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# 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

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

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
- 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 *N*_{1} from GL_{1}(*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

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: 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

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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