Dieudonné determinant

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.

If K is a division ring, then the Dieudonné determinant is a group homomorphism from the group GLn(K&hairsp;) of invertible n-by-n matrices over K onto the abelianization K&hairsp;×/&hairsp;[K&hairsp;×,&thinsp;K&hairsp;×] of the multiplicative group K&hairsp;× of K.

For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in K&hairsp;×/&hairsp;[K&hairsp;×,&thinsp;K&hairsp;×], of


 * $$\det \left({\begin{array}{*{20}c} a & b \\ c & d \end{array}}\right) =

\left\lbrace{\begin{array}{*{20}c} -cb & \text{if } a = 0 \\ ad - aca^{-1}b & \text{if } a \ne 0. \end{array}}\right. $$

Properties
Let R be a local ring. There is a determinant map from the matrix ring GL(R&hairsp;) to the abelianised unit group R&hairsp;×ab with the following properties:
 * The determinant is invariant under elementary row operations
 * The determinant of the identity matrix is 1
 * If a row is left multiplied by a in R&hairsp;× then the determinant is left multiplied by a
 * The determinant is multiplicative: det(AB) = det(A)det(B)
 * If two rows are exchanged, the determinant is multiplied by −1
 * If R is commutative, then the determinant is invariant under transposition

Tannaka–Artin problem
Assume that K is finite over its center F. The reduced norm gives a homomorphism Nn from GLn(K&hairsp;) to F&hairsp;×. We also have a homomorphism from GLn(K&hairsp;) to F&hairsp;× obtained by composing the Dieudonné determinant from GLn(K&hairsp;) to K&hairsp;×/&hairsp;[K&hairsp;×,&thinsp;K&hairsp;×] with the reduced norm N1 from GL1(K&hairsp;) = K&hairsp;× to F&hairsp;× via the abelianization.

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