On the Derived Functors of Destabilization and of Iterated Loop Functors

  • Geoffrey PowellEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2194)


These notes explain how to construct small functorial chain complexes which calculate the derived functors of destabilization (respectively iterated loop functors) in the theory of modules over the mod 2 Steenrod algebra; this shows how to unify results of Singer and of Lannes and Zarati.



The author is grateful to the anonymous referee for their careful reading of the manuscript and for their suggestions.


  1. [Cur75]
    E.B. Curtis, The Dyer-Lashof algebra and the \(\Lambda\)-algebra. Ill. J. Math. 19, 231–246 (1975). MR 0377885Google Scholar
  2. [EKMM97]
    A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, Rings, modules, and algebras in stable homotopy theory, in Mathematical Surveys and Monographs, vol. 47 (American Mathematical Society, Providence, RI, 1997). With an appendix by M. Cole. MR 1417719Google Scholar
  3. [Goe86]
    P.G. Goerss, Unstable projectives and stable Ext: with applications. Proc. Lond. Math. Soc. (3) 53(3), 539–561 (1986). MR 868458 (88d:55011)Google Scholar
  4. [GS13]
    G. Gaudens, L. Schwartz, Applications depuis \(K(\mathbb{Z}/p,2)\) et une conjecture de N. Kuhn. Ann. Inst. Fourier (Grenoble) 63(2), 763–772 (2013). MR 3112848Google Scholar
  5. [HM89]
    J.R. Harper, H.R. Miller, Looping Massey-Peterson towers, in Advances in Homotopy Theory (Cortona, 1988). London Mathematical Society Lecture Note Series, vol. 139 (Cambridge University Press, Cambridge, 1989), pp. 69–86. MR 1055869 (91c:55032)Google Scholar
  6. [HM16]
    R. Haugseng, H. Miller, On a spectral sequence for the cohomology of infinite loop spaces. Algebr. Geom. Topol. 16(5), 2911–2947 (2016). MR 3572354Google Scholar
  7. [HP16]
    N.H.V. Hưng, G. Powell, The A-decomposability of the Singer construction (2016). arXiv:1606.09443Google Scholar
  8. [HQT14]
    N.H.V. Hưng, V.T.N. Quỳnh, N.A. Tuấn, On the vanishing of the Lannes-Zarati homomorphism. C. R. Math. Acad. Sci. Paris 352(3), 251–254 (2014). MR 3167575Google Scholar
  9. [HS95]
    N.H.V. Hưng, N. Sum, On Singer’s invariant-theoretic description of the lambda algebra: a mod p analogue. J. Pure Appl. Algebra 99(3), 297–329 (1995). MR 1332903 (96c:55024)Google Scholar
  10. [HT15]
    N.H.V. Hưng, N.A. Tuấn, The generalized algebraic conjecture on spherical classes. preprint 1564 (2015)
  11. [Hưn97]
    N.H.V. Hưng, Spherical classes and the algebraic transfer. Trans. Am. Math. Soc. 349(10), 3893–3910 (1997). MR 1433119 (98e:55020)Google Scholar
  12. [Hưn99]
    N.H.V. Hưng, The weak conjecture on spherical classes. Math. Z. 231(4), 727–743 (1999). MR 1709493Google Scholar
  13. [Hưn03]
    N.H.V. Hưng, On triviality of Dickson invariants in the homology of the Steenrod algebra. Math. Proc. Camb. Philos. Soc. 134(1), 103–113 (2003). MR 1937796Google Scholar
  14. [KM13]
    N.J. Kuhn, J. McCarty, The mod 2 homology of infinite loopspaces. Algebr. Geom. Topol. 13(2), 687–745 (2013). MR 3044591Google Scholar
  15. [Kuh14]
    N.J. Kuhn, Adams filtration and generalized Hurewicz maps for infinite loopspaces (2014). arXiv:1403.7501Google Scholar
  16. [Kuh15]
    N.J. Kuhn, The Whitehead conjecture, the tower of S 1 conjecture, and Hecke algebras of type A. J. Topol. 8(1), 118–146 (2015). MR 3335250Google Scholar
  17. [Lan88]
    J. Lannes, Sur le n-dual du n-ème spectre de Brown-Gitler. Math. Z. 199(1), 29–42 (1988). MR 954749Google Scholar
  18. [Lan92]
    J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un p -groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math. 75, 135–244 (1992). With an appendix by Michel Zisman. MR 1179079 (93j:55019)Google Scholar
  19. [LZ83]
    J. Lannes, S. Zarati, Invariants de Hopf d’ordre supérieur et suite spectrale d’Adams. C. R. Acad. Sci. Paris Sér. I Math. 296(15), 695–698 (1983). MR 705694 (85a:55009)Google Scholar
  20. [LZ84]
    J. Lannes, S. Zarati, Invariants de Hopf d’ordre supérieur et suite spectrale d’Adams. Preprint (1984)Google Scholar
  21. [LZ87]
    J. Lannes, S. Zarati, Sur les foncteurs dérivés de la déstabilisation. Math. Z. 194(1), 25–59 (1987). MR MR871217 (88j:55014)Google Scholar
  22. [Mar83]
    H.R. Margolis, Spectra and the Steenrod Algebra. North-Holland Mathematical Library, vol. 29 (North-Holland Publishing Co, Amsterdam, 1983). Modules over the Steenrod algebra and the stable homotopy category. MR 738973 (86j:55001)Google Scholar
  23. [MM65]
    J.W. Milnor, J.C. Moore, On the structure of Hopf algebras. Ann. Math. (2) 81, 211–264 (1965). MR 0174052 (30 #4259)Google Scholar
  24. [Mùi75]
    H. Mùi, Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22(3), 319–369 (1975). MR 0422451 (54 #10440)Google Scholar
  25. [Mùi86]
    H. Mùi, Cohomology operations derived from modular invariants. Math. Z. 193(1), 151–163 (1986). MR 852916 (88e:55015)Google Scholar
  26. [Pow10]
    G.M.L. Powell, Module structures and the derived functors of iterated loop functors on unstable modules over the Steenrod algebra. J. Pure Appl. Algebra 214(8), 1435–1449 (2010). MR 2593673Google Scholar
  27. [Pow12]
    G.M.L. Powell, On unstable modules over the Dickson algebras, the Singer functors R s and the functors Fixs. Algebr. Geom. Topol. 12, 2451–2491 (2012) [electronic]Google Scholar
  28. [Pow14]
    G.M.L. Powell, On the derived functors of destabilization at odd primes. Acta Math. Vietnam. 39(2), 205–236 (2014). MR 3212661Google Scholar
  29. [Pri70]
    S.B. Priddy, Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970). MR 0265437 (42 #346)Google Scholar
  30. [Sch94]
    L. Schwartz, Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1994). MR MR1282727 (95d:55017)Google Scholar
  31. [Sin78]
    W.M. Singer, Iterated loop functors and the homology of the Steenrod algebra. J. Pure Appl. Algebra 11(1–3), 83–101 (1977/1978). MR MR0478155 (57 #17644)Google Scholar
  32. [Sin80]
    W.M. Singer, Iterated loop functors and the homology of the Steenrod algebra. II. A chain complex for \(\Omega _{s}^{k}M\). J. Pure Appl. Algebra 16(1), 85–97 (1980). MR MR549706 (81b:55040)Google Scholar
  33. [Sin81]
    W.M. Singer, A new chain complex for the homology of the Steenrod algebra. Math. Proc. Camb. Philos. Soc. 90(2), 279–292 (1981). MR MR620738 (82k:55018)Google Scholar
  34. [Sin83]
    W.M. Singer, Invariant theory and the lambda algebra. Trans. Am. Math. Soc. 280(2), 673–693 (1983). MR MR716844 (85e:55029)Google Scholar
  35. [Sin89]
    W.M. Singer, The transfer in homological algebra. Math. Z. 202(4), 493–523 (1989). MR 1022818 (90i:55035)Google Scholar
  36. [Wil83]
    C. Wilkerson, A primer on the Dickson invariants, in Proceedings of the Northwestern Homotopy Theory Conference (Evanston, III, 1982). Contemporary Mathematics, vol. 19 (American Mathematical Society, Providence, RI, 1983), pp. 421–434. MR 711066 (85c:55017)Google Scholar
  37. [Zar84]
    S. Zarati, Dérivés du foncteur de déstabilisation en caractéristique impaire et applications. Thèse d’état, Université Paris Sud (1984)Google Scholar
  38. [Zar90]
    S. Zarati, Derived functors of the destabilization and the Adams spectral sequence. Astérisque 191(8), 285–298 (1990). International Conference on Homotopy Theory (Marseille-Luminy, 1988). MR MR1098976 (92c:55020)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratoire angevin de recherches en mathématiques (LAREMA), CNRSUniversité d’Angers, Université Bretagne LoireAngers Cedex 01France

Personalised recommendations