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COHOMOLOGY AND OVERCONVERGENCE FOR REPRESENTATIONS OF POWERS OF GALOIS GROUPS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 2 / March 2021
- Published online by Cambridge University Press:
- 11 April 2019, pp. 361-421
- Print publication:
- March 2021
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- Article
- Export citation
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We show that the Galois cohomology groups of
$p$-adic representations of a direct power of 
$\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable 
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. Using Tate duality and a pairing for multivariable 
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules we extend this to analogues of the Iwasawa cohomology. We show that all 
$p$-adic representations of a direct power of 
$\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable 
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.
