Arithmetic on two dimensional local rings

p-adic étale Tate twists and arithmetic duality☆

“…[Sa1], p. 413). However our proof is not new, because we use Artin's proper base-change theorem for Brauer groups.…”

On Swan conductors for Brauer groups of curves over local fields

“…Similar to (3), there exists for each closed point x ∈ X a canonical isomorphism Finally, for a closed point x ∈ X and w ∈ Br(X), we have a commutative diagram (cf. [5], 2.7)…”

“…This diagram is known to be anti-commutative (cf. [5] p. 411). We will briefly review the proof of this fact at the end of this section.…”

“…First, we have an explicit description of ψ (cf. [5], Section 7). For each y ∈ Y 0 , we have a composite map ψ y :…”

“…Since κ(x) is a finite extension of k, we have Cor κ(x)/k (w x ) ∈ Br(k) ∼ = Q/Z, where the last isomorphism is given by local class field theory. Then we define the pairing (1) by Here Hom c denotes the group of all continuous homomorphisms of finite order, and Pic(X) is endowed with a certain topology defined in [5], 9.4 (cf. Lemma 3.2).…”