Abstract
A major conjecture in classical K-theory is the Bass–Suslin conjecture. This asks if a unimodular row of length \((r + 1)\) over a polynomial extension of a local ring R is completable to an invertible matrix if \(1/r! \in R\). We give two approaches to this problem.
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References
H. Bass, Some Problems in “Classical” Algebraic \(K\)-Theory. Algebraic \(K\)-Theory, II: “Classical” Algebraic \(K\)-Theory and Connections with Arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 342 (Springer, Berlin, 1973), pp. 3–73
Selby Jose, Ravi A. Rao, A structure theorem for the elementary unimodular vector group. Trans. Am. Math. Soc. 358(7), 3097–3112 (2006)
S. Jose, R.A. Rao, A fundamental property of Suslin matrices. J. \(K\)-Theory 5(3), 407–436 (2010)
S. Jose, R.A. Rao, A Witt group structure on Suslin matrices, in preparation
W. van der Kallen, A module structure on certain orbit sets of unimodular rows. J. Pure Appl. Algebr. 57(3), 281–316 (1989)
S.M. Bhatwadekar, R.A. Rao, On a question of Quillen. Trans. Am. Math. Soc. 279(2), 801–810 (1983)
H. Lindel, On the Bass–Quillen conjecture concerning projective modules over polynomial rings. Invent. Math. 65(2), 319–323 (1981/82)
D. Popescu, General Néron desingularization. Nagoya Math. J. 100, 97–126 (1985)
D. Quillen, Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976)
R.A. Rao, The Bass–Quillen conjecture in dimension three but characteristic \(\ne 2\),\(3\) via a question of A. Suslin. Invent. Math. 93(3), 609–618 (1988)
R.A. Rao, On completing unimodular polynomial vectors of length three. Trans. Am. Math. Soc. 325(1), 231–239 (1991)
M. Roitman, On stably extended projective modules over polynomial rings. Proc. Am. Math. Soc. 97(4), 585–589 (1986)
A.A. Suslin, L.N. Vaserstein, Serre’s problem on projective modules over polynomial rings and algebraic K-theory. Math. USSR Izvestija 10, 937–1001 (1976)
A.A. Suslin, Stably free modules. (Russian) Mat. Sb. (N.S.) 102 (144)(4), 537–550 (1977)
A.A. Suslin, The structure of the special linear group over rings of polynomials. Izv. Akad. Nauk SSSR Ser. Mat. 41, 235–252 (1977)
R.G. Swan, R.A. Rao, J. Fasel, A regenerative property of a fibre of invertible alternating matrices. see Excerpts on home page of R.G. Swan at http://math.uchicago.edu/~swan/
Acknowledgements
The second author thank the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India, for the funding of project MTR/2017/000875 under Mathematical Research Impact Centric Support (MATRICS).
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Rao, R.A., Jose, S. (2020). Two Approaches to the Bass–Suslin Conjecture. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_11
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DOI: https://doi.org/10.1007/978-981-15-1611-5_11
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