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Published in:
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2020 | OriginalPaper | Chapter

1. Cohomology of Finite Groups: Basic Properties

Author : David Harari

Published in: Galois Cohomology and Class Field Theory

Publisher: Springer International Publishing

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Abstract

This chapter gives the first properties of the cohomology of a finite group, which will be essential in the whole book.

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next chapter Groups Modified à la Tate, Cohomology of Cyclic Groups
Footnotes
1
When G is not assumed finite, the reverse convention is often adopted (cf.for example in Chap. VII of [47]) that is to call co-induced the modules of the form \(I_G(A)\) and induced the modules of the form \(\mathbf{Z}[G] \otimes A\). The terminology that we use is compatible with the traditional terminology used for profinite groups that we will encounter in Chap. 4.
 
2
I thank Joël Riou who suggested this method to me which avoids invoking Theorem A.52 of the Appendix, cf.Remark 1.22 below.
 
3
As J. Riou pointed out to me, the assumption that G is finite is important in this statement. For example, there is no analogue of this statement (already at the level of the \(H^1\)) for \(G=\mathbf{Z}^{(\mathbf{N})}\) acting trivially on an infinite family of abelian groups.
 
4
One can also say « u and f are compatible », as in [47], Chap. VII, Sect. 5.
 
5
We can also use Theorem 1.21 along with the fact that a resolution of \(\mathbf{Z}\) by the projective \(\mathbf{Z}[G]\)-modules provides it with a resolution by the projective \(\mathbf{Z}[H]\)-modules.
 
Metadata
Title
Cohomology of Finite Groups: Basic Properties
Author
David Harari
Copyright Year
2020
Book
Galois Cohomology and Class Field Theory

Print ISBN: 978-3-030-43900-2

Electronic ISBN: 978-3-030-43901-9

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-43901-9_1

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