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We prove for a large family of rings R that their lambda-pure globaldimension is greater than one for each infinite regular cardinal lambda. Thisanswers in negative a problem posed by Rosicky. The derived categories of suchrings then do not satisfy the Adams lambda-representability for morphisms forany lambda. Equivalently, they are examples of well generated triangulatedcategories whose lambda-abelianization in the sense of Neeman is not a fullfunctor for any lambda. In particular we show that given a compactly generatedtriangulated category, one may not be able to find a Rosicky functor among thelambda-abelianization functors.

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