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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.

##### MSC:
 11S25 Galois cohomology 11F80 Galois representations 11S20 Galois theory 12G05 Galois cohomology
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