Matsumoto's theorem

The set of Matsumoto is a lemma from the mathematical sub-region of the K-theory , it is an explicit description for the second algebraic K theory of a body to. The sentence is named after the Japanese mathematician Matsumoto Hideya .

Matsumoto's theorem

Let it be a body and its second algebraic K-theory, then: ${\displaystyle F}$ ${\displaystyle K_{2}(F)}$

${\displaystyle K_{2}(F)=F^{\times }\otimes _{\mathbf {Z} }F^{\times }/\langle a\otimes (1-a)\mid a\not =0,1\rangle .}$.

In other words: is isomorphic to the coke of the Dehn invariant ${\displaystyle K_{2}(F)}$

${\displaystyle D\colon \mathbb {Z} \left[F\setminus \left\{0,1\right\}\right]\to F^{\times }\otimes _{\mathbf {Z} }F^{\times }}$

${\displaystyle D(a):=a\otimes (1-a)}$

Milnor's K-theory (history)

Motivated by Matsumoto's theorem, Milnor defined Milnor 's K-theory of bodies, which was later named after him ${\displaystyle K^{M}(F)}$

${\displaystyle K_{*}^{M}(F):=T^{*}F^{\times }/(a\otimes (1-a))}$,

thus as graduated components of the quotient of the tensor algebra over the Abelian group F ^× according to the two-sided ideal, that of the elements of form

${\displaystyle a\otimes (1-a)}$

for a  ≠ 0.1 is generated. There is a picture

${\displaystyle K_{n}^{M}(F)\to K_{n}(F),}$

which for and after Matsumoto's theorem is also for an isomorphism. ${\displaystyle n=0,1}$ ${\displaystyle n=2}$

For is , however, no isomorphism of Coker ${\displaystyle n>2}$ ${\displaystyle K_{n}^{M}(F)\to K_{n}(F)}$

${\displaystyle K_{n}^{ind}(F):=\mathrm {Kokern} (K_{n}^{M}(F)\to K_{n}(F))}$

is the so-called incomposable K-theory , which is the same in the case of number fields . ${\displaystyle K_{n}(F)}$

For is modulo 2 torsion isomorphic to the Bloch group . ${\displaystyle n=3}$ ${\displaystyle K_{3}^{ind}(F)}$

literature

• J. Rosenberg: Algebraic K-theory and its applications . (= Graduate Texts in Mathematics. 147). Springer Verlag, Berlin et al. 1996, ISBN 3-540-94248-3 .
• H. Matsumoto: Sur les sous-groupes arithmétiques des groupes semi-simple déployés. In: Ann.Sci.École Norm.Sup. Series 4, Volume 2, 1969, pp. 1-62. (French)

Web links

• Dalawat: Some aspects of the functor K ₂ of fields