Some remarks on symplectic injective stability
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- by Rabeya Basu, Pratyusha Chattopadhyay and Ravi A. Rao PDF
- Proc. Amer. Math. Soc. 139 (2011), 2317-2325 Request permission
Abstract:
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 Um$_{d+1}(A) = e_1\textrm {Sp}_{d+1}(A)$. 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 $\textrm {Sp}_d(A) \cap \textrm {ESp}_{d+2}(A) = \textrm {ESp}_d(A)$. This result is a partial analogue for even-dimensional algebras of the one obtained by Basu and Rao for odd-dimensional algebras earlier.References
- Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
- H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\textrm {SL}_{n}\,(n\geq 3)$ and $\textrm {Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 244257
- H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
- Rabeya Basu and Ravi A. Rao, Injective stability for $K_1$ of classical modules, J. Algebra 323 (2010), no. 4, 867–877. MR 2578583, DOI 10.1016/j.jalgebra.2009.12.012
- Ravi A. Rao, Rabeya Basu, and Selby Jose, Injective stability for $K_1$ of the orthogonal group, J. Algebra 323 (2010), no. 2, 393–396. MR 2564846, DOI 10.1016/j.jalgebra.2009.09.022
- P. Chattopadhyay, R.A. Rao, Elementary symplectic orbits and improved $K_1$-stability, to appear in Journal of $K$-theory, doi:10.1017/is010002021jkt109, published online by Cambridge University Press, 10 June 2010.
- Anuradha S. Garge and Ravi A. Rao, A nice group structure on the orbit space of unimodular rows, $K$-Theory 38 (2008), no. 2, 113–133. MR 2366558, DOI 10.1007/s10977-007-9011-4
- Jean Fasel, Stably free modules over smooth affine threefolds, arXiv:0911.3495v2 [math.AC], to appear in Duke Mathematical Journal.
- Selby Jose and Ravi A. Rao, A structure theorem for the elementary unimodular vector group, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3097–3112. MR 2216260, DOI 10.1090/S0002-9947-05-03794-3
- Selby Jose, Ravi A. Rao, A fundamental property of Suslin matrices, Journal of $K$-theory: $K$-theory and its Applications to Algebra, Geometry, and Topology 5 (2010), no. 3, 407–436.
- Ravi A. Rao and Selby Jose, A group structure on squares, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1181–1191. MR 2367092, DOI 10.1090/S0002-9939-07-09065-X
- Wilberd van der Kallen, A group structure on certain orbit sets of unimodular rows, J. Algebra 82 (1983), no. 2, 363–397. MR 704762, DOI 10.1016/0021-8693(83)90158-8
- Wilberd van der Kallen, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra 57 (1989), no. 3, 281–316. MR 987316, DOI 10.1016/0022-4049(89)90035-2
- Ravi A. Rao and Wilberd van der Kallen, Improved stability for $SK_1$ and $WMS_d$ of a non-singular affine algebra, Astérisque 226 (1994), 11, 411–420. $K$-theory (Strasbourg, 1992). MR 1317126
- Ravi A. Rao, A stably elementary homotopy, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3637–3645. MR 2529870, DOI 10.1090/S0002-9939-09-09949-3
- Ravi A. Rao, Rabeya Basu, and Selby Jose, Injective stability for $K_1$ of the orthogonal group, J. Algebra 323 (2010), no. 2, 393–396. MR 2564846, DOI 10.1016/j.jalgebra.2009.09.022
- L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245
- A. A. Suslin, Stably free modules, Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537–550, 632 (Russian). MR 0441949
- A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477 (Russian). MR 0472792
- A. A. Suslin, Mennicke symbols and their applications in the $K$-theory of fields, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 334–356. MR 689382
- A. A. Suslin, The cancellation problem for projective modules, and related questions, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 323–330 (Russian). MR 562623
- A. A. Suslin, Cancellation for affine varieties, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 187–195, 222 (Russian). Modules and algebraic groups. MR 669571
- L. N. Vaseršteĭn, On the stabilization of the general linear group over a ring, Math. USSR-Sb. 8 (1969), 383–400. MR 0267009
- L. N. Vaseršteĭn, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N.S.) 81 (123) (1970), 328–351 (Russian). MR 0269722
- Leonid N. Vaserstein, Operations on orbits of unimodular vectors, J. Algebra 100 (1986), no. 2, 456–461. MR 840588, DOI 10.1016/0021-8693(86)90088-8
Additional Information
- Rabeya Basu
- Affiliation: Indian Institute of Science Education and Research, Kolkata 741 252, India
- Email: rbasu@iiserkol.ac.in
- Pratyusha Chattopadhyay
- Affiliation: Institute of Mathematical Sciences, Chennai 600 113, India
- Email: pratyusha@imsc.res.in
- Ravi A. Rao
- Affiliation: Tata Institute of Fundamental Research, Mumbai 400 005, India
- Email: ravi@math.tifr.res.in
- Received by editor(s): January 9, 2010
- Received by editor(s) in revised form: January 12, 2010, June 9, 2010, and June 18, 2010
- Published electronically: December 16, 2010
- Communicated by: Martin Lorenz
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2317-2325
- MSC (2000): Primary 13C10, 13H05, 19B14, 19B99, 55R50
- DOI: https://doi.org/10.1090/S0002-9939-2010-10654-8
- MathSciNet review: 2784796