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Transactions of the American Mathematical Society

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Kähler Differentials, the $T$-functor,
and a Theorem of Steinberg

Authors: W. G. Dwyer and C. W. Wilkerson
Journal: Trans. Amer. Math. Soc. 350 (1998), 4919-4930
MSC (1991): Primary 55N99, 13D99
MathSciNet review: 1621741
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $T$ be the functor on the category of unstable algebras over the Steenrod algebra constructed by Lannes. We use an argument involving Kähler differentials to show that $T$ preserves polynomial algebras. This leads to new and relatively simple proofs of some topological and algebraic theorems.

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  • 1. J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian 𝑝-groups, Topology 24 (1985), no. 4, 435–460. MR 816524, 10.1016/0040-9383(85)90014-X
  • 2. D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge University Press, 1994.
  • 3. A. Borel, Groupes linéaires algébriques, Annals of Mathematics 64(2) (195), 20-82. MR 19:1195k
  • 4. Armand Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Math. J. (2) 13 (1961), 216–240 (French). MR 0147579
  • 5. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • 6. H.E.A. Campbell, J.P. Hughes, and J. Shank, Preliminary notes on rigid reflection groups, preprint (1995).
  • 7. Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 0072877
  • 8. H. S. M. Coxeter, Discrete groups generated by reflections., Ann. Math. 35 (1934), 588-621.
  • 9. W.G. Dwyer and C.W. Wilkerson, Torsion in the cohomology of classifying spaces, hour address by Clarence Wilkerson, Holyoke College (1994).
  • 10. William G. Dwyer and Clarence W. Wilkerson, Smith theory and the functor 𝑇, Comment. Math. Helv. 66 (1991), no. 1, 1–17. MR 1090162, 10.1007/BF02566633
  • 11. W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442. MR 1274096, 10.2307/2946585
  • 12. W. G. Dwyer and C. W. Wilkerson, The center of a 𝑝-compact group, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 119–157. MR 1320990, 10.1090/conm/181/02032
  • 13. David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
  • 14. G. Kemper and G. Malle, The finite irreducible linear groups with polynomial rings of invariants, Universität Heidelberg preprint 96-38 (August 1996).
  • 15. J. Lannes, Cohomology of groups and function spaces, preprint (1986).
  • 16. Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un 𝑝-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079
  • 17. Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • 18. Haruhisa Nakajima, Regular rings of invariants of unipotent groups, J. Algebra 85 (1983), no. 2, 253–286. MR 725082, 10.1016/0021-8693(83)90094-7
  • 19. Lionel 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 1282727
  • 20. J.-P. Serre, Groupes finis d'automorphismes d'anneaux locaux réguliers, Colloq. d'Alg. E.N.S. (1967).
  • 21. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304. MR 15:6000b
  • 22. Robert Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392–400. MR 0167535, 10.1090/S0002-9947-1964-0167535-3
  • 23. Hirosi Toda, Cohomology 𝑚𝑜𝑑 3 of the classifying space 𝐵𝐹₄ of the exceptional group 𝐹₄, J. Math. Kyoto Univ. 13 (1973), 97–115. MR 0321086

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Additional Information

W. G. Dwyer
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

C. W. Wilkerson
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Received by editor(s): August 1, 1996
Additional Notes: The authors were supported in part by the National Science Foundation
Article copyright: © Copyright 1998 American Mathematical Society