On a non-vanishing Ext

The existence of valuation domains admitting non-standard uniserial modules for which certain Exts do not vanish was proved in [1] under Jensen’s Diamond Principle. In this note, the same is verified using the ZFC axioms alone. In the construction of large indecomposable divisible modules over certain valuation domainsR, the first author used the property thatR satisfied Ext1R(Q,U) 6= 0, where Q stands for the field of quotients of R (viewed as an R-module) and U denotes any uniserial divisible torsion R-module, for instance, the module K = Q/R; see [1]. However, both the existence of such a valuation domain R and the nonvanishing of Ext were established only under Jensen’s Diamond Principle ♦ (which holds true, e.g., in Gödel’s Constructible Universe). Our present goal is to get rid of the Diamond Principle, that is, to verify in ZFC the existence of valuation domains R that admit divisible non-standard uniserial modules and also satisfy Ext1R(Q,U) 6= 0 for several uniserial divisible torsion R-modules U . (For the proof of Corollary 3, one requires only 6 such U .) We start by recalling a few relevant definitions. By a valuation domain we mean a commutative domain R with 1 in which the ideals form a chain under inclusion. A uniserial R-module U is defined similarly as a module whose submodules form a chain under inclusion. K = Q/R is a uniserial torsion R-module, it is divisible in the sense that rK = K holds for all 0 6= r ∈ R. A divisible uniserial R-module is called standard if it is an epic image of the uniserial module Q; otherwise it is said to be non-standard. The existence of valuation domains admitting non-standard uniserials has been established in ZFC; see e.g. [3], [2], and the literature quoted there. As the R-module Q is uniserial, it can be represented as the union of a wellordered ascending chain of cyclic submodules: R = Rr0 < Rr −1 1 < . . . < Rr α < . . . < ⋃ α<κRr −1 α = Q (α < κ), where r0 = 1, rα ∈ R, and κ denotes an infinite cardinal and also the initial ordinal of the same cardinality. As a consequence, K = ⋃ α<κ(Rr −1 α /R) where Rr −1 α /R ∼= 1 The second author was supported by the German-Israeli Foundation for Scientific Research and Development. Publication 766.