Bochner–Martinelli formula: Difference between revisions

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In [[linear algebra]], the '''Dieudonné determinant''' is a generalization of the [[determinant of a matrix]] to matrices over [[division ring]]s and [[local ring]]s. It was introduced by {{harvs|txt|last=Dieudonné|authorlink=Jean Dieudonné|year=1943}}.
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If ''K'' is a division ring, then the Dieudonné determinant is a [[homomorphism]] of groups from the group GL<sub>''n''</sub>(''K'') of invertible ''n'' by ''n'' matrices over ''K'' onto the [[abelianization]] ''K''<sup>*</sup>/[''K''<sup>*</sup>, ''K''<sup>*</sup>] of the multiplicative group ''K''<sup>*</sup> of ''K''.
 
For example, the Dieudonné determinant for a 2-by-2 matrix is
 
:<math> \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''<sup>&lowast;</sup><sub>ab</sub> with the following properties:<ref name=Ros64>Rosenberg (1994) p.64</ref>
* The determinant is invariant under [[elementary row operation]]s
* The determinant of the identity is 1
* If a row is left multiplied by ''a'' in ''R''<sup>&lowast;</sup> 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''<sub>''n''</sub> from GL<sub>''n''</sub>(''K'') to ''F''<sup>*</sup>.  We also have a homomorphism from GL<sub>''n''</sub>(''K'') to ''F''<sup>*</sup> obtained by composing the Diedonné determinant from GL<sub>''n''</sub>(''K'') to ''K''<sup>*</sup>/[''K''<sup>*</sup>, ''K''<sup>*</sup>] with the reduced norm ''N''<sub>1</sub> from GL<sub>''1''</sub>(''K'') = ''K''<sup>*</sup> to ''F''<sup>*</sup> via the abelianization.
 
The '''Tannaka–Artin problem''' is whether these two maps have the same kernel SL<sub>''n''</sub>(''K'').  This is true when ''F'' is locally compact<ref>{{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 }}</ref> but false in general.<ref>{{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 }}</ref>
 
==See also==
*[[Moore determinant over a division algebra]]
 
==References==
{{reflist}}
*{{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=http://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}}.  [http://www-users.math.umd.edu/~jmr/KThy_errata2.pdf 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}}
 
{{DEFAULTSORT:Dieudonne determinant}}
[[Category:Linear algebra]]
[[Category:Determinants]]

Latest revision as of 17:00, 16 December 2014

Kitchen knives are a have to if you ever plan to cook a substantial meal in your dwelling. Each kitchen must have an element of efficiency. The block for these is not, in our opinion, the most appealing of knife blocks but it definitely does the job in a space effective manner and matches the monochromatic look of the knives. This knife block set is from a firm that has been in the company for virtually two hundred years and they remain as devoted to the creation of kitchen knives as ever. As is popular with German knives, F. Dick Premier knives are forged.



Customer Reports ranked two Henckels sets—the 7-piece Zwilling J.A. Henckels Twin Experienced S” and the 9-piece Zwilling J.A. Henckels Twin Profection #33049—and the 7-piece Wüsthof Classic Ikon #8347 set higher than the Wüsthof Classic set. Having said that, in most tests (for chef's knives, at least), such as those by Cooking for Engineers and America's Test Kitchen , Wüsthof knives regularly beat Henckels. If you enjoyed this information and you would certainly like to get more facts regarding Greatest Japanese Deba Knife (http://Thebestkitchenknivesreviews.com) kindly see the page. I consider most house cooks would be extra than happy with this acquire.

Bread knives give a extended, serrated blade that handles the job like no other knife. Far from a one-job utensil, nonetheless, serrated bread knives also make neat, brief operate of cutting a cake or slicing a juicy tomato. Now, for the rest of you, study on to hear far more about what are very possibly the finest steak knives that have ever graced the Earth. The initial issue that you see with these steak knives is how striking they are how elegantly designed and however so powerfully engineered.

Study the revaluation just in case grant the client to apprize of a standard Wusthof Classic eight Piece Knife Block Set (All-natural Block) disadvantages or professionals. The appealing, ergonomic black handles are designed for comfort and handle, and are produced of a seamless, synthetic material for a hygienic grip. Read our in-depth evaluation of the Wusthof Classic Knife Block Set. The most effective-selling 8 piece kitchen cutlery set in wood block.

Most cooks, even the extra experienced, only use 3 or four knives on a regular basis. Rather than obtaining a complete set, which can price hundreds of dollars, these three or 4 will be the workhorses of the kitchen. Additional knives with other specialized functions can often be purchased later. The chef's knife is the standard and most versatile of kitchen knives. And to hone knives among sharpening, a sharpening steel is necessary. Right care of kitchen knives is vital. And in no way wash knives in a dishwasher.

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