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

Part of:Abelian groupsCategories and theoriesHomological algebraModules, bimodules and idealsRepresentation theory of rings and algebrasApplied homological algebra and category theoryAbelian categories

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: bazzoni@math.unipd.it)
Jan Šťovíček
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83, 186 75 Praha 8, Czech Republic (email: stovicek@karlin.mff.cuni.cz)
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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.

Keywords

purityhigher pure global dimensionderived categoryAdams representabilityabelianizationRosický functor

MSC classification

Secondary:16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality18G25: Relative homological algebra, projective classes16G99: None of the above, but in this section18C35: Accessible and locally presentable categories18E30: Derived categories, triangulated categories20K25: Direct sums, direct products, etc.55U35: Abstract and axiomatic homotopy theory

Information

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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