Stability theorems in algebraic K-theory

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)

Comments

Many groups in algebraic -theory are defined as direct limits. For example, [a1], for any associative ring with . The corresponding stability theorem asserts that the sequence is eventually stable, i.e., the mappings become isomorphisms starting from some point. In the above example, for , where is the Bass stable rank of [a1]–[a3]. See [a4] for a similar result for higher -functors. For the -functor, a stability result is the so-called cancellation theorem [a1]. A similar result for modules with quadratic forms is known as the Witt theorem.

The most common meaning of "stability theorem" is that given in the last sentence of the main article above (i.e. stabilization of -functors under transfer from stable to unstable objects), cf. [a3].

The stability theorem for Whitehead groups, or Bass–Heller–Swan theorem, was generalized to all -groups by D. Quillen, [a4].

References

[a1]	H. Bass, "-theory and stable algebra" Publ. Math. IHES , 22 (1964) pp. 485–544
[a2]	L.N. Vaserstein, "-theory and the congruence subgroup problem" Math. Notes , 5 (1969) pp. 141–148 Mat. Zametki , 5 (1969) pp. 233–244
[a3]	A. Suslin, "Stability in algebraic -theory" R.K. Dennis (ed.) , Algebraic -theory (Oberwolfach, 1980) , Lect. notes in math. , 966 , Springer (1982) pp. 304–333
[a4]	D. Quillen, "Higher algebraic -theory I" H. Bass (ed.) , Algebraic -theory I (Battelle Inst. Conf.) , Lect. notes in math. , 341 , Springer (1973) pp. 85–147