A group structure on squares

Rao, Ravi; Jose, Selby

doi:10.1090/S0002-9939-07-09065-X

A group structure on squares
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by Ravi A. Rao and Selby Jose PDF

Proc. Amer. Math. Soc. 136 (2008), 1181-1191 Request permission

Abstract:

We show that there is an abelian group structure on the orbit set of “squares” of unimodular rows of length $n$ over a commutative ring of stable dimension $d$ when $d = 2n - 3$, $n$ odd and also an abelian group structure on the orbit set of “fourth powers” of unimodular rows of length $n$ over a commutative ring of stable dimension $d$ when $d = 2n - 3$, $n$ even.

References

J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632. MR 139178, DOI 10.2307/1970213
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
S. M. Bhatwadekar and Raja Sridharan, The Euler class group of a Noetherian ring, Compositio Math. 122 (2000), no. 2, 183–222. MR 1775418, DOI 10.1023/A:1001872132498
S. M. Bhatwadekar and Raja Sridharan, On Euler classes and stably free projective modules, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 139–158. MR 1940666
A.S. Gadre and R.A. Rao, A nice group structure on the orbit space of unimodular rows, (preprint 2006), accepted and to appear in K-theory, January 2008.
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 and Ravi A. Rao, A local global principle for the elementary unimodular vector group, Commutative algebra and algebraic geometry, Contemp. Math., vol. 390, Amer. Math. Soc., Providence, RI, 2005, pp. 119–125. MR 2187329, DOI 10.1090/conm/390/07298
S. Jose and R.A. Rao, A Fundamental Property of Suslin matrices, preprint 2007.
Ravi A. Rao, The Bass-Quillen conjecture in dimension three but characteristic $\not =2,3$ via a question of A. Suslin, Invent. Math. 93 (1988), no. 3, 609–618. MR 952284, DOI 10.1007/BF01410201
Ravi A. Rao, On completing unimodular polynomial vectors of length three, Trans. Amer. Math. Soc. 325 (1991), no. 1, 231–239. MR 991967, DOI 10.1090/S0002-9947-1991-0991967-0
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
Ravi A. Rao, An abelian group structure on orbits of “unimodular squares” in dimension $3$, J. Algebra 210 (1998), no. 1, 216–224. MR 1656421, DOI 10.1006/jabr.1998.7543
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
A. A. Suslin, Stably free modules, Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537–550, 632 (Russian). MR 0441949
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
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
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
L. N. Vaseršteĭn, The stable range of rings and the dimension of topological spaces, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 17–27 (Russian). MR 0284476
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
Wilberd van der Kallen, From Mennicke symbols to Euler class groups, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 341–354. MR 1940672