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 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 the residue class, in K ×/ [K ×, K ×], of

Properties

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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

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Assume that K is finite over its center 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

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References

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  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 (2): 227–261. Bibcode:1976IzMat..10..211P. doi:10.1070/IM1976v010n02ABEH001686. Zbl 0338.16005.