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
[edit]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
[edit]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
[edit]References
[edit]- ^ Rosenberg (1994) p.64
- ^ 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.
- ^ 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