Image of the Brauer group under a field extension

Question

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k M_n(k)\cong B\otimes_k M_m(k))$, with the group operation given by the tensor product $\otimes_k$.

Given a Galois field extension $k\subset K$, we get a natural map $Br(k)\rightarrow Br(K)$. The kernel of this map, $Br(K\vert k)$, is just the subgroup of $Br(k)$ of algebras which split over $K$; this is a pretty snappy description.

My question is:

Is there an equally snappy description of the image of $Br(k)$ in $Br(K)$?

(I'm asking this here, as opposed to MO, since I suspect the answer is pretty simple and I just haven't run across it.)

I've tagged this question with the "algebraic geometry" and "group theory" tags; I'm not sure they are appropriate, though, so feel free to delete/replace them.

Answer 1

Let $G$ be the Galois group of $K$ over $k$. The image is the kernel of a map $Br(K)^G \to H^3(G, K^*)$. Look in books on Galois cohomology, preferably one written in French...

Answer 2

Using the cohomological description of Br(k) as $H^2$($k^s$/k, $k^s$ $^*$) (where $k^s$ denotes a separable closure), one can give a first approach to your problem for a Galois extension K/k with group G, via the Hochschild-Serre spectral sequence: because of Hilbert 90, we get a 5 term exact sequence which shows that the kernel Br(K/k) is isomorphic to $H^2$(G, $K^*$), and the image of Br(k) in Br(K) is the kernel of the so called transgression map which appears in Count Dracula's answer. See Serre's "Corps Locaux" (there's an English translation now), chapter VII, end of §6. To the best of my knowledge, this is all we can say in full generality.

Particular cases :

• fields with trivial Brauer groups: a list is given in op. cit., end of chapter X (finite fields, $C_1$-fields, the abelian closure of Q, etc.) But in view of your question, these are not very interesting
• global and local fields: let us concentrate on number fields k and their completions $k_v$. At archimedean places, we have Br(C)=0 and Br(R)=Z/2Z. For the completion $k_v$ at a non archimedean place v, local class field theory shows that Br($k_v$) is isomorphic to Q/Z. An important point here is that for an extension $K_w$/$k_v$, the map from Br($k_v$) to Br($K_w$) corresponds to the multiplication by the degree of $K_w$/$k_v$ in Q/Z. Global class field theory shows that Br(k) injects canonically into the direct sum of all the $k_v$'s (archimedean or not), with cokernel isomorphic to Q/Z. This answers your question in this case. ¤