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


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


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


  • 1. I. T. Adamson, Cohomology theory for non-normal subgroups and non-normal fields, Proc. Glasgow Math. Assoc. 2 (1954), 66--76. MR0065546.
  • 2. D. Barnes, J. P. C. Greenlees, M. Kedziorek, and B. Shipley, Rational $SO(2)$-equivariant spectra, Preprint available at :, 2015.
  • 3. J. E. Bergner, Homotopy limits of model categories and more general homotopy theories, Bull. Lond. Math. Soc. 44 (2012), no. 2, 311--322. DOI 10.1112/blms/bdr095; zbl 1242.55006; MR2914609; arxiv 1010.0717.
  • 4. A. K. Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003), 1001--1053 (electronic). DOI 10.2140/gt.2003.7.1001; zbl 1065.55012; MR2026537; arxiv math/0312531.
  • 5. K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419--458. DOI 10.1090/S0002-9947-1973-0341469-9; zbl 0245.55007; MR0341469.
  • 6. W. Chachólski, W. Pitsch, and J. Scherer, Injective classes of modules, J. Algebra Appl. 12 (2013), no. 4, 1250188, 13. DOI 10.1142/S0219498812501885; zbl 1282.13019; MR3037263.
  • 7. W. Chachólski and J. Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. 155 (2002), no. 736, x+90. zbl 1006.18015; MR1879153; arxiv math/0110316.
  • 8. C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85--124. DOI 10.2307/1990637; zbl 0031.24803; MR0024908.
  • 9. J. D. Christensen and M. Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, 261--293. DOI 10.1017/S0305004102006126; zbl 1016.18008; MR1912401; arxiv math/0011216.
  • 10. W. G. Dwyer, P. S. Hirschhorn, D. M. Kan, and J. H. Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs, vol. 113, American Mathematical Society, Providence, RI, 2004. zbl 1072.18012; MR2102294.
  • 11. W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73--126. zbl 0869.55018; MR1361887.
  • 12. S. Eilenberg and J. C. Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. No. 55 (1965), 39. zbl 0129.01101; MR0178036.
  • 13. E. Enochs and J. A. López-Ramos, Kaplansky classes, Rend. Sem. Mat. Univ. Padova 107 (2002), 67--79. zbl 1099.13019; MR1926201.
  • 14. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, de Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. zbl 0952.13001; MR1753146.
  • 15. E. Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996. DOI 10.1007/BFb0094429; zbl 0842.55001; MR1392221.
  • 16. J. P. C. Greenlees and B. Shipley, Homotopy theory of modules over diagrams of rings, Proc. Amer. Math. Soc. Ser. B 1 (2014), 89--104. DOI 10.1090/S2330-1511-2014-00012-2; zbl1350.55022; arxiv 1309.6997 MR3254575.
  • 17. A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957), no. 2, 119--221. zbl 0118.26104; MR0102537.
  • 18. L. Gruson and C. U. Jensen, Modules algébriquement compacts et foncteurs $\varprojlim^{(i)}$, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1651--A1653. zbl 0259.18015; MR0320112.
  • 19. Y. Harpaz and M. Prasma, The Grothendieck construction for model categories, Adv. Math. 281 (2015), 1306--1363. DOI 10.1016/j.aim.2015.03.031; zbl 1333.18024; MR3366868; arxiv 1404.1852.
  • 20. G. Hochschild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246--269. DOI 10.2307/1992988; zbl 0070.26903; MR0080654.
  • 21. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167--193. DOI 10.1016/j.jpaa.2003.11.007; zbl 1050.16003; MR2038564.
  • 22. S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, and U. Walther, Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. zbl 1129.13001; MR2355715.
  • 23. M. Kashiwara and P. Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. zbl 1118.18001; DOI 10.1007/3-540-27950-4; MR2182076.
  • 24. R. Kiełpiński and D. Simson, On pure homological dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 1--6. zbl 0303.16016; MR0407089.
  • 25. S. Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition. MR1344215.
  • 26. E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511--528. DOI 10.2140/pjm.1958.8.511; zbl 0084.26601; MR0099360.
  • 27. M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons, New York-London, 1962. zbl 0123.03402; MR0155856.
  • 28. A. Neeman, The chromatic tower for $D(R)$, Topology 31 (1992), no. 3, 519--532, With an appendix by Marcel Bökstedt. DOI 10.1016/0040-9383(92)90047-L; zbl 0793.18008; MR1174255.
  • 29. A. Neeman, Non-left-complete derived categories, Math. Res. Lett. 18 (2011), no. 5, 827--832. DOI 10.4310/MRL.2011.v18.n5.a2; zbl 1244.18009; MR2875857; arxiv 1103.5539.
  • 30. M. Prest, Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009. zbl 1205.16002; MR2530988.
  • 31. D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin, 1967. DOI 10.1007/BFb0097438; zbl 0168.20903; MR0223432.
  • 32. J.-E. Roos, Derived functors of inverse limits revisited, J. London Math. Soc. (2) 73 (2006), no. 1, 65--83. DOI 10.1112/S0024610705022416; zbl 1089.18007; MR2197371.
  • 33. C. Serpé, Resolution of unbounded complexes in Grothendieck categories, J. Pure Appl. Algebra 177 (200), no. 1, 103--112. DOI 10.1016/S0022-4049(02)00075-0; zbl 1033.18007; MR1948842.
  • 34. N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121--154. zbl 0636.18006; MR0932640.
  • 35. S. Virili, On the exactness of products in the localization of AB$4^\ast$ Grothendieck categories, to appear in J. of Algebra (2017). DOI 10.1016/j.jalgebra.2016.08.019: zbl 1372.18013; MR3565424.


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