Stable range condition

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

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

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

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

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

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

• 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]

• Bass' stable range condition for principal ideal domains