词条 | Stable range condition |
释义 |
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 v0,v1, ... , vn in R generate the unit ideal (they form a unimodular row), there exist some t1, ... , tn in R such that the elements vi - v0ti for 1 ≤ i ≤ 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 rangeThe Bass stable range condition SRm refers to precisely the same notion, but for historical reasons it is indexed differently: a ring R satisfies SRm if for any v1, ... , vm in R generating the unit ideal there exist t2, ... , tm in R such that vi - v1ti for 2 ≤ i ≤ m generate the unit ideal. Comparing with the above definition, a ring with stable range n satisfies SRn+1. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension d satisfies SRd+2. (For this reason, one often finds hypotheses phrased as "Suppose that R satisfies Bass's stable range condition SRd+2...") Stable range relative to an idealLess 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 v0, ... , vn in R that generate the unit ideal and satisfy vn {{math|≡}} 1 mod I and vi {{math|≡}} 0 mod I for 0 ≤ i ≤ n-1, there exist t1, ... , tn in R such that vi - v0ti for 1 ≤ i ≤ n also generate the unit ideal. As above, in this case we say that (R,I) satisfies the Bass stable range condition SRn+1. By definition, the stable range of (R,I) is always less than or equal to the stable range of R. References
H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011. External links
1 : K-theory |
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