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 Template:Harvs.

If K is a division ring, then the Dieudonné determinant is a group homomorphism from the group GL_n(KTemplate:Hairsp) of invertible n-by-n matrices over K onto the abelianization K^{Template:Hairsp×}/Template:Hairsp[K^{Template:Hairsp×}, K^{Template:Hairsp×}] of the multiplicative group K^{Template:Hairsp×} of K.

For example, the Dieudonné determinant for a 2-by-2 matrix is the residue class, in K^{Template:Hairsp×}/Template:Hairsp[K^{Template:Hairsp×}, K^{Template:Hairsp×}], of

$det\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\{\begin{array}{cc}-cb& \text{if }a=0\\ ad-ac{a}^{-1}b& \text{if }a\ne 0.\end{array}$

Properties

Let R be a local ring. There is a determinant map from the matrix ring GL(RTemplate:Hairsp) to the abelianised unit group R^{Template:Hairsp×}_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^{Template:Hairsp×} 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(KTemplate:Hairsp) to F^{Template:Hairsp×}. We also have a homomorphism from GL_n(KTemplate:Hairsp) to F^{Template:Hairsp×} obtained by composing the Dieudonné determinant from GL_n(KTemplate:Hairsp) to K^{Template:Hairsp×}/Template:Hairsp[K^{Template:Hairsp×}, K^{Template:Hairsp×}] with the reduced norm N₁ from GL₁(KTemplate:Hairsp) = K^{Template:Hairsp×} to F^{Template:Hairsp×} via the abelianization.

The Tannaka–Artin problem is whether these two maps have the same kernel SL_n(KTemplate:Hairsp). This is true when F is locally compact but false in general.

See also

• Moore determinant over a division algebra

References

• Template:Citation
• Template:Citation. Errata
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• Template:Eom