Purity in compactly generated derivators and t-structures with Grothendieck hearts

  • Rosanna LakingEmail author


We study t-structures with Grothendieck hearts on compactly generated triangulated categories \({\mathcal {T}}\) that are underlying categories of strong and stable derivators. This setting includes all algebraic compactly generated triangulated categories. We give an intrinsic characterisation of pure triangles and the definable subcategories of \({\mathcal {T}}\) in terms of directed homotopy colimits. For a left nondegenerate t-structure \(\mathbf{t}=({\mathcal {U}},{\mathcal {V}})\) on \({\mathcal {T}}\), we show that \({\mathcal {V}}\) is definable if and only if \(\mathbf{t}\) is smashing and has a Grothendieck heart. Moreover, these conditions are equivalent to \(\mathbf{t}\) being homotopically smashing and to \(\mathbf{t}\) being cogenerated by a pure-injective partial cosilting object. Finally, we show that finiteness conditions on the heart of \(\mathbf{t}\) are determined by purity conditions on the associated partial cosilting object.


T-structure Cosilting Cotilting Purity Definable Reduced product Derivator Homotopically smashing Locally coherent Locally noetherian 

Mathematics Subject Classification

18E15 18E30 03C20 



The author would like to thank Moritz Groth and Gustavo Jasso for many interesting and helpful conversations about derivators and other higher categorical structures. She would like to thank Lidia Angeleri Hügel, Frederik Marks and Jorge Vitória for discussions regarding the definition of partial cosilting, which also led to the contents of Example 4.4. Particular thanks are extended to Prof. Angeleri Hügel for her ongoing support of this project. During the later stages of this project, the author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 797281.


  1. 1.
    Angeleri Hügel, L.: On the abundance of silting modules. In: Surveys in Representation Theory of Algebras, Contemp. Math., vol. 716, pp. 1–23. American Mathematical Society, Providence (2018)Google Scholar
  2. 2.
    Angeleri Hügel, L., Coelho, F.U.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13(2), 239–250 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Angeleri Hügel, L., Marks, F., Vitória, J.: Torsion pairs in silting theory. Pac. J. Math. 291(2), 257–278 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bazzoni, S.: The $t$-structure induced by an $n$-tilting module. Trans. Am. Math. Soc. 371(9), 6309–6340 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces. I (Luminy, 1981), Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)Google Scholar
  6. 6.
    Beligiannis, A.: Relative homological algebra and purity in triangulated categories. J. Algebra 227(1), 268–361 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bondarko, M.V.: On torsion pairs, (well generated) weight structures, adjacent t-structures, and related (co)homological functors. arXiv:1611.00754 (2016)Google Scholar
  8. 8.
    Chang, C.C., Keisler, H.J.: Model Theory, Studies in Logic and the Foundations of Mathematics, vol. 73, 3rd edn. North-Holland, Amsterdam (1990)Google Scholar
  9. 9.
    Colpi, R., Mantese, F., Tonolo, A.: Cotorsion pairs, torsion pairs, and $\Sigma $-pure-injective cotilting modules. J. Pure Appl. Algebra 214(5), 519–525 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Crawley-Boevey, W.: Infinite-dimensional modules in the representation theory of finite-dimensional algebras. In: Algebras and Modules, I (Trondheim, 1996), CMS Conf. Proc., vol. 23, pp. 29–54. American Mathematical Society, Providence (1998)Google Scholar
  11. 11.
    Garkusha, G., Prest, M.: Triangulated categories and the Ziegler spectrum. Algebr. Represent. Theory 8(4), 499–523 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Groth, M.: Derivators, pointed derivators and stable derivators. Algebr. Geom. Topol. 13(1), 313–374 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Groth, M.: Revisiting the canonicity of canonical triangulations. Theory Appl. Categ. 33, 350–389, Paper No. 14 (2018)Google Scholar
  14. 14.
    Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120(575), viii+ 88 (1996)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Harada, M.: Perfect categories. V. Osaka J. Math. 10, 585–596 (1973)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Herzog, I.: The Ziegler spectrum of a locally coherent Grothendieck category. Proc. Lond. Math. Soc. (3) 74(3), 503–558 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Keller, B., Vossieck, D.: Aisles in derived categories. Bull. Soc. Math. Belg. Sér. A 40(2), 239–253 (1988). Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987)Google Scholar
  19. 19.
    Krause, H.: The spectrum of a locally coherent category. J. Pure Appl. Algebra 114(3), 259–271 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krause, H.: Exactly definable categories. J. Algebra 201(2), 456–492 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Krause, H.: Smashing subcategories and the telescope conjecture–an algebraic approach. Invent. Math. 139(1), 99–133 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Krause, H.: Coherent functors in stable homotopy theory. Fundam. Math. 173(1), 33–56 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lurie, J.: Higher algebra. (2006)
  25. 25.
    Marks, F., Vitória, J.: Silting and cosilting classes in derived categories. J. Algebra 501(2), 526–544 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nicolás, P., Saorín, M., Zvonareva, A.: Silting theory in triangulated categories with coproducts. J. Pure Appl. Algebra 223(6), 2273–2319 (2019)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Parra, C.E., Saorín, M.: Direct limits in the heart of a t-structure: the case of a torsion pair. J. Pure Appl. Algebra 219(9), 4117–4143 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Parra, C.E., Saorín, M.: Addendum to “Direct limits in the heart of a t-structure: the case of a torsion pair” [J. Pure Appl. Algebra 219 (9) (2015) 4117–4143] [MR3336001]. J. Pure Appl. Algebra 220(6), 2467–2469 (2016)Google Scholar
  29. 29.
    Prest, M.: Model Theory and Modules, London Mathematical Society Lecture Note Series, vol. 130. Cambridge University Press, Cambridge (1988)Google Scholar
  30. 30.
    Prest, M.: Purity, Spectra and Localisation, Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  31. 31.
    Prest, M.: Definable additive categories: purity and model theory. Mem. Am. Math. Soc. 210(987), vi+109 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Psaroudakis, C., Vitória, J.: Realisation functors in tilting theory. Math. Z. 288(3), 965–1028 (2018)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Saorín, M.: On locally coherent hearts. Pac. J. Math. 287(1), 199–221 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Saorín, M., Št’ovíek, J., Virili, S.: t-structures on stable derivators and Grothendieck hearts. arXiv:1708.07540 (2018)Google Scholar
  35. 35.
    Stenström, B.: Coherent rings and $F\, P$-injective modules. J. Lond. Math. Soc. 2(2), 323–329 (1970)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Šťovíček, J.: Derived equivalences induced by big cotilting modules. Adv. Math. 263, 45–87 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Informatica, Settore di MatematicaUniversitá degli Studi di VeronaVeronaItaly

Personalised recommendations