Almost fully decomposable infinite rank lattices over orders

Let ΛΛ be an order over a Dedekind domain RR with quotient field KK. An object of Λ-Lat, the category of RR-projective ΛΛ-modules, is said to be fully decomposable   if it admits a decomposition into (finitely generated) ΛΛ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105–128], we give a necessary and sufficient criterion for RR-orders ΛΛ in a separable KK algebra AA with the property that every L∈Λ-Lat is fully decomposable. In the present paper, we assume that A/RadA is separable, but that the pp-adic completion ApAp is not semisimple for at least one p∈SpecR. We show that there exists an L∈Λ-Lat, such that KLKL admits a decomposition KL=M0⊕M1KL=M0⊕M1 with M0∈A-mod finitely generated, where L∩M1L∩M1 is fully decomposable, but LL itself is not fully decomposable.