Difference between revisions of "Stability theorems in algebraic K-theory"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
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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]]. | 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 | + | 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]]]. | 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==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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 {{MR|0174604}} {{ZBL|}} </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 {{MR|246941}} {{ZBL|}} </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 {{MR|}} {{ZBL|}} </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 {{MR|338129}} {{ZBL|}} </TD></TR></table> |
Revision as of 14:52, 24 March 2012
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 MR0174605 Zbl 0248.18026 |
[2] | V.P. Platonov, "Reduced ![]() |
[3] | V.P. Platonov, V.I. Yanchevskii, "![]() |
[4] | V.I. Yanchevskii, "Reduced unitary ![]() |
[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 MR1044048 |
[6] | H. Bass, "Algebraic ![]() |
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, "![]() |
[a2] | L.N. Vaserstein, "![]() |
[a3] | A. Suslin, "Stability in algebraic ![]() ![]() |
[a4] | D. Quillen, "Higher algebraic ![]() ![]() |
Stability theorems in algebraic K-theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stability_theorems_in_algebraic_K-theory&oldid=21943