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

Part of: Applied homological algebra and category theory Homological algebra Abelian categories Representation theory of rings and algebras Modules, bimodules and ideals Categories and theories Abelian groups

Published online by Cambridge University Press:  06 November 2012

Silvana Bazzoni  and
Jan Šťovíček
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

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.

Keywords

purityhigher pure global dimensionderived categoryAdams representabilityabelianizationRosický functor

MSC classification

Secondary: 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality 18G25: Relative homological algebra, projective classes 16G99: None of the above, but in this section 18C35: Accessible and locally presentable categories 18E30: Derived categories, triangulated categories 20K25: Direct sums, direct products, etc. 55U35: Abstract and axiomatic homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 125 - 147
Copyright
Copyright © The Author(s) 2012

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