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 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 the residue class, in K^×/ [K^×, K^×], of

[math]\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 \ne 0. \end{array}}\right. }[/math]

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:

• 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 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 but false in general.

See also

• Moore determinant over a division algebra

References

• 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 
• 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, https://books.google.com/books?id=TtMkTEZbYoYC . Errata
• Serre, Jean-Pierre (2003), Trees, Springer, p. 74, ISBN 3-540-44237-5 
• Hazewinkel, Michiel, ed. (2001), "Determinant", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=D/d031410