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Farrell cohomology and Brown theorems for profinite groups

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Abstract

LetG be a profinite group which has an open subgroupH such that the cohomologicalp-dimensiond≔cdp(H) is finite (p is a fixed prime). The main result of this paper expresses thep-primary part of high degree cohomology ofG in terms of the elementary abelianp-subgroups ofG: From the latter one constructs a natural profinite simplicial setA G, on whichG acts by conjugation. ThenH n(G,M)≅H nG (AG,M) holds forn≧d+r and everyp-primary discreteG-moduleM (r≔p-rank ofG). If one uses profinite Farrell cohomology, which is introduced in this paper, the analogous fact holds in all degrees. These results are the profinite analogues of theorems by K.S. Brown for discrete groups.

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References

  • [Bn] D. J. Benson: Representations and Cohomology. (2 volumes) Cambridge Univ. Press, Cambridge, 1991

    Google Scholar 

  • [Br1] K.S. Brown: Groups of virtually finite dimension. In: Homological Group Theory, ed. C.T.C. Wall, London Math. Soc. Lect. Notes36, Cambridge, 1979, pp. 27–70

  • [Br2] K. S. Brown: Cohomology of Groups. Graduate Texts in Mathematics87, Springer, New York Heidelberg Berlin, 1982

    MATH  Google Scholar 

  • [Bru] A. Brumer: Pseudocompact algebras, profinite groups and class formations. J. Algebra4, 442–470 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  • [De] P. Deligne: Théorie de Hodge III. Publ. Math. Inst. Hautes Etudes Scient.44, 5–77 (1974)

    MATH  Google Scholar 

  • [F] F. T. Farrell: An extension of Tate cohomology to a class of infinite groups. J. Pure Applied Algebra10, 153–161 (1977)

    Article  MathSciNet  Google Scholar 

  • [GS] O. Goldman, C.-H. Sah: On a special class of locally compact rings. J. Algebra4, 71–95 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  • [G] A. Grothendieck: sur quelques points d'algèbre homologique. Tôhoku Math. J.9, 119–221 (1957)

    MATH  MathSciNet  Google Scholar 

  • [EGA] A. Grothendieck: Éléments de Géométrie Algébrique. Chap III, 1ère Partie. Publ. Math. IHES11, Paris, 1961

  • [Q1] D. Quillen: The spectrum of an equivariant cohomology ring: I. Ann. Math.94, 549–572 (1971)

    Article  MathSciNet  Google Scholar 

  • [Q2] D. Quillen: Homotopy properties of the poset of nontrivialp-subgroups of a group. Adv. Math.28, 101–128 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • [Sch1] C. Scheiderer: Quasi-augmented simplicial spaces, with an application to cohomological dimension. J. Pure Applied Algebra81, 293–311 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • [Sch2] C. Scheiderer: Real and Étale Cohomology. Lect. Notes Math.1588, Springer, Berlin Heidelberg New York, 1994

    MATH  Google Scholar 

  • [S1] J.-P. Serre: Cohomologie Galoisienne. Cinquième édition. Lect. Notes Math.5, Springer, Berlin Heidelberg New York, 1994

    Google Scholar 

  • [S2] J.-P. Serre: Sur la dimension cohomologique des groupes profinis. Topology3, 413–420 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  • [S3] J.-P. Serre: Cohomologie des groupes discrets. Ann. Math. Studies70, 77–169 (1971)

    MathSciNet  Google Scholar 

Download references

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  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Claus Scheiderer

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  1. Claus Scheiderer
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Scheiderer, C. Farrell cohomology and Brown theorems for profinite groups. Manuscripta Math 91, 247–281 (1996). https://doi.org/10.1007/BF02567954

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