On the abelianization of derived categories and a negative solution to Rosicky's problem

We prove for a large family of rings R that their lambda-pure global dimension is greater than one for each in finite regular cardinal lambda. This answers in the negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy, for any lambda, the Adams lambda-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose lambda-abelianization in the sense of Neeman is not a full functor for any lambda. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosicky functor among the lambda-abelianization functors.