Duality and local group cohomology

Author:
P. R. Hewitt

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1909-1914

MSC (1991):
Primary 20J05

MathSciNet review:
1468193

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Abstract: Let be a group, let be a field, and let be a *local system* - an upwardly directed collection of subgroups whose union is . In this paper we give a short, elementary proof of the following result: If either is a --bimodule, or else is finite dimensional over its center, then . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.

**[AM]**Alejandro Adem and R. James Milgram,*Cohomology of finite groups*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, Springer-Verlag, Berlin, 1994. MR**1317096****[B]**D. J. Benson,*Representations and cohomology. I*, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR**1110581****[CPS]**Edward Cline, Brian Parshall, and Leonard Scott,*Cohomology of finite groups of Lie type. I*, Inst. Hautes Études Sci. Publ. Math.**45**(1975), 169–191. MR**0399283****[M]**U Meierfrankenfeld,*A note on the cohomology of finitary modules*, Proc AMS (to appear).**[Ph]**R. E. Phillips,*Finitary linear groups: a survey*, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 111–146. MR**1362808****[ShW]**M. Shirvani and B. A. F. Wehrfritz,*Skew linear groups*, London Mathematical Society Lecture Note Series, vol. 118, Cambridge University Press, Cambridge, 1986. MR**883801****[S]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[W-1]**B. A. F. Wehrfritz,*Complete reducibility in skew linear groups*, J. London Math. Soc. (2)**28**(1983), no. 2, 301–309. MR**713385**, 10.1112/jlms/s2-28.2.301**[W-2]**B Wehrfritz,*The complete reducibility of locally completely reducible finitary linear groups*, Bull LMS**29**(1997), 173-176. CMP**97:06**

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Additional Information

**P. R. Hewitt**

Affiliation:
Department of Mathematics, University of Toledo, Toledo, Ohio 43606

DOI:
https://doi.org/10.1090/S0002-9939-98-04543-2

Received by editor(s):
December 12, 1996

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1998
American Mathematical Society