Relative homological algebra via truncations

Wojciech Chachólski, Amnon Neeman, Wolfgang Pitsch, Jérôme Scherer

Research output: Contribution to journalArticleResearchpeer-review


© 2018 Deutsche Mathematiker Vereinigung. To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spal-tenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class of "injective objects" I. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between un- bounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A; I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.
Original languageEnglish
Pages (from-to)895-937
JournalDocumenta Mathematica
Publication statusPublished - 1 Jan 2018


  • Injective class
  • Krull dimension
  • Local cohomology
  • Model approximation
  • Model category
  • Noetherian ring
  • Relative homological algebra
  • Relative resolution
  • Truncation

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