Chachólski, Wojciech; Neeman, Amnon; Pitsch, Wolfgang; Scherer, Jérôme

Relative Homological Algebra via Truncations

Doc. Math. 23, 895-937 (2018)
DOI: 10.25537/dm.2018v23.895-937

Summary

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein 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 $\Cal{A}$ and fix a class of "injective objects" $\Cal{I}$. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded 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 $D(\Cal{A};\Cal{I})$, we need more: the split error term must vanish. This is the case when $\Cal I$ 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$^\ast$-$n$ axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

Mathematics Subject Classification

55U15, 55U35, 18E40, 13D45

Keywords/Phrases

relative homological algebra, relative resolution, injective class, model category, model approximation, truncation, Noetherian ring, Krull dimension, local cohomology

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Affiliation

Chachólski, Wojciech
Department of Mathematics, KTH, S 10044 Stockholm, Sweden
Neeman, Amnon
Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
Pitsch, Wolfgang
Dept. de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Cerdanyola del Vallés), Spain
Scherer, Jérôme
Institute of Mathematics EPFL, Station 8 CH - 1015 Lausanne, Switzerland

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