Exercise 17, § 16

Suppose that the field K is complete for a discrete valuation $v$, assumed normed (cf. Exercise 12 of VIII, p. 272); denote the ring of $v$ by A and its residue field by $\kappa$. The homomorphism $r_{\kappa /A}:$ Br(A) $\rightarrow$ Br($\kappa$ ) is bijective (Exercise 14); denote by $\iota :$ Br($\kappa$ )$\rightarrow$ Br(K) the homomorphism $r_{A/K}\circ r_{\kappa /A}^{-1}$.

a) Let L be an unramified Galois extension of K of finite degree, with Galois group G. Let B be the valuation ring of $v_L$ extending $v$ and $\kappa_L$ be its residue field. The homomorphism $\iota$ induces a homomorphism $\iota_L:$ Br($\kappa_L/\kappa$ )$\rightarrow$ Br(L$/K$). Construct a split exact sequence

$0\rightarrow$ Br($\kappa_L/\kappa$ )$\longrightarrow^{\iota_L}$ Br(L$/K$)$\rightarrow H^2(G,\mathbf{Z})\rightarrow 0$,

where $\mathbf{Z}$ is endowed with the trivial action of G (observe that the exact sequence of $\mathbf{Z}$[G]-modules $0\rightarrow B^*\rightarrow L^*\longrightarrow^{v_L}\mathbf{Z}$ splits, and use Exercise 14). b) Let G be a group. Using the exact sequence $0\rightarrow \mathbf{Z}\rightarrow \mathbf{Q}\rightarrow \mathbf{Q}/\mathbf{Z}\rightarrow 0$, prove that the linking homomorphism of Exercise 1 induces an isomorphism from Hom(G$,\mathbf{Q}/\mathbf{Z}$) to $H^2(G,\mathbf{Z})$.

c) Conclude that we have a split exact sequence

$0\rightarrow$ Br($\kappa$ )$\rightarrow^{\iota}$ Br(K) $\rightarrow \Xi (\mathfrak{g},\mathbf{Q}/\mathbf{Z})\rightarrow 0$,

where $\mathfrak{g}$ is the Galois group of a separable closure of $\kappa$ over $\kappa$ and $\Xi (\mathfrak{g})$ is the group of continuous homomorphisms from $\mathfrak{g}$ to $\mathbf{Q}/\mathbf{Z}$, endowed with the discrete topology (pass to the direct limit using Exercise 12 of VIII, p. 272 and the fact that every finite Galois extension of $\kappa$ is the residue field extension of an unramified finite Galois extension of K, unique up to isomorphism).

d) Deduce that Br(K) $=\{0\}$ if $\kappa$ is algebraically closed.

e) Suppose that the field $\kappa$ is finite, in other words, that K is a finite extension of a $p$-adic field or a field of formal power series over a finite field. Construct an isomorphism from Br(K) to $\mathbf{Q}/\mathbf{Z}.*$