃ their respective fields of quotients. If L is a prime ideal of R, as in [5], we define the total defect at L, dR(L), the completion defect at L, cR(L), as the rank of the torsion-free R/L-module (̂R/L) and the rank of the torsion-free R/L-module (̃R/L), respectively. Recall that a local ring R isHenselian if each indecomposable module-finite R-algebra is local and a valuation domain is strongly discrete if it has no non-zero idempotent prime ideal. The aim of this paper is to study valuation domains R for which dR(0) < ∞. The first example of a such valuation domain was given by Nagata [11]; it is a Henselian rank-one discrete valuation domain of characteristic p > 0 for which dR(0) = p. By using a generalization of Nagata’s idea, Facchini and Zanardo gave other examples of characteristic p > 0, which are Henselian and strongly discrete. More precisely: Example 0.1. [5, Example 6] For each prime integer p and for each finite sequence of integers l(0) = 1, l(1), . . . , l(m) there exists a strongly discrete valuation domain R with prime ideals P = L0 ⊃ L1 ⊃ · · · ⊃ Lm = 0 such that cR(Li) = p, ∀i, 1 ≤ i ≤ m. So, dR(0) = p ( Pi=m i=1 l(i)) by [5, Corollary 4]. Theorem 0.2. [5, Theorem 8] Let α be an ordinal number, l : α+1 → N∪ {∞} a mapping with l(0) = 1 and p a prime integer. Then there exists a strongly discrete valuation domain R and an antiisomorphism α+ 1 → Spec(R), λ 7→ Lλ, such that cR(Lλ) = p , ∀λ ≤ α. So, if l(λ) = 0, ∀λ ≤ α, except for a finite subset, then dR(0) < ∞ by [5, Corollary 4]. 2000 Mathematics Subject Classification. Primary 13F30, 13C11.