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Stable rank

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Let be an associative ring with unit element. A sequence of elements ( a _ {1} \dots a _ {n} ) is called left unimodular if the left ideal generated by the a _ {i} , i= 1 \dots n , is all of R .

The left stable rank of R is the least integer n such that for each m> n and left unimodular sequence ( a _ {1} \dots a _ {m} ) there are r _ {1} \dots r _ {m-} 1 such that ( a _ {1} ^ \prime \dots a _ {m-} 1 ^ \prime ) with a _ {i} ^ \prime = a _ {i} + r _ {i} a _ {m} , i= 1 \dots m- 1 , is also left unimodular.

The right stable rank of R 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 R , denoted by \textrm{ st.r. } ( R) .

By writing a left unimodular sequence ( a _ {1} \dots a _ {n} ) as a column, there is a natural left action of \mathop{\rm GL} _ {n} ( R ) on U _ {c} ( n, R ) , the set of all left unimodular sequences of length n . The general linear rank, \mathop{\rm glr} ( R) , of R is the least integer n such that \mathop{\rm GL} _ {m} ( R) acts transitively on U _ {c} ( m, R) for all m> n . This is equivalent to the property that all right stably-free modules of rank \geq n are free, [a2].

Recall that P is stably free if P\oplus R ^ {n} \simeq R ^ {m} for some n, m ; the rank of P is then defined as m- n . This is well-defined if R has the invariant basis property (i.e. R ^ {n} \simeq R ^ {m} if and only if n= m ). This property holds, e.g., if R is commutative or right Noetherian.

One has \mathop{\rm glr} ( R) \leq \textrm{ st.r. } ( R) , so that any stably-free module of rank \geq \textrm{ st.r. } ( R) is free.

For a field k one has \mathop{\rm glr} ( k[ X _ {1} \dots X _ {n} ])= 1 for all n .

Let k be a field of transcendence degree t over its prime subfield k _ {0} . The Kronecker dimension of k is then defined as t+ 1 if \mathop{\rm char} k= 0 and as t otherwise. For n\leq Kronecker dimension of k , \textrm{ st.r. } ( k[ X _ {1} \dots X _ {n} ]) = n+ 1 . If R is commutative of Krull dimension m< \infty ( cf. also Dimension of an associative ring), then \textrm{ st.r. } ( R[ X _ {1} \dots X _ {n} ]) \leq m+ n+ 1 ( Bass' theorem).

Let X be a topological space, Y a metric space and f: X \rightarrow Y a continuous mapping. A point y \in Y is a stable value of f if it is in f( X) and if there is an \epsilon such that for every continuous mapping g: X \rightarrow Y with \| f( x)- g( x) \| < \epsilon for all x \in X it is still true that y \in g( X) . The mapping dimension of a topological space X , d( X) , is the largest integer d for which there exists a mapping X \rightarrow \mathbf R ^ {d} for which the origin is a stable value. (If no such d exists, d( X) is set equal to \infty .) For nice spaces, e.g., metrizable, separable, X , 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 C( X) be the ring of real-valued continuous functions on a topological space X and C _ {b} ( X) \subset C( X) the subring of bounded functions. Then \textrm{ st.r. } ( C( X)) = \textrm{ st.r. } ( C _ {b} ( X))= d( X)+ 1 ( Vaserstein's theorem).

Both Bass' and Vaserstein's theorem indicate that \textrm{ st.r. } ( R) - 1 is a good dimension concept for rings.

More generally, the stable rank is defined for subrings and ideals of an associative ring R with unit.

Let R be an associative ring with unit, and let \mathfrak q be a subring (possibly without unit) of R . A sequence of elements ( a _ {1} \dots a _ {n} ) is left \mathfrak q - unimodular if it is left unimodular (in R ) and, moreover, a _ {1} - 1 \in \mathfrak q , a _ {i} \in \mathfrak q , i= 2 \dots n . The stable rank of the subring \mathfrak q is the least number n such that for each left \mathfrak q - unimodular sequence ( a _ {1} \dots a _ {m} ) of length m> n there are q _ {i} \in \mathfrak q , i= 1 \dots m- 1 , such that ( a _ {1} ^ \prime \dots a _ {m-} 1 ^ \prime ) , with a _ {i} ^ \prime = a _ {i} + q _ {i} a _ {m} , is a left \mathfrak q - unimodular sequence of length m- 1 . (Such a property is referred to as a stable range condition, cf., e.g., [a4]). The stable rank of \mathfrak q does not depend on the ambient ring R . 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 -theory" , Springer (1981) pp. §4.1 MR1007302 MR0842441
[a4] H. Bass, "Algebraic -theory" , Benjamin (1968) pp. Chapt. V, §3 MR249491
[a5] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
How to Cite This Entry:
Stable rank. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_rank&oldid=48799