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Groups Modified à la Tate, Cohomology of Cyclic Groups

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Galois Cohomology and Class Field Theory

Part of the book series: Universitext ((UTX))

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Abstract

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

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Notes

  1. 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. 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.

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Correspondence to David Harari .

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Harari, D. (2020). Groups Modified à la Tate, Cohomology of Cyclic Groups. In: Galois Cohomology and Class Field Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-43901-9_2

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