Dieudonné determinant

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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 GLn(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

 \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[edit]

Let R be a local ring. There is a determinant map from the matrix ring GL(R) to the abelianised unit group Rab with the following properties:[1]

  • 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
  • The determinant is invariant under transposition

Tannaka–Artin problem[edit]

Assume that K is finite over its centre F. The reduced norm gives a homomorphism Nn from GLn(K) to F*. We also have a homomorphism from GLn(K) to F* obtained by composing the Dieudonné determinant from GLn(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 SLn(K). This is true when F is locally compact[2] but false in general.[3]

See also[edit]

References[edit]

  1. ^ Rosenberg (1994) p.64
  2. ^ Nakayama, Tadasi; Matsushima, Yozô (1943). "Über die multiplikative Gruppe einer p-adischen Divisionsalgebra". Proc. Imp. Acad. Tokyo (in German) 19: 622–628. doi:10.3792/pia/1195573246. Zbl 0060.07901. 
  3. ^ Platonov, V.P. (1976). "The Tannaka-Artin problem and reduced K-theory". Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian) 40: 227–261. Zbl 0338.16005.