# Stability theorems in algebraic K-theory

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Assertions on the invariance of the groups or their subgroups, given certain special extensions of the ground ring (see Algebraic -theory).

The following are the best-known stability theorems. Let be a regular ring (cf. Regular ring (in commutative algebra)) and let be the ring of polynomials in the variables over . The stability theorem for Whitehead groups under the transfer from to , , states that the natural homomorphism imbedding in induces an isomorphism between and (cf. also Whitehead group).

In the case of a skew-field that is finite-dimensional over its centre , one can define a reduced-norm homomorphism of the multiplicative group of into the multiplicative group of its centre. The kernel of this homomorphism, usually written as , determines the reduced Whitehead group of : (see Special linear group), which is a subgroup in . If is the field of rational functions in over , then the algebra is a skew-field, and the natural imbedding of in induces a homomorphism The stability theorem for reduced Whitehead groups states that the homomorphism is bijective (, see also ). Similar statements are also true in unitary and spinor algebraic -theories , .

Theorems on stabilization for -functors under the transfer from the stable objects to unstable ones are also called stability theorems (see ).

How to Cite This Entry:
Stability theorems in algebraic K-theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_theorems_in_algebraic_K-theory&oldid=12794
This article was adapted from an original article by V.I. Yanchevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article