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Relative Homological Algebra Via Truncations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Australian Natl Univ, Math Sci Inst, Ctr Math & Its Applicat, Canberra, ACT 0200, Australia..
Univ Autonoma Barcelona, Dept Matemat, Bellaterra 08193, Cerdanyola Del, Spain..
Ecole Polytech Fed Lausanne, Inst Math, Stn 8, CH-1015 Lausanne, Switzerland..
2018 (English)In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 23, p. 895-937Article in journal (Refereed) Published
Abstract [en]

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 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(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.

Place, publisher, year, edition, pages
FIZ Karlsruhe – Leibniz-Institut für Informationsinfrastruktur , 2018. Vol. 23, p. 895-937
Keywords [en]
relative homological algebra, relative resolution, injective class, model category, model approximation, truncation, Noetherian ring, Krull dimension, local cohomology
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-252939ISI: 000468272500029Scopus ID: 2-s2.0-85068177745OAI: oai:DiVA.org:kth-252939DiVA, id: diva2:1322952
Note

QC 20190611

Available from: 2019-06-11 Created: 2019-06-11 Last updated: 2019-08-02Bibliographically approved

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