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 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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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) to the abelianised unit group R^∗_ab with the following properties:

• The determinant is invariant under elementary row operations
• The determinant of the identity is 1
• If a row is left multiplied by a in R^∗ 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 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.

See also

• Moore determinant over a division algebra

References

• . Errata