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Some remarks on symplectic injective stability. (English) Zbl 1223.19001

Let \(R\) be a commutative ring with \(1\) of Krull dimension \(d\) and let \(E_n(R)\) denotes the elementary subgroup of \(\text{GL}_n(R)\). It was conjectured by H. Bass, J. W. Milnor and J.-P. Serre [Publ. Math., Inst. Hautes Étud. Sci. 33, 59–137 (1967; Zbl 0174.05203)] and proved by L. N. Vaserstein [Math. USSR, Sb. 8, 383–400 (1969); translation from Mat. Sb., n. Ser. 79(121), 405–424 (1969; Zbl 0238.20057)] that the sequence of quotient groups \(\text{GL}_n(R)/E_n(R)\) stabilizes for \(n\geq d+2\). In [Math. USSR, Sb. 10, 307–326 (1970); translation from Mat. Sb., n. Ser. 81(123), 328–351 (1970; Zbl 0253.20066)] L. N. Vaserstein extended his result to the even orthogonal, symplectic and unitary groups with the stabilization bound given by \(2n\geq 2d+4\). In [R. Basu and R. A. Rao, J. Algebra 323, No. 4, 867–877 (2010; Zbl 1185.19001)], it was shown that for \(\text{Sp}_{2n}(R)\), where \(R\) is a non-singular affine algebra over a perfect \(C_1\), the quotients \(\text{Sp}_{2n}(R)/E\text{Sp}_{2n}(R)\) stabilize already for \(2n\geq d+1\).
In the paper under review it is shown that if \(A\) is an affine algebra of odd dimension \(d\) over an infinite field of cohomological dimension at most one, with \((d + 1)!A = A\), and with \(4|(d-1)\), then \(\text{Um}_{d+1}(A) = e_1\text{Sp}_{d+1}(A)\). As a consequence it is shown that if \(A\) is a non-singular affine algebra of dimension \(d\) over an infinite field of cohomological dimension at most one, and \(d!A = A\), and \(4| d\), then \(\text{Sp}_d(A)\cap \text{ESp}_{d+2}(A) = \text{ESp}_d(A)\).

MSC:

19B14 Stability for linear groups
13C10 Projective and free modules and ideals in commutative rings
13H05 Regular local rings
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
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