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Surjective stability for the \(K_ 1\)-functor for some exponential Chevalley groups. (English. Russian original) Zbl 0802.19003

J. Sov. Math. 64, No. 1, 751-766 (1993); translation from Zap. Nauchn. Semin. POMI 198, 65-88 (1991).
Let \(R\) be a commutative ring with 1, \(\Phi\) an irreducible root system, \(G(\Phi)\) the Chevalley-Demazure group scheme, \(E(\Phi,R)\) the subgroup of \(G(\Phi,R)\) generated by all root elements. For several inclusions \(\Delta\subset \Phi\) of root systems, M. Stein proved that \(G(\Phi, R)= E(\Phi, R)G(\Delta, R)\) under certain conditions on \(R\). Also using basic representations, weight diagrams, and appropriate absolute stable rank conditions on \(R\), the author obtains this equality (referred to as the surjective stability in the title) for other inclusions.

MSC:

19G05 Stability for quadratic modules
20G30 Linear algebraic groups over global fields and their integers
19B14 Stability for linear groups
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References:

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