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 GL_n(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

$\det \left({{\begin{array}{*{20}c}a&b\\c&d\end{array}}}\right)=\left\lbrace {{\begin{array}{*{20}c}-cb&{\text{if }}a=0\\ad-aca^{{-1}}b&{\text{if }}a\neq 0\end{array}}}\right..$

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 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 N_n from GL_n(K) to F^*. We also have a homomorphism from GL_n(K) to F^* obtained by composing the Dieudonné determinant from GL_n(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 SL_n(K). This is true when F is locally compact^[2] but false in general.^[3]

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.

• 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), "Determinant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4