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On symplectic groups over polynomial rings. (English) Zbl 0725.20038

Let A be a locally principal ring, which means that the localization \(A_{\mu}\) of A at every maximal A-ideal \(\mu\) is a principal ideal ring. The principal results of this paper are as follows. Let \(R=A[x_ 1,...,x_ m]\), where \(m\geq 0\). Then
(i) \(Sp_{2n}R=Sp_{2n}A\cdot Ep_{2n}R\), for all \(n\geq 2,\)
(ii) \(SL_ nR=SL_ nA\cdot E_ nR\), for all \(n\geq 3,\)
where \(Ep_{2n}R\) is the subgroup of the symplectic group \(Sp_{2n}R\) generated by the (symplectic) elementary matrices and \(E_ nR\) is the subgroup of \(SL_ nR\) generated by the elementary matrices (n\(\geq 2)\). It follows that \(K_ 1Sp(A)=K_ 1Sp(R)\) and that \(SK_ 1(A)=SK_ 1(R)\). These results extend many earlier results of Suslin, Kopejko, Bass and others. The proofs are based on an effective localization and patching technique which reduces the problem to the case where A is a local principal ideal ring. The proofs then make use of symplectic symbols of Mennicke type.
Using similar methods the authors also prove some results for the Laurent polynomial ring \(R'=A[x,x^{-1}]\). When A is a local (principal ideal) ring it is proved that (i) \(Sp_{2n}R'=Ep_{2n}R'\), (ii) \(SL_{n+1}R'=E_{n+1}R'\), for all \(n\geq 2\). Finally it is proved that when A is a principal ideal domain \(Sp_{2n}R'=Sp_{2n}A\cdot Ep_{2n}R'\), for all \(n\geq 2\). It follows that \(K_ 1Sp(A)=K_ 1Sp(R')\).

MSC:

20H25 Other matrix groups over rings
19B14 Stability for linear groups
13F10 Principal ideal rings
20G35 Linear algebraic groups over adèles and other rings and schemes
16S34 Group rings
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References:

[1] Abe, E.: Whitehead groups of Chevalley groups over polynomial rings. Commun. Algebra11, 1271–1307 (1983) · Zbl 0513.20030
[2] Atiyah, M., Macdonald, I.G.: Introduction to commutative algebra. Reading, M.A.: Addison-Wesley 1969 · Zbl 0175.03601
[3] Bachmuth, S., Mochizuki, H.Z.: IA-Automorphisms of the free metabelian group of rank 3. J. Algebra55, 106–115 (1978) · Zbl 0401.20033
[4] Bass, H.:K-theory and stable algebra. Publ. Math., Inst. Hautes Étud. Sci.22, 485–544 (1964) · Zbl 0248.18025
[5] Bass, H., Milnor, J., Serre, J.-P.: Solution of the congruence subgroup problem for SL n A(n) and Sp 2n A(n). Publ. Math. Inst. Hautes Étud. Sci.33, 59–137 (1967) Erratum: On a functorial property of power residue symbols, Publ. Math. Inst. Hautes Étud. Sci.44, 241–244 (1974) · Zbl 0174.05203
[6] Bass, H.: AlgebraicK-theory. Benjamin, New York, 1968 · Zbl 0174.30302
[7] Bass, H., Heller, A., Swan, R.G.: The Whitehead group of a polynomial extension, Publ. Math. Hautes Étud. Sci.22, 61–79 (1964) · Zbl 0248.18026
[8] Bourbaki, N.: Algebre commutative. Chap. 2, Localisation. Paris: Herman 1961
[9] Farber, M.Sh.: Exactness of the Novikov inequalities. Funk. An.19, 40–48 (1985) · Zbl 0603.58030
[10] Grunewald, F., Mennicke, J., Vaserstein, L.: On the groups SL2(\(\mathbb{Z}\)[x]) and SL2(K[x,y]). Preprint · Zbl 0805.20042
[11] Karoubi, M.: Périodicité de laK-théorie hermitienne. (Lecture Notes Math. vol. 343, 301–411) Berlin Heidelberg New York: Springer 1973
[12] Kopeiko, V.I.: Stabilization of symplectic groups over polynomial rings. Math. Sbornik106, 94–107 (1978)
[13] Lam, T.Y.: Serre’s problem. (Lecture Notes Math. vol. 635) Berlin Heidelberg New York: Springer 1978
[14] Mennicke, J.L.: Zur Theorie der Siegelschen Modulgruppe. Math. Ann.159, 115–129 (1965) · Zbl 0134.26502
[15] Serre, J.-P.: Trees. Berlin Heidelberg New York: Springer 1980
[16] Stein, M.: Stability theorems forK 1,K 2, and related functors modeled on Chevalley groups. Japan J. Math., New Ser.4, 77–108 (1978) · Zbl 0403.18010
[17] Suslin, A.: On the structure of the special linear group over polynomial rings. Izv. Akad. Nauk SSSR, Ser. Mat.41, 235–252 (1977) (Translated as: Math. USSR, Izv.11, 221–238 (1977) · Zbl 0354.13009
[18] Vaserstein, L.N.: Stabilization for unitary and orthogonal groups over a ring with involution. Mat. Sbornik81, 328–351 (1970) (Translated as: Math. USSSR, Sb.10, 307–326 (1970)
[19] Vaserstein, L.N.: Stabilization for classical groups over rings. Mat. Sbornik93, 268–295 (1974) (Translated as: Math. USSR, Sb.22, 271–303 (1974)
[20] Vaserstein, L.N.: On the normal subgroups of GL n over a ring. (Lecture Notes Math. vol. 854 pp. 454–465 Berlin Heidelberg New York: Springer 1981 · Zbl 0464.20030
[21] Vaserstein, L.N., Suslin, A.A.: Serre’s problem on projective modules over polynomial rings and algebraicK-theory. Izv. Akad. Nauk SSSR Ser. Mat.40, 993–1054 (1976) (Translated as Math. USSR, Izv.5, 931–1001 (1971) · Zbl 0338.13015
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