Derived completion for comodules

  • Tobias Barthel
  • Drew Heard
  • Gabriel ValenzuelaEmail author
Open Access


The objective of this paper is to introduce and study completions and local homology of comodules over Hopf algebroids, extending previous work of Greenlees and May in the discrete case. In particular, we relate module-theoretic to comodule-theoretic completion, construct various local homology spectral sequences, and derive a tilting-theoretic interpretation of local duality for modules. Our results translate to quasi-coherent sheaves over global quotient stacks and feed into a novel approach to the chromatic splitting conjecture.

Mathematics Subject Classification

55P60 (13D45, 14B15, 55U35) 



  1. 1.
    Baker, A.: \(I_n\)-local Johnson-Wilson spectra and their Hopf algebroids. Doc. Math. 5, 351–364 (2000). (electronic)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981), volume 100 of Astérisque, pp. 5–171. Soc. Math. France, Paris (1982)Google Scholar
  3. 3.
    Bican, L., El Bashir, R., Enochs, E.: All modules have flat covers. Bull. Lond. Math. Soc. 33(4), 385–390 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barthel, T., Beaudry, A., Peterson, E.: The homology of inverse limits and the algebraic chromatic splitting conjecture (in preparation)Google Scholar
  5. 5.
    Barthel, T., Frankland, M.: Completed power operations for Morava \(E\)-theory. Algebr. Geom. Topol. 15(4), 2065–2131 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Barthel, T., Heard, D., Valenzuela, G.: Local duality in algebra and topology. Adv. Math. 335, 563–663 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304. Springer, Berlin-New York (1972)Google Scholar
  8. 8.
    Brodmann, M.P., Sharp, R.Y.: Local Cohomology, volume 136 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2013). An algebraic introduction with geometric applicationsGoogle Scholar
  9. 9.
    Devinatz, E.S.: Morava’s change of rings theorem. In: The Čech Centennial (Boston, MA, 1993), volume 181 of Contemp. Math., pp. 83–118. Amer. Math. Soc., Providence (1995)Google Scholar
  10. 10.
    Dwyer, W.G., Greenlees, J.P.C.: Complete modules and torsion modules. Am. J. Math. 124(1), 199–220 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Enochs, E.: Flat covers and flat cotorsion modules. Proc. Am. Math. Soc. 92(2), 179–184 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Greenlees, J.P.C., May, J.P.: Derived functors of \(I\)-adic completion and local homology. J. Algebra 149(2), 438–453 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Greenlees, J.P.C., May, J.P.: Localization and completion theorems for \(M{{\rm U}}\)-module spectra. Ann. Math. (2) 146(3), 509–544 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hovey, M.: Homotopy theory of comodules over a Hopf algebroid. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic \(K\)-theory, volume 346 of Contemp. Math., pp. 261–304. Amer. Math. Soc., Providence (2004)Google Scholar
  15. 15.
    Hovey, M., Palmieri, J.H., Strickland, N.P.: Axiomatic stable homotopy theory. Mem. Am. Math. Soc. 128(610), x+114 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hovey, M.A., Strickland, N.P.: Morava \({K}\)-theories and localisation. Mem. Am. Math. Soc. 139(666), viii+100–100 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hovey, M., Strickland, N.: Comodules and Landweber exact homology theories. Adv. Math. 192(2), 427–456 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hovey, M., Strickland, N.: Local cohomology of \({{\rm BP}}_{*}{{\rm BP}}\)-comodules. Proc. Lond. Math. Soc. (3) 90(2), 521–544 (2005)CrossRefGoogle Scholar
  19. 19.
    Jannsen, U.: Continuous étale cohomology. Math. Ann. 280(2), 207–245 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lam, T.Y.: Lectures on Modules and Rings, volume 189 of Graduate Texts in Mathematics. Springer, New York (1999)Google Scholar
  22. 22.
    Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)Google Scholar
  23. 23.
    Lurie, J.: Proper Morphisms, Completions, and the Grothendieck Existence Theorem (2011). Draft available from author’s website as
  24. 24.
    Lurie, J.: Higher Algebra (2017). Draft available from author’s website as
  25. 25.
    Naumann, N.: The stack of formal groups in stable homotopy theory. Adv. Math. 215(2), 569–600 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pstrągowski, P.: Synthetic spectra and the cellular motivic category (2018). arXiv:1803.01804
  27. 27.
    Porta, M., Shaul, L., Yekutieli, A.: On the homology of completion and torsion. Algebr. Represent. Theory 17(1), 31–67 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ravenel, D.C.: Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, vol. 121. Academic Press Inc, Orlando (1986)Google Scholar
  29. 29.
    Rezk, C.: Analytic Completion (2013). Draft available at
  30. 30.
    Schenzel, P.: Proregular sequences, local cohomology, and completion. Math. Scand. 92(2), 161–180 (2003)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sitte. T.: Local cohomology sheaves on algebraic stacks. Ph.D. thesis (2014)Google Scholar
  32. 32.
    Valenzuela, G.: Homological algebra of complete and torsion modules. Ph.D. thesis, Wesleyan University (2015)Google Scholar
  33. 33.
    Weibel, C.A.: An Introduction to Homological Algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)Google Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  3. 3.Max-Planck-Institut für MathematikBonnGermany

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