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 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

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 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 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]

gollark: We should be evaluating it on how well it does what we want it to, not how well the designers *claim it does*.
gollark: Oh, right.
gollark: What?
gollark: A lot of the time explanations are basically just rationalised after the fact to justify something you're already doing.
gollark: The purpose written down somewhere doesn't really matter if people with different preferences try and shape it in their way, or if it doesn't actually work well at satisfying that purpose.

See also
  • Moore determinant over a division algebra

References
  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.
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