Exercise 7, § 16

Let L be an extension of K. For any integer $n\geqslant 1$, denote by $\mathscr{A}_n(L/K)$ the set of isomorphism classes of central simple K-algebras of degree $n^2$ split by L.

a) Prove that the kernel Br(L$/K$) of the canonical homomorphism Br(K) $\rightarrow$ Br(L) is isomorphic to the direct limit of the sets $\mathscr{A}_n(L/K)$; specify a system of mappings. b) Assume from now on that L is a finite Galois extension of K, with Galois group G. Deduce a bijection $\theta_n:\mathscr{A}_n(L/K)\rightarrow H^1(G,\mathbf{P}\mathbf{G}\mathbf{L}_n(L))$ from Exercise 6.

c) Denote by $\partial_n: H^1(G,\mathbf{P}\mathbf{G}\mathbf{L}_n(L))\rightarrow H^2(G,L^*)$ the mapping deduced from the exact sequence $1\rightarrow L^*\rightarrow \mathbf{G}\mathbf{L}_n(L)\rightarrow \mathbf{P}\mathbf{G}\mathbf{L}_n(L)\rightarrow 1$ (Exercise 5, c)) and by $\delta_n:\mathscr{A}_n(L/K)\rightarrow H^2(G,L^*)$ the mapping $\partial_n\circ \theta_n$. If $A\in \mathscr{A}_n(L/K)$, then we have $\delta_n(A) = 0$ if and only if A is a matrix algebra over K; if $A\in \mathscr{A}_p(L/K)$ and $B\in \mathscr{A}_q(L/K)$, then we have $\delta_{pq}(A\otimes_KB) =\delta_p(A) +\delta_q(B)$. Deduce an injective group homomorphism $\delta_{L/K}:$ Br(L$/K$)$\rightarrow H^2(G,L^*)$ by passing to the direct limit. d) Prove that $\partial_n$ is surjective for $n=$ Card(G). (Let $c: G\times G\rightarrow L^*$ be an element of $Z^2(G,L^*)$. For every $\sigma \in G$, consider the endomorphism $u(\sigma )$ of $L^G$ that sends $e_{\tau}$ to $c(\sigma , \tau )e_{\sigma \tau}$ for every $\tau \in G$, and determine $u(\sigma )^{\sigma}u(\tau )u(\sigma \tau )^{-1}.$)

e) Compare $\delta_{L/K}$ and $\Phi_{L/K}$.

f ) Prove that the group Br(K) can be identified with the limit of the direct system of groups $(H^2$(Gal $(L/K),L^*$)) with respect to the set (ordered by inclusion) of finite Galois extensions of K contained in a fixed algebraic closure of K, where the homomorphisms between these groups are the (injective) inflation homomorphisms. g) Let F be an extension of K contained in L and Γ the subgroup of G that leaves F invariant. Prove that the following diagram commutes:

Br(L$/K$)$^{r_{F/K}}$ // Br(F$/L$)

$\delta_{L/K}\delta_{F/L}$

$H^2(G,L^*)^{Res^2}/$/ $H^2(\Gamma ,L^*)$.

Deduce a canonical isomorphism from Br(F$/K$) to the kernel of the homomorphism Res$^2: H^2(G,L^*)\rightarrow H^2(\Gamma ,L^*)$.