Corestriction of Brauer group of local field is an isomorphism

Brauer group of any field is the set of finite dimensional central simple algebras over that field modulo an equivalence relation, called the Brauer equivalence. It turns out to be an abelian group under the operation \otimes.  Every equivalence class has a unique central division algebra as a representative. So if we can classify all central division algebras over the field, we know its Brauer group.

Another way to describe the Brauer group of a field is via the profinite group cohomology of the Galois group of K:

H^2(K, \mathbb{Q}/\mathbb{Z})

Thus for (finite) field extensions L/K, we have the following morphisms :

  1. Restriction : Res : Br(K)\to Br(L), which at the central simple algebra level is just a base change  from K to L
  2. Corestriction : Cor : Br(L)\to Br(K)

If K is an algebraically closed field, then there are no nontrivial finite dimensional central division algebras over it (If D is one such, then it must contain a nontrivial field extension of finite degree over K, which is not possible). Thus Br(K) = (0).

If K is a finite field, then there are still no nontrivial finite dimensional central division algebras over it (This is not as simple as the case of an algebraically closed field though, but the proof is still very elementary. One elegant way of showing this is to show K is a C_1 field. A C_1 field is any field such that any homogeneous polynomial with degree strictly less than the number of variables used has a nontrivial zero. And it is easy to see that there are no nontrivial central division algebras over such a field by simply observing that the reduced norm polynomial is a homogeneous polynomial with degree = \sqrt{[D:K]} in [D:K] number of variables where D is the central division algebra over C_1 field K) Thus Br(K)=(0) here also.

The field \mathbb{R} has exactly one nontrivial central division algebra over it. It is the famous Hamiltonian quaternions. Thus Br(\mathbb{R})\cong \mathbb{Z}/{2\mathbb{Z}}. I have forgotten how to prove this ..

We now come to K, a local field (examples are \mathbb{Q}_p and k((t)) where k is a finite field). Any finite extension of a local field is also a local field. Local class field theory tells us that Br(K)\cong \mathbb{Q}/{\mathbb{Z}}. In fact, it gives us the following very useful commutative diagram

\begin{array}{ccc} Br(K) & \to^{Res} & Br(L) \\ \downarrow & . &\downarrow \\ \frac{\mathbb{Q}}{\mathbb{Z}} & \to ^{[L:K]} & \frac{\mathbb{Q}}{\mathbb{Z}} \end{array}

The vertical arrows are isomorphisms. These maps which give you the isomorphism of the Brauer groups of local fields with \frac{\mathbb{Q}}{\mathbb{Z}} are called the Hasse-invariants or simply inv.

To see the usefulness of this diagram, here is a nice application :

If K is a local field and central division algebra D/K has exponent r (that is r[D]=0\in BR(K)), then any field L/K of degree r splits D.

Thus checking whether a field is a splitting field of a division algebra becomes as trivial as checking the degree of the field extension. The proof easily follows from the diagram :

\begin{array}{ccc} D & \to^{Res} & D\otimes L \\ \downarrow & . &\downarrow \\ a & \to ^{[L:K]} & {ra=0} \end{array}

Thus the restriction map Res : Br(K)\to Br(L) can be simply described as multiplication map \frac{\mathbb{Q}}{\mathbb{Z}} \to \frac{\mathbb{Q}}{\mathbb{Z}}, sending p/q \to np/q where n=[L:K]. This is clearly surjective.

Thus restriction map between Brauer groups of local fields is a surjection

What about the corestriction Cor : Br(L)\to Br(K) ? It corresponds to the identity map of \frac{\mathbb{Q}}{\mathbb{Z}} and in fact fits into the following commutative diagram :

\begin{array}{ccc} Br(L) & \to^{Cor} & Br(K) \\ \downarrow & &\downarrow \\ \frac{\mathbb{Q}}{\mathbb{Z}} & \to ^{id} & \frac{\mathbb{Q}}{\mathbb{Z}} \end{array}

Here is a proof :

Once we show that the above diagram commutes, then since the vertical maps are isomorphisms and so is the bottom map, Cor will be an isomorphism.

Let [D]\in Br(L). Since retsriction map is surjective, it comes from [C]\in BR(K). The following is a well-known and very useful fact from finite group cohomology, namely if H is a subgroup of $G$ of index n, then Cor\circ Res : H^i(G,-)\to H^i(H,-)\to H^i(G,-) is simply multiplication by n.

Thus Cor([D]) = Cor(Res([C])) = n[C] where n=[L:K]. Let inv([C])=a and inv([D])=b. The following diagram will make things clear !

\begin{array}{ccccc} C & \to^{Res} & D & \to^{Cor} & nC\\| & . & | & . & |\\a & \to & b & \to & na \end{array}

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s