Stable rank

Let $ R $ 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)