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Cohomology and duality for \(({\varphi}, {\Gamma})\)-modules over the Robba ring. (English) Zbl 1248.11093
Given a \(p\)-adic representation \(V\) of the Galois group of a local field \(K\), the author proves that the Galois cohomology groups \(H^i(K,V)\) may be computed using the associated étale (\({\varphi}, {\Gamma}\))-module over the Robba ring. A similar prior result is due to L. Herr, who proved in [Bull. Soc. Math. Fr. 126, No. 4, 563–600 (1998; Zbl 0967.11050)] that the Galois cohomology groups may be recovered from the associated étale (\({\varphi}, {\Gamma}\))-module over Fontaine’s larger ring \({\mathcal{E}_K}\). The author then obtains analogous results for \(({\varphi}, {\Gamma})\)-modules over the Robba ring which are not necessarily étale, and further deduces the Euler-Poincaré characteristic formula and Tate local duality for the cohomology groups.

11S25 Galois cohomology
11F80 Galois representations
11S20 Galois theory
12G05 Galois cohomology
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