Unbounded Derived Categories

Abstract

AbstractIn this chapter we study the unbounded derived categories of Grothendieck categories, using the results of Chap. 9. We prove the existence of enough homotopically injective objects in order to define unbounded right derived functors, and we prove that these triangulated categories satisfy the hypotheses of the Brown representability theorem. We also study unbounded derived functors in particular for pairs of adjoint functors. We start this study in the framework of abelian categories with translation, then we apply it to the case of the categories of unbounded complexes in abelian categories.Many of the results in this Chapter are not new and many authors have contributed to the results presented here, in particular, Spaltenstein [65] who first considered unbounded complexes and unbounded derived functors. Other contributions are due to [2, 6, 21, 41, 44], [53]. Note that many of the ideas encountered here come from Topology, and the names of Adams, Bousfield, Kan, Thomason, and certainly many others, should be mentioned.KeywordsExact SequenceCommutative RingFull SubcategoryAdditive FunctorAbelian CategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.