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 group homomorphism 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 the residue class, in K^×/ [K^×, K^×], of

${\displaystyle \det \left({\begin{array}{*{20}c}a&b\\c&d\end{array}}\right)=\left\lbrace {\begin{array}{*{20}c}-cb&{\text{if }}a=0\\ad-aca^{-1}b&{\text{if }}a\neq 0.\end{array}}\right.}$

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:^[1]

• 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^× 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 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^[2] but false in general.^[3]

See also

• Moore determinant over a division algebra