View source for Stability theorems in algebraic K-theory

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Assertions on the invariance of the groups  $  K _{i} (R) $
or their subgroups, given certain special extensions of the ground ring  $  R $(
see [[Algebraic K-theory|Algebraic  $  K $-
theory]]).

The following are the best-known stability theorems. Let  $  R $
be a regular ring (cf. [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]) and let  $  R[t _{1} \dots t _{n} ] $
be the ring of polynomials in the variables  $  t _{1} \dots t _{n} $
over  $  R $. 
The stability theorem for Whitehead groups under the transfer from  $  R $
to  $  R[t _{1} \dots t _{n} ] $, 
[[#References|[1]]], states that the natural homomorphism imbedding  $  R $
in  $  R[t _{1} \dots t _{n} ] $
induces an isomorphism between  $  K _{1} (R) $
and  $  K _{1} (R[t _{1} \dots t _{n} ]) $(
cf. also [[Whitehead group|Whitehead group]]).

In the case of a skew-field  $  R $
that is finite-dimensional over its centre  $  Z(R) $, 
one can define a reduced-norm homomorphism  $   \mathop{\rm Nrd}\nolimits _{R} : \  R ^ \star  \rightarrow Z(R) ^ \star  $
of the multiplicative group  $  R ^ \star  $
of  $  R $
into the multiplicative group  $  Z(R) ^ \star  $
of its centre. The kernel of this homomorphism, usually written as  $   \mathop{\rm SL}\nolimits (1,\  R) $, 
determines the reduced Whitehead group  $  SK _{1} (R) $
of  $  R $:
$$ 
SK _{1} (R)   \simeq   { \mathop{\rm SL}\nolimits (1,\  R)} / {[R ^ \star  ,\  R ^ \star  ]}
$$(
see [[Special linear group|Special linear group]]), which is a subgroup in  $  K _{1} (R) $. 
If  $  Z(R)(t _{1} \dots t _{n} ) $
is the field of rational functions in  $  t _{1} \dots t _{n} $
over  $  Z(R) $, 
then the algebra $$ 
R(t _{1} \dots t _{n} )   =   R \otimes _{Z(R)} Z(R)(t _{1} \dots t _{n} )
$$
is a skew-field, and the natural imbedding  $  \phi _ {t _{1} \dots t _ n} $
of  $  R $
in  $  R(t _{1} \dots t _{n} ) $
induces a homomorphism $$ 
\psi _ {t _{1} \dots t _ n} ^ \prime  : \  SK _{1} (R)   \rightarrow  
SK _{1} (R(t _{1} \dots t _{n} )).
$$
The stability theorem for reduced Whitehead groups states that the homomorphism  $  \psi _ {t _{1} \dots t _ n} ^ \prime  $
is bijective ([[#References|[2]]], see also [[#References|[3]]]). Similar statements are also true in unitary and spinor algebraic  $  K $-
theories [[#References|[4]]], [[#References|[5]]].

Theorems on stabilization for  $  K _{i} $-
functors under the transfer from the stable objects  $  K _{i} (R) $
to unstable ones are also called stability theorems (see [[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Bass, A. Heller, R. Swan, "The Whitehead group of a polynomial extension" ''Publ. Math. IHES'' : 22 (1964) pp. 61–79 {{MR|0174605}} {{ZBL|0248.18026}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.P. Platonov, "Reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706038.png" />-theory and approximation in algebraic groups" ''Proc. Steklov Inst. Math.'' , '''142''' (1976) pp. 213–224 ''Trudy Mat. Inst. Steklov.'' , '''142''' (1976) pp. 198–207 {{MR|568310}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.P. Platonov, V.I. Yanchevskii, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706039.png" /> 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 {{MR|0553335}} {{ZBL|0437.16015}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.I. Yanchevskii, "Reduced unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706040.png" />-theory. Applications to algebraic groups" ''Math. USSR Sb.'' , '''38''' (1981) pp. 533–548 ''Mat. Sb.'' , '''110''' : 4 (1979) pp. 579–596 {{MR|1331389}} {{MR|0919253}} {{MR|0772116}} {{MR|0684770}} {{MR|0549289}} {{MR|0562210}} {{MR|0509375}} {{MR|0508832}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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 {{MR|1044048}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Bass, "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706041.png" />-theory" , Benjamin (1968) {{MR|249491}} {{ZBL|}} </TD></TR></table>



====Comments====
Many groups in algebraic  $  K $-
theory are defined as direct limits. For example, [[#References|[a1]]],  $  K _{1} (R) =  \mathop{\rm lim}\nolimits \   \mathop{\rm GL}\nolimits _{n} (R) / E _{n} (R) $
for any associative ring  $  R $
with  $  1 $. 
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,  $  K _{1} (R) =  \mathop{\rm GL}\nolimits _{n} (R) / E _{n} (R) $
for  $  n \geq  \mathop{\rm sr}\nolimits (R) +1 $, 
where  $   \mathop{\rm sr}\nolimits (R) $
is the Bass stable rank of  $  R $[[#References|[a1]]]–[[#References|[a3]]]. See [[#References|[a4]]] for a similar result for higher  $  K $-
functors. For the  $  K _{0} $-
functor, a stability result is the so-called cancellation theorem [[#References|[a1]]]. A similar result for modules with quadratic forms is known as the [[Witt theorem|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  $  K _{i} $-
functors under transfer from stable to unstable objects), cf. [[#References|[a3]]].

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

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Bass, "K-theory and stable algebra" ''Publ. Math. IHES'' , '''22''' (1964) pp. 485–544 {{MR|0174604}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> L.N. Vaserstein, "$K_1$-theory and the congruence subgroup problem" ''Math. Notes'' , '''5''' (1969) pp. 141–148 ''Mat. Zametki'' , '''5''' (1969) pp. 233–244 {{MR|246941}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Suslin, "Stability in algebraic K-theory" R.K. Dennis (ed.) , ''Algebraic K-theory (Oberwolfach, 1980)'' , ''Lect. notes in math.'' , '''966''' , Springer (1982) pp. 304–333 {{MR|}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Quillen, "Higher algebraic K-theory I" H. Bass (ed.) , ''Algebraic K-theory I (Battelle Inst. Conf.)'' , ''Lect. notes in math.'' , '''341''' , Springer (1973) pp. 85–147 {{MR|338129}} {{ZBL|}} </TD></TR>
</table>