Tannaka–Artin problem

In linear algebra, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division rings and local rings. It was introduced by Dieudonné (1943).

If K is a division ring, then the Dieudonné determinant is a homomorphism of groups from the group GL_n(K) of invertible n by n matrices over K onto the abelianization K^*/[K^*, K^*] of the multiplicative group K^* of K.

For example, the Dieudonné determinant for a 2-by-2 matrix is

Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group R^∗_ab with the following properties:

Assume that K is finite over its centre F. The reduced norm gives a homomorphism N_n from GL_n(K) to F^*. We also have a homomorphism from GL_n(K) to F^* obtained by composing the Dieudonné determinant from GL_n(K) to K^*/[K^*, K^*] with the reduced norm N₁ from GL₁(K) = K^* to F^* via the abelianization.

The Tannaka–Artin problem is whether these two maps have the same kernel SL_n(K). This is true when F is locally compact but false in general.

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