Milnor ring

In the mathematical discipline known as K-theory, the Milnor ring of a field F, named after John Milnor, is defined ^[1] as the graded ring $K_*^M (F)$ with unit, generated by symbols $\ell(a)$ (for $a \in F -\{0\}$ of degree one, with relations

$\ell(ab)=\ell(a)+\ell(b),\quad \ell(a)\ell(1-a)=0. \,$

One can show that $K_0^M (F)={\mathbb Z}$, $K_1^M (F)=F -\{0\}$.

The Milnor ring appears as one side of the Milnor conjecture.

References

1. ^ T.A. Springer, A remark on the Milnor ring, Inventiones mathematicae, 1970

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: K-theory Algebra stubs Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Milnor K-theory Milnrow Cricket Club Look at other dictionaries: Milnor number — In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either an… …   Wikipedia Milnor–Thurston kneading theory — The Milnor–Thurston kneading theory is a mathematical theory which analyzes the iterates of piecewise monotone mappings of an interval into itself. The emphasis is on understanding the properties of the mapping that are invariant under… …   Wikipedia Algebraic K-theory — In mathematics, algebraic K theory is an important part of homological algebra concerned with defining and applying a sequence Kn(R) of functors from rings to abelian groups, for all integers n. For historical reasons, the lower K groups K0 and… …   Wikipedia Genus of a multiplicative sequence — In mathematics, the genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.DefinitionA genus phi; assigns a number phi;( X ) to …   Wikipedia Differentiable manifold — A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the middle chart the Tropic of Cancer is a smooth curve, whereas in the first it has a sharp… …   Wikipedia List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia Complex cobordism — In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using… …   Wikipedia Steenrod algebra — In algebraic topology, a branch of mathematics, the Steenrod algebra is a structure occurring in the theory of cohomology operations. It is an object of great importance, most especially to homotopy theorists. More precisely, for a given prime… …   Wikipedia K-Theorie — Das mathematische Teilgebiet der K Theorie beschäftigt sich mit dem Studium von Vektorbündeln auf topologischen Räumen (topologische K Theorie) oder Ringen bzw. Schemata (algebraische K Theorie). (Der Name K Theorie wurde von Alexander… …   Deutsch Wikipedia Complex projective space — The Riemann sphere, the one dimensional complex projective space, i.e. the complex projective line. In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”