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Zap. Nauchn. Sem. POMI, 2016, Volume 452, Pages 5–31 (Mi znsl6354)

Local-global principle for general quadratic and general Hermitian groups and the nilpotence of $\mathrm{KH}_1$
R. Basu

References

1. E. Abe, “Chevalley groups over local rings”, Tôhoku Math. J. (2), 21 (1969), 474–494  crossref  mathscinet  zmath
2. A. Bak, $\mathrm K$-Theory of forms, Annals of Mathematics Studies, 98, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981  mathscinet
3. A. Bak, “Nonabelian $\mathrm K$-theory: the nilpotent class of $\mathrm K_1$ and general stability”, $\mathrm K$-Theory, 4:4 (1991), 363–397  crossref  mathscinet  zmath
4. A. Bak, G. Tang, “Stability for Hermitian $\mathrm K_1$”, J. Pure Appl. Algebra, 150 (2000), 107–121  crossref  mathscinet  zmath  isi
5. A. Bak, V. Petrov, G. Tang, “Stability for Quadratic $\mathrm K_1$”, $\mathrm K$-Theory, 29 (2003), 1–11  crossref  mathscinet
6. A. Bak, R. Basu, R. A. Rao, “Local-global principle for transvection groups”, Proc. Amer. Math. Soc., 138:4 (2010), 1191–1204  crossref  mathscinet  zmath  isi  elib
7. A. Bak, R. Hazrat, N. Vavilov, “Localization-completion strikes again: Relative $\mathrm K_1$ is nilpotent-by-abelian”, J. Pure Appl. Algebra, 213 (2009), 1075–1085  crossref  mathscinet  zmath  isi  elib
8. A. Bak, N. Vavilov, “Structure of hyperbolic unitary groups I, elementary subgroups”, Alg. Colloquium, 7:2 (2000), 159–196  crossref  mathscinet  zmath
9. H. Bass, Algebraic K-Theory, Benjamin, New York–Amsterdam, 1968  mathscinet  zmath
10. H. Bass, “Unitary algebraic $\mathrm K$-theory”, Algebraic K-theory, Proc. Conf. (Battelle Memorial Inst., Seattle, Wash., 1972), v. III, Lecture Notes in Mathematics, 343, Hermitian $\mathrm K$-theory and geometric applications, Springer, Berlin, 1973, 57–265  crossref  mathscinet
11. H. Bass, “Quadratic modules over polynomial rings”, Contribution to Algebra, Collection of papers dedicated to Ellis Kolchin, Academic Press, N.Y., 1977, 1–23  crossref  mathscinet
12. R. Basu, R. A. Rao, R. Khanna, “On Quillen's local-global principle”, Commutative Algebra and Algebraic Geometry (Bangalore, India, 2003), Contemp. Math., 390, AMS, Providence, RI, 2005, 17–30  crossref  mathscinet  zmath
13. A. J. Berrick, M. E. Keating, An Introduction to Rings and Modules with K-theory in view, Cambridge Studies Adv. Math., 65, Cambridge Univ. Press, 2000  mathscinet  zmath
14. P. Chattopadhyay, R. A. Rao, “Elementary symplectic orbits and improved $\mathrm K_1$-stability”, J. K-Theory, 7:2 (2011), 389–403  crossref  mathscinet  zmath  isi  elib
15. J. Fasel, R. A. Rao, R. G. Swan, “On stably free modules over affine algebras”, Publ. Math. Inst. Hautes Études Sci., 116 (2012), 223–243  crossref  mathscinet  zmath
16. Fu An Li, “The structure of orthogonal groups over arbitrary commutative rings”, Chinese Ann. Math. Ser. B, 10 (1989), 341–350  mathscinet  zmath  isi
17. R. Hazrat, “Dimension theory and nonstable $\mathrm K_1$ of quadratic modules”, $\mathrm K$-Theory, 27:4 (2002), 293–328  crossref  mathscinet  zmath
18. R. Hazrat, N. Vavilov, “$\mathrm K_1$ of Chevalley groups are nilpotent”, J. Pure Appl. Algebra, 179:1–2 (2003), 99–116  crossref  mathscinet  zmath  isi
19. R. Hazrat, N. Vavilov, “Bak's work on $\mathrm K$-theory of rings. On the occasion of his 65th birthday”, J. $\mathrm K$-Theory, 4:1 (2009), 1–65  crossref  mathscinet  zmath
20. R. Hazrat, N. Vavilov, Z. Zhang, “Relative unitary commutator calculus, and applications”, J. Algebra, 343 (2011), 107–137  crossref  mathscinet  zmath  isi  elib
21. J. Math. Sci. (N.Y.), 179:6 (2011), 662–678  mathnet  crossref  mathscinet  zmath
22. A. J. Hahn, O. T. O'Meara, The Classical groups and $\mathrm K$-theory, With a foreword by J. Dieudonné, Grundlehren Math. Wiss., 291, Springer-Verlag, Berlin, 1989  crossref  mathscinet
23. N. Jacobson, Lectures on Quadratic Jordan algebras, Tata Istitute of Fundamental Research, Bombay, 1969  mathscinet  zmath
24. I. S. Klein, A. V. Mikhalev, “The Orthogonal Steinberg group over a ring with involution”, Algebra Logika, 9 (1970), 145–166  mathnet  mathscinet
25. I. S. Klein, A. V. Mikhalev, “The Unitary Steinberg group over a ring with involution”, Algebra Logika, 9 (1970), 510–519  mathnet  mathscinet
26. V. I. Kopeiko, “The stabilization of Symplectic groups over a polynomial ring”, Math. USSR. Sbornik, 34:5 (1978), 655–669  mathnet  crossref  mathscinet  zmath
27. K. McCrimmon, “A general theory of Jordan rings”, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1072–1079  crossref  mathscinet  zmath
28. R. Parimala, “Failure of Quadratic analog of Serre's Conjecture”, Bull. Amer. Math. Soc., 82 (1976), 962–964  crossref  mathscinet  zmath
29. R. Parimala, “Failure of Quadratic analog of Serre's Conjecture”, Amer. J. Math., 100 (1978), 913–924  crossref  mathscinet  zmath
30. V. A. Petrov, “Odd unitary groups”, J. Math. Sci. (N.Y.), 130:3 (2005), 4752–4766  mathnet  crossref  mathscinet  zmath
31. St. Petersburg Math. J., 20:4 (2009), 625–644  mathnet  crossref  mathscinet  zmath  isi
32. R. A. Rao, W. van der Kallen, “Improved stability for $\mathrm{SK}_1$ and $\mathrm{WMS}_d$ of a non-singular affine algebra”, $\mathrm K$-theory (Strasbourg, 1992), Astérisque, 226, 1994, 411–420  mathscinet  zmath
33. R. A. Rao, R. Basu, S. Jose, “Injective Stability for $\mathrm K_1$ of the Orthogonal group”, J. Algebra, 323 (2010), 393–396  crossref  mathscinet  zmath  isi
34. Sergei Sinchuk, “Injective stability for unitary $\mathrm K_1$, revisited”, J. K-Theory, 11:2 (2013), 233–242  mathscinet  zmath
35. A. A. Suslin, “On the structure of special Linear group over polynomial rings”, Math. USSR. Izv., 11:2 (1977), 221–238  mathnet  crossref  mathscinet  zmath
36. M. R. Stein, “Stability theorems for $\mathrm K_1$, $\mathrm K_2$ and related functors modeled on Chevalley groups”, Japan. J. Math. (N.S.), 4:1 (1978), 77–108  mathscinet  zmath
37. A. A. Suslin, V. I. Kopeiko, “Quadratic modules and Orthogonal groups over polynomial rings”, Zap. Nauchn. Sem. LOMI, 71, 1978, 216–250  mathnet  mathscinet  zmath
38. A. A. Suslin, L. N. Vaserstein, “Serre's problem on projective modules over polynomial rings, and algebraic $\mathrm K$-theory”, Izv. Akad SSSR. Ser. Mat., 40:5 (1976), 993–1054  mathnet  mathscinet  zmath
39. Guoping Tang, “Hermitian groups and $K$-theory”, $\mathrm K$-Theory, 13:3 (1998), 209–267  crossref  mathscinet  zmath
40. G. Taddei, “Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau”, Application of Algebraic $\mathrm K$-Theory to Algebraic Geometry and Number Theory, Part II (Boulder, Colo., 1983), Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986, 693–710  crossref  mathscinet
41. M. S. Tulenbaev, “Schur multiplier of a group of elementary matrices of finite order”, Zap. Nauchn. Semin. LOMI, 86, 1979, 162–169  mathnet  mathscinet  zmath
42. Math. USSR-Sbornik, 8:3 (1969), 383–400  mathnet  crossref  mathscinet  zmath
43. L. N. Vaserstein, “Stabilization of Unitary and Orthogonal Groups over a Ring with Involution”, Mat. Sbornik, 81(123):3 (1970), 328–351  mathnet  crossref  mathscinet  zmath
44. L. N. Vaserstein, “Stabilization for Classical groups over rings”, Mat. Sb. (N.S.), 93(135):2 (1974), 268–295, 327 (in Russian)  mathnet  mathscinet  zmath
45. L. N. Vaserstein, “On the normal subgroups of $\mathrm{GL}_n$ over a ring”, Algebraic $K$-theory, Proc. Conf. (Northwestern Univercisy, Evanston, Ill., 1980), Lecture Notes in Mathematics, 854, Springer, Berlin–New York, 1981, 456–465  crossref  mathscinet
46. Weibe Yu, “Stability for odd unitary $\mathrm K_1$ under the $\Lambda$-stable range condition”, J. Pure Appl. Algebra, 217 (2013), 886–891  crossref  mathscinet  isi
47. J. Wilson, “The normal and subnormal structure of general linear groups”, Proc. Camb. Phil. Soc., 71 (1972), 163–177  crossref  mathscinet  zmath


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