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

det ( a b c d ) = { − c b if  a = 0 a d − a c a − 1 b if  a ≠ 0. {\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 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 (2): 227–261. Bibcode:1976IzMat..10..211P. doi:10.1070/IM1976v010n02ABEH001686. Zbl 0338.16005.
  • Dieudonné, Jean (1943), "Les déterminants sur un corps non commutatif", Bulletin de la Société Mathématique de France, 71: 27–45, doi:10.24033/bsmf.1345, ISSN 0037-9484, MR 0012273, Zbl 0028.33904
  • Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, vol. 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", Encyclopedia of Mathematics, EMS Press