Stable range condition

In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring $R$ is the smallest integer $n$ such that whenever $v_{0},v_{1},...,v_{n}$ in $R$ generate the unit ideal (they form a unimodular row), there exist some $t_{1},...,t_{n}$in $R$ such that the elements $v_{i}-v_{0}t_{i}$ for $1\leq i\leq n$ also generate the unit ideal.

If $R$ is a commutative Noetherian ring of Krull dimension $d$ , then the stable range of $R$ is at most $d+1$ (a theorem of Bass).

Bass stable range

The Bass stable range condition $SR_{m}$ refers to precisely the same notion, but for historical reasons it is indexed differently: a ring $R$ satisfies $SR_{m}$ if for any $v_{1},...,v_{m}$ in $R$ generating the unit ideal there exist $t_{2},...,t_{m}$ in $R$ such that $v_{i}-v_{1}t_{i}$ for $2\leq i\leq m$ generate the unit ideal.

Comparing with the above definition, a ring with stable range $n$ satisfies $SR_{n+1}$. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension $d$ satisfies $SR_{d+2}$. (For this reason, one often finds hypotheses phrased as "Suppose that $R$ satisfies Bass's stable range condition $SR_{d+2}$...")

Stable range relative to an ideal

Less commonly, one has the notion of the stable range of an ideal $I$ in a ring $R$. The stable range of the pair $(R,I)$ is the smallest integer $n$ such that for any elements $v_{0},...,v_{n}$ in $R$ that generate the unit ideal and satisfy $v_{n}\equiv 1$ mod $I$ and $v_{i}\equiv 0$ mod $I$ for $0\leq i\leq n-1$, there exist $t_{1},...,t_{n}$ in $R$ such that $v_{i}-v_{0}t_{i}$ for $1\leq i\leq n$ also generate the unit ideal. As above, in this case we say that $(R,I)$ satisfies the Bass stable range condition $SR_{n+1}$.

By definition, the stable range of $(R,I)$ is always less than or equal to the stable range of $R$.

References

• Charles Weibel, The K-book: An introduction to algebraic K-theory

• H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011. [1]

External links

• Bass' stable range condition for principal ideal domains