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Published in:
Cohomology of Finite Groups: Basic Properties Read chapter Read first chapter

2020 | OriginalPaper | Chapter

2. Groups Modified à la Tate, Cohomology of Cyclic Groups

Author : David Harari

Published in: Galois Cohomology and Class Field Theory

Publisher: Springer International Publishing

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Abstract

This chapter introduces the modified Tate groups and cup-products.

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previous chapter Cohomology of Finite Groups: Basic Properties
next chapter p-Groups, the Tate–Nakayama Theorem
Footnotes
1
As J. Riou pointed out to me, this convention is different from the one used, for example, in [12], which would be indispensable if we have replaced A by a complex of \(G\)-modules.
 
2
There is a subtlety here (pointed out by J. Riou): the \(\varphi _{p, q}\) do not generally induce morphisms of complexes \(X_{{\scriptscriptstyle \bullet }} \rightarrow X_{{\scriptscriptstyle \bullet }} \otimes X_{{\scriptscriptstyle \bullet }}\) because \(X_{{\scriptscriptstyle \bullet }}\) is not à priori bounded from either side, and hence the image of \(X_n\) in \(\prod _{p+q=n} X_p \otimes X_q\) is not contained in \(\bigoplus _{p+q=n} X_p \otimes X_q\). Nevertheless, the passage to \({{\,\mathrm{Hom}\,}}_G(., C)\) allows to define the desired homomorphism in an evident way.
 
Metadata
Title
Groups Modified à la Tate, Cohomology of Cyclic Groups
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_2

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