# Stability theorems in algebraic K-theory

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 , [1], 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 ([2], see also [3]). Similar statements are also true in unitary and spinor algebraic -theories [4], [5].

Theorems on stabilization for -functors under the transfer from the stable objects to unstable ones are also called stability theorems (see [6]).

#### References

 [1] H. Bass, A. Heller, R. Swan, "The Whitehead group of a polynomial extension" Publ. Math. IHES : 22 (1964) pp. 61–79 [2] V.P. Platonov, "Reduced -theory and approximation in algebraic groups" Proc. Steklov Inst. Math. , 142 (1976) pp. 213–224 Trudy Mat. Inst. Steklov. , 142 (1976) pp. 198–207 [3] V.P. Platonov, V.I. Yanchevskii, " for division rings of noncommutative rational functions" Soviet Math. Dokl. , 20 : 6 (1976) pp. 1393–1397 Dokl. Akad. Nauk SSSR , 249 : 5 (1979) pp. 1064–1068 [4] V.I. Yanchevskii, "Reduced unitary -theory. Applications to algebraic groups" Math. USSR Sb. , 38 (1981) pp. 533–548 Mat. Sb. , 110 : 4 (1979) pp. 579–596 [5] A.P. Monastyrnyi, V.I. Yanchevskii, "Whitehead groups of spinor groups" Math. USSR Izv. , 54 : 1 (1991) pp. 61–100 Izv. Akad. Nauk SSSR Ser. Mat. , 54 : 1 (1990) pp. 60–96 [6] H. Bass, "Algebraic -theory" , Benjamin (1968)