NONHENSELIAN VALUATION DOMAINS AND THE KRULL-SCHMIDT PROPERTY FOR TORSION-FREE MODULES

Let R be an henselian valuation domain; then the Krull-Schmidt propert y holds for finite rank torsion-free R-modules, namely direct decompos itions of such modules into indecomposable summands are unique up to i somorphism. Vamos gave examples of non-henselian valution domains such that the Krull-Schmidt property holds. We show that the Krull-Schmidt property fails for finite rank torsion-free R-modules, whenever the v aluation domain R contains a prime ideal P such that R/P is nonhenseli an and [K(R/P)<^>:K(R/P)] greater than or equal to 6; K(R/P) is the fi eld of fractions of R/P, and K(R/P)<^> denotes its completion in the t opology of the valuation. We examine nonhenselian valuation domains to which the preceding result is not applicable. If [K(R/P)<^>:K(R/P)] < infinity for every prime ideal P of R, and Spec (R) is well-ordered b y the reverse inclusion, we prove that R has characteristic zero and c ontains a prime ideal J such that [K(R/P)<^>:K(R/P)] is 2 if P = J, an d 1 otherwise. Moreover, K(R/J)<^> is algebraically closed, and K(R/P) is algebraically closed if and only if P superset of J.