Difference between revisions of "Stable rank"
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By writing a left unimodular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713017.png" /> as a column, there is a natural left action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713019.png" />, the set of all left unimodular sequences of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713020.png" />. The general linear rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713021.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713022.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713024.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713026.png" />. This is equivalent to the property that all right stably-free modules of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713027.png" /> are free, [[#References|[a2]]]. | By writing a left unimodular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713017.png" /> as a column, there is a natural left action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713019.png" />, the set of all left unimodular sequences of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713020.png" />. The general linear rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713021.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713022.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713024.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713026.png" />. This is equivalent to the property that all right stably-free modules of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713027.png" /> are free, [[#References|[a2]]]. | ||
− | Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713028.png" /> is stably free if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713029.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713030.png" />; the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713031.png" /> is then defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713032.png" />. This is well-defined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713033.png" /> has the invariant basis property (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713034.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713035.png" />). This property holds, e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713036.png" /> is commutative or right Noetherian. | + | Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713028.png" /> is [[Stably free module|stably free]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713029.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713030.png" />; the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713031.png" /> is then defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713032.png" />. This is well-defined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713033.png" /> has the invariant basis property (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713034.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713035.png" />). This property holds, e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713036.png" /> is commutative or right Noetherian. |
One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713037.png" />, so that any stably-free module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713038.png" /> is free. | One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713037.png" />, so that any stably-free module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713038.png" /> is free. |
Revision as of 18:37, 10 September 2013
Let be an associative ring with unit element. A sequence of elements
is called left unimodular if the left ideal generated by the
,
, is all of
.
The left stable rank of is the least integer
such that for each
and left unimodular sequence
there are
such that
with
,
, is also left unimodular.
The right stable rank of is defined analogously by replacing left with right everywhere. The left and right stable ranks are equal [a1], cf. also, e.g., [a2], §11.3, and both are therefore called the stable rank of
, denoted by
.
By writing a left unimodular sequence as a column, there is a natural left action of
on
, the set of all left unimodular sequences of length
. The general linear rank,
, of
is the least integer
such that
acts transitively on
for all
. This is equivalent to the property that all right stably-free modules of rank
are free, [a2].
Recall that is stably free if
for some
; the rank of
is then defined as
. This is well-defined if
has the invariant basis property (i.e.
if and only if
). This property holds, e.g., if
is commutative or right Noetherian.
One has , so that any stably-free module of rank
is free.
For a field one has
for all
.
Let be a field of transcendence degree
over its prime subfield
. The Kronecker dimension of
is then defined as
if
and as
otherwise. For
Kronecker dimension of
,
. If
is commutative of Krull dimension
(cf. also Dimension of an associative ring), then
(Bass' theorem).
Let be a topological space,
a metric space and
a continuous mapping. A point
is a stable value of
if it is in
and if there is an
such that for every continuous mapping
with
for all
it is still true that
. The mapping dimension of a topological space
,
, is the largest integer
for which there exists a mapping
for which the origin is a stable value. (If no such
exists,
is set equal to
.) For nice spaces, e.g., metrizable, separable,
, this concept of dimension coincides with other notions of dimension, such as inductive dimension, [a5], Chapt. VI, §1 (cf. Dimension theory). It always coincides with the notion of dimension defined by essential mappings (cf. Dimension theory), [a5], Chapt. VI, §3.
Let be the ring of real-valued continuous functions on a topological space
and
the subring of bounded functions. Then
(Vaserstein's theorem).
Both Bass' and Vaserstein's theorem indicate that is a good dimension concept for rings.
More generally, the stable rank is defined for subrings and ideals of an associative ring with unit.
Let be an associative ring with unit, and let
be a subring (possibly without unit) of
. A sequence of elements
is left
-unimodular if it is left unimodular (in
) and, moreover,
,
,
. The stable rank of the subring
is the least number
such that for each left
-unimodular sequence
of length
there are
,
, such that
, with
, is a left
-unimodular sequence of length
. (Such a property is referred to as a stable range condition, cf., e.g., [a4]). The stable rank of
does not depend on the ambient ring
. Again it is true that the notion of stable rank is left/right symmetric, [a1].
References
[a1] | L.N. Vaserstein, "Stable ranks of rings and dimensionality of topological spaces" Funct. Anal. Appl. , 5 (1971) pp. 102–110 Funkts. Anal. i Prilozhen. , 5 : 2 (1970) pp. 17–27 |
[a2] | J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) MR934572 |
[a3] | A.J. Hahn, O.T. O'Meara, "The classical groups and ![]() |
[a4] | H. Bass, "Algebraic ![]() |
[a5] | W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) |
Stable rank. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_rank&oldid=30517