Cohomology and Duality for (φ, Γ)-modules over the Robba Ring

By Ruochuan Liu. 2007

105 citations recorded by the source corpus.

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale (φ,Γ)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (φ,Γ)-modules over the Robba ring, of the Euler-Poincaré characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e655b98a49b953d206696d48ede69c42a443f49

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In Geometry and Topology

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Citations in the corpus

Relative P-adic Hodge Theory: Foundations (2013, methodology): “3, one can similarly check that our computation of proétale cohomology in terms of φ-cohomology agrees with the analogous computations in the language of classical (φ,Γ)-modules made by Herr [72, 73] and the second author [96].”
Relative p-adic Hodge theory, II: Imperfect period rings (2016, background): “For example, in the context of classical p-adic Hodge theory, the pseudocoherent φ-modules coincide with the generalized (φ,Γ)-modules considered in [86] and elsewhere.”
Analytic families of finite-slope Selmer groups (2013, methodology): “= dimQp H 1 f (G, F ), using the local Euler–Poincaré formulas of [Liu 2007] and Equation (3-2) for F, the computation H2(G, F)= H0(G, (F)(1)) = 0 by local duality for (φ, 0)modules [Liu 2007], and that F has only negative Hodge–Tate weights.”
Cohomology of arithmetic families of (,Γ)-modules (2012, methodology): “This argument only works if both Iwasawa H(2) groups vanish; the remainder of the argument is a dévissage to reduce to this case, in the spirit of some arguments from [36] but not quite identical to any of them.”