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]

See also

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.
  • Dieudonné, Jean (1943), "Les déterminants sur un corps non commutatif", Bulletin de la Société Mathématique de France, 71: 27–45, ISSN 0037-9484, MR 0012273, Zbl 0028.33904
  • Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, 147, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001 . Errata
  • Serre, Jean-Pierre (2003), Trees, Springer, p. 74, ISBN 3-540-44237-5, Zbl 1013.20001
  • Suprunenko, D.A. (2001) [1994], "Determinant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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