Cohomological Hasse principle for the ring $\mathbb{F}_{p}((t))[[X,Y]]$

Curves over higher local fields

“…2) If n = 2, we obtain a duality which is analogue to a duality for scheme associated to two-dimensional local ring [2]. For general n, the analogy is explained in [3].…”

The arithmetic of curves over two dimensional local fields

“…Hence W 0 = H 0 (Y [1] , Q ℓ )/ Im{H 0 (Y [0] , Q ℓ ) −→ H 0 (Y [1] , Q ℓ )}. Thus, it suffices to prove the vanishing of the composing map H 0 (Y [1] ,…”

“…for all v ∈ P. Let z v be the 0− cycle in Y obtained by specializing v, which induces a map z [1] v −→ Y [1] .…”

“…If we apply the same method of Saito to study curves over two-dimensional local fields, we need class field theory of two-dimensional local ring having one-dimensional local field as residue field. This is done by myself in [1]. Hence, one can follow Saito 's method to obtain the same results.…”

“…We will prove that W 0 (H 1 Y , Q l ) ⊆ Kerα. The group W 0 = W 0 (H 1 Y , Q l ) is calculated as the homologie of the complex [1] , Q ℓ )}. Thus, it suffices to prove the vanishing of the composing map…”