Relative Vorst theorem and relative Karoubi sequence yield improved injective stability bounds for relative K1 and K1Sp groups over regular rings.
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Proves that unimodular rows of length d over affine algebras R of dim d>=4 over algebraic closure of finite field, with 1/(d-1)! in R, can be mapped to factorial rows by elementary transformations.
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Improved injective stability for relative $\mathrm{K_1Sp}$-groups
Relative Vorst theorem and relative Karoubi sequence yield improved injective stability bounds for relative K1 and K1Sp groups over regular rings.
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Unimodular rows over affine algebras over algebraic closure of a finite field
Proves that unimodular rows of length d over affine algebras R of dim d>=4 over algebraic closure of finite field, with 1/(d-1)! in R, can be mapped to factorial rows by elementary transformations.