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 .

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

:$\det \left(\right) = \left\lbrace\right.$

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:Rosenberg (1994) p.64
* 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 ''N''₁ 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

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Errata
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{{DEFAULTSORT:Dieudonne determinant
Linear algebra
Determinants