Algebraic curves over \(n\)-dimensional general local fields (Q5954052)

Let \(k\) be an \(n\)-dimensional local field over a pseudofinite residue field \(k_{0}\) and \(\mathcal X\) be a complete smooth absolutely irreducible curve over \(k\) with successive good reduction. Let \(p\) be a prime, \(p\not= \operatorname {char} k_{0},\) suppose that \(k\) contains all \(p\)-th roots of unity and denote by \(\mu _{p}\) the sheet of \(p\)-th roots of unity. The main result of the paper: for any \(n\leq 3\) there exists a nondegenerate pairing of stable cohomology groups NEWLINE\[NEWLINEH^{n+2}({\mathcal X},\mu _{p})\times H^{1}({\mathcal X}, \mu _{p})\to \mu _{p},NEWLINE\]NEWLINE such that the groups \(H^{n+2}({\mathcal X}, \mu _{p})\) and \(H^{1}({\mathcal X},\mu _{p})\) are dual. This is the analogue for \(n\)-dimensional pseudolocal fields contained in the result of \textit{J.-C. Douai} [J. Algebra 125, No. 1, 181-196 (1989; Zbl 0703.11064)] in the case of curves over \(n\)-dimensional local fields.