“Dieudonné determinant”的意思、由来-开放百科全书

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Dieudonné determinant

释义

Properties
Tannaka–Artin problem
See also
References

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 {{harvs|txt|last=Dieudonné|authorlink=Jean Dieudonné|year=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

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]

References

1. ^Rosenberg (1994) p.64
2. ^{{cite journal | zbl=0060.07901 | last1=Nakayama | first1=Tadasi | last2=Matsushima | first2=Yozô | title=Über die multiplikative Gruppe einer p-adischen Divisionsalgebra | language=German | journal=Proc. Imp. Acad. Tokyo | volume=19 | pages=622–628 | year=1943 | doi=10.3792/pia/1195573246}}
3. ^{{cite journal | zbl=0338.16005 | last=Platonov | first=V.P. | authorlink=Vladimir Platonov | title=The Tannaka-Artin problem and reduced K-theory | language=Russian | journal=Izv. Akad. Nauk SSSR, Ser. Mat. | volume=40 | pages=227–261 | year=1976 }}

{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Les déterminants sur un corps non commutatif | url=http://www.numdam.org/item?id=BSMF_1943__71__27_0 | mr=0012273 | year=1943 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=71 | pages=27–45 | zbl=0028.33904 }}
{{Citation | last1=Rosenberg | first1=Jonathan | authorlink=Jonathan Rosenberg (mathematician) | title=Algebraic K-theory and its applications | url=https://books.google.com/books?id=TtMkTEZbYoYC | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94248-3 | mr=1282290 | zbl=0801.19001 | year=1994 | volume=147}}. Errata
{{citation | title=Trees | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | publisher=Springer | year=2003 | isbn=3-540-44237-5 | zbl=1013.20001 | page=74 }}
{{eom|id=D/d031410|title=Determinant|first=D.A. |last=Suprunenko}}

2 : Linear algebra|Determinants

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Properties

Tannaka–Artin problem

See also

References