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On the abelianization of derived categories and a negative solution to Rosický’s problem

Published online by Cambridge University Press:  06 November 2012

Silvana Bazzoni
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy (email: [email protected])
Jan Šťovíček
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83, 186 75 Praha 8, Czech Republic (email: [email protected])
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Abstract

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

Type
Research Article
Copyright
Copyright © The Author(s) 2012

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