View source for Stability theorems in algebraic K-theory

Assertions on the invariance of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870602.png" /> or their subgroups, given certain special extensions of the ground ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870603.png" /> (see [[Algebraic K-theory|Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870604.png" />-theory]]).

The following are the best-known stability theorems. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870605.png" /> be a regular ring (cf. [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870606.png" /> be the ring of polynomials in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870607.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870608.png" />. The stability theorem for Whitehead groups under the transfer from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s0870609.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706010.png" />, [[#References|[1]]], states that the natural homomorphism imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706012.png" /> induces an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706014.png" /> (cf. also [[Whitehead group|Whitehead group]]).

In the case of a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706015.png" /> that is finite-dimensional over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706016.png" />, one can define a reduced-norm homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706017.png" /> of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706019.png" /> into the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706020.png" /> of its centre. The kernel of this homomorphism, usually written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706021.png" />, determines the reduced Whitehead group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706023.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706024.png" /></td> </tr></table>

(see [[Special linear group|Special linear group]]), which is a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706026.png" /> is the field of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706027.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706028.png" />, then the algebra

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706029.png" /></td> </tr></table>

is a skew-field, and the natural imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706032.png" /> induces a homomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png" /></td> </tr></table>

The stability theorem for reduced Whitehead groups states that the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706034.png" /> is bijective ([[#References|[2]]], see also [[#References|[3]]]). Similar statements are also true in unitary and spinor algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706035.png" />-theories [[#References|[4]]], [[#References|[5]]].

Theorems on stabilization for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706036.png" />-functors under the transfer from the stable objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706037.png" /> 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</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</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</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</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</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)</TD></TR></table>



====Comments====
Many groups in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706042.png" />-theory are defined as direct limits. For example, [[#References|[a1]]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706043.png" /> for any associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706045.png" />. 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706046.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706048.png" /> is the Bass stable rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706049.png" /> [[#References|[a1]]]–[[#References|[a3]]]. See [[#References|[a4]]] for a similar result for higher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706050.png" />-functors. For the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706051.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706052.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706053.png" />-groups by D. Quillen, [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bass,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706054.png" />-theory and stable algebra"  ''Publ. Math. IHES'' , '''22'''  (1964)  pp. 485–544</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.N. Vaserstein,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706055.png" />-theory and the congruence subgroup problem"  ''Math. Notes'' , '''5'''  (1969)  pp. 141–148  ''Mat. Zametki'' , '''5'''  (1969)  pp. 233–244</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Suslin,   "Stability in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706056.png" />-theory"  R.K. Dennis (ed.) , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706057.png" />-theory (Oberwolfach, 1980)'' , ''Lect. notes in math.'' , '''966''' , Springer  (1982)  pp. 304–333</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Quillen,   "Higher algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706058.png" />-theory I"  H. Bass (ed.) , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706059.png" />-theory I (Battelle Inst. Conf.)'' , ''Lect. notes in math.'' , '''341''' , Springer  (1973)  pp. 85–147</TD></TR></table>