A stably elementary homotopy

Rao, Ravi

doi:10.1090/S0002-9939-09-09949-3

A stably elementary homotopy
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Proc. Amer. Math. Soc. 137 (2009), 3637-3645 Request permission

Abstract:

If $R$ is an affine algebra of dimension $d$ over a perfect C$_1$ field and $\sigma \in SL_{d + 1}(R)$ is a stably elementary matrix, we show that there is a stably elementary matrix $\sigma (X) \in SL_{d + 1}(R[X])$ with $\sigma (1) = \sigma$ and $\sigma (0) = I_{d + 1}$.

References

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
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
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
Hartmut Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981/82), no. 2, 319–323. MR 641133, DOI 10.1007/BF01389017
N. Mohan Kumar and M. Pavaman Murthy, Algebraic cycles and vector bundles over affine three-folds, Ann. of Math. (2) 116 (1982), no. 3, 579–591. MR 678482, DOI 10.2307/2007024
Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171. MR 427303, DOI 10.1007/BF01390008
Ravi A. Rao, An elementary transformation of a special unimodular vector to its top coefficient vector, Proc. Amer. Math. Soc. 93 (1985), no. 1, 21–24. MR 766519, DOI 10.1090/S0002-9939-1985-0766519-0
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
Moshe Roitman, On unimodular rows, Proc. Amer. Math. Soc. 95 (1985), no. 2, 184–188. MR 801320, DOI 10.1090/S0002-9939-1985-0801320-0
Moshe Roitman, On stably extended projective modules over polynomial rings, Proc. Amer. Math. Soc. 97 (1986), no. 4, 585–589. MR 845969, DOI 10.1090/S0002-9939-1986-0845969-9
Jean-Pierre Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin-New York, 1973. Cours au Collège de France, Paris, 1962–1963; Avec des textes inédits de J. Tate et de Jean-Louis Verdier; Quatrième édition. MR 0404227
A. A. Suslin, Stably free modules, Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537–550, 632 (Russian). MR 0441949
A.A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izv. 11, 221–238, 1977.
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
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
R.G. Swan, Some stably free modules which are not self dual. See unpublished papers on homepage of R.G. Swan at http://www.math.uchicago.edu/.
Richard G. Swan, A cancellation theorem for projective modules in the metastable range, Invent. Math. 27 (1974), 23–43. MR 376681, DOI 10.1007/BF01389963
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
Ton Vorst, The general linear group of polynomial rings over regular rings, Comm. Algebra 9 (1981), no. 5, 499–509. MR 606650, DOI 10.1080/00927878108822596