Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T17:54:23.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Bak's work on the K-theory of rings

Published online by Cambridge University Press:  03 September 2009

Roozbeh Hazrat  and
Nikolai Vavilov
Roozbeh Hazrat
Affiliation:
Dept. of Pure Mathematics, Queen's University, Belfast BT7 1NN, United Kingdom, [email protected].
Nikolai Vavilov
Affiliation:
Dept. of Mathematics and Mechanics, Saint-Petersburg State University, Saint-Petersburg 198904, Russia, [email protected].
Get access

Abstract

This paper studies the work of Bak in algebra and (lower) algebraic K-theory and some later developments stimulated by them. We present an overview of his work in these areas, describe the setup and problems as well as the methods he introduced to attack these problems and state some of the crucial theorems. The aim is to analyse in detail some of his methods which are important and promising for further work in the subject. Among the topics covered are, unitary/general quadratic groups over form rings, structure theory and stability for such groups, quadratic K2 and the quadratic Steinberg groups, nonstable K-theory and localisation-completion, intermediate subgroups, congruence subgroup problem, dimension theory and surgery theory.

Keywords

form parameterclassical-like groupK-theory of formsnonstable K-theorydimension theorysurgery theory
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 1 , August 2009 , pp. 1 - 65
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aravire, Roberto; Baeza, Ricardo, Milnor's K-theory and quadratic forms over fields of characteristic two. Comm. Algebra 20 (1992), no. 4, 1087–1107.CrossRefGoogle Scholar
2.Artin, Emil, Geometric Algebra. Wiley Classics Library, 1988, reprint of the 1957 original.CrossRefGoogle Scholar
3.Aschbacher, Michael; Some multilinear forms with large isometry groups. Geom. dedic. 25 (1988), 417–465.CrossRefGoogle Scholar
4.Atiyah, Michael; K-theory. Notes by D.W. Anderson. Second edition. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.Google Scholar
5.Baeza, Ricardo, Some algebraic aspects of quadratic forms over fields of characteristic two. Proc. Conf Quadratic Forms and Related Topics (Baton Rouge–2001), Doc.Math. 2001, Extra Volume, 49–63.Google Scholar
6.Bak, Anthony; The stable structure of quadratic modules, Thesis Columbia Univ., 1969Google Scholar
7.Bak, Anthony, On modules with quadratic forms, Lecture Notes Math. 108 (1969), 55–66.CrossRefGoogle Scholar
8.Bak, Anthony; The computation of surgery groups of odd torsion groups; Bull. Amer. Math. Soc. 80 (1976), no. 6, 1113–1116.CrossRefGoogle Scholar
9.Bak, Anthony; Odd dimension surgery groups of odd torsion groups vanish. Topology 14 (1975), no. 4, 367–374.CrossRefGoogle Scholar
10.Bak, Anthony; The computation of surgery groups of finite groups with abelian 2-hyperelementary subgroups. Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), 384–409. Lecture Notes in Math., Vol. 551, 1976, Springer, Berlin.Google Scholar
11.Bak, Anthony; Grothendieck groups of modules and forms over commutative rings; Amer. J. Math., 99 (1977); no. 1, 107–120.CrossRefGoogle Scholar
12.Bak, Anthony; The computation of even dimension surgery groups of odd torsion groups; Comm. Algebra. 6 (1978), no. 14, 1393–1458.CrossRefGoogle Scholar
13.Bak, Anthony; Arf theorem for trace Noetherian and other rings. J. Pure Appl. Algebra 14 (1979), no. 1, 1–20.CrossRefGoogle Scholar
14.Bak, Anthony, K-theory of Forms. Annals of Mathematics Studies, 98. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981Google Scholar
15.Bak, Anthony; Le problème des sous-groupes de congruence et le problème métaplectique pour les groupes classiques de rang > 1. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 5, 307–310.Google Scholar
16.Bak, Anthony, Subgroups of the general linear group normalized by relative elementary subgroup. Algebraic K-theory, Part II (Oberwolfach, 1980), 1–22. Lecture Notes in Math., Vol. 967, 1982, Springer, Berlin.Google Scholar
17.Bak, Anthony, A norm theorem for K 2 of global fields. Algebraic topology (Aarhus, 1982), 1–7. Lecture Notes in Math., Vol. 1051, 1982, Springer, Berlin.Google Scholar
18.Bak, Anthony, Nonabelian K-theory: the nilpotent class of K 1 and general stability. K-Theory 4 (1991), no. 4, 363–397.CrossRefGoogle Scholar
19.Bak, Anthony, Lectures on dimension theory, group valued functors, and nonstable K-theory, Buenos Aires (1995), PreprintGoogle Scholar
20.Bak, Anthony; Induction for finite groups revisited J. Pure Appl. Algebra 104 (1995), no. 3, 235–241.CrossRefGoogle Scholar
21.Bak, Anthony; Global actions: an algebraic double of a topological space. Uspehi Mat. Nauk, (1997), no. 5, 71–112 (Russian, English transl. Russian Math. Surveys 52 (1997), no. 5, 955–996).Google Scholar
22.Bak, Anthony; Topological methods in Algebra. Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), 43–54, Lect. Notes Pure Appl. Math., 967, 1998, Marcel Dekker, N.Y.Google Scholar
23.Bak, Anthony; Kolster, Manfred; The computation of odd-dimensional projective surgery groups of finite groups. Topology 21 (1982), 35–63.CrossRefGoogle Scholar
24.Bak, Anthony; Morimoto, Masaharu; Cancellation over rings of dimension ≤ 1. Bull. Soc. Math. Belg., Sér A 45 (1993), no. 1–2, 29–37.Google Scholar
25.Bak, Anthony; Morimoto, Masaharu; Equivariant surgery and applications. Topology Hawaii (Honolulu, 1990) World Sci., 1992, 13–25.Google Scholar
26.Bak, Anthony; Morimoto, Masaharu; K-theoretical groups with positioning map and equivariant surgery. Proc. Japan. Acad. Scu, Ser. A Math. Sci. 70 (1994), no. 1, 6–11.Google Scholar
27.Bak, Anthony; Morimoto, Masaharu; Equivariant surgery with middle-dimensional singular sets. I. Forum Math. 8 (1996), no. 3, 267–302.Google Scholar
28.Bak, Anthony; Morimoto, Masaharu; The dimension of spheres with smooth one fixed point actions. Forum Math. 17 (2005) no. 2, 199–216.CrossRefGoogle Scholar
29.Bak, Anthony; Morimoto, Masaharu; Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets, J. K-Theory 2 (2008), no. 3, 507–600.CrossRefGoogle Scholar
30.Bak, Anthony; Muranov, Yuri; Splitting along submanifolds, and -spectra. Sovrem. Mat. Prilozh., no. 1, 2003, 3–18 (Russian, English transl. J. Math. Sci. 123 (2004), no. 4, 4169–4184).Google Scholar
31.Bak, Anthony; Petrov, Viktor; Tang, Guoping; Stability for quadratic K 1, K-Theory 30 (2003), no. 1, 1–11.CrossRefGoogle Scholar
32.Bak, Anthony; Rehmann, Ulf; Le problème des sous-groupes de congruence dans SLn≥2 sur un corps gauche. C. R. Acad. Sci. Paris, Sér. A–B 289 (1979), no. 3, 151.Google Scholar
33.Bak, Anthony; Rehmann, Ulf; The congruence subgroup and metaplectic problems for SLn≥2 of division algebras. J. Algebra 78 (1982), no. 2, 475–547.CrossRefGoogle Scholar
34.Bak, Anthony; Rehmann, Ulf; K 2-analogs of Hasse's norm theorems. Comment.Math. Helv. 59 (1984), no. 1, 1–11.CrossRefGoogle Scholar
35.Bak, Anthony; Scharlau, Winfried; Grothendieck and Witt groups of orders and finite groups Invent.Math. 23 (1974), 207–240.CrossRefGoogle Scholar
36.Bak, Anthony; Stepanov, Alexei, Dimension theory and nonstable K-theory for net groups. Rend. Sem. Mat. Univ. Padova 106 (2001), 207–253.Google Scholar
37.Bak, Anthony, Tang, Guoping, Stability for hermitian K 1 J. Pure Appl. Algebra 150 (2000), no. 2, 109–121.CrossRefGoogle Scholar
38.Bak, Anthony; Tang, Guoping, Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups. Adv. Math. 189 (2004), no. 1, 1–37.CrossRefGoogle Scholar
39.Bak, Anthony; Vavilov, NikolaiNormality for elementary subgroup functors. Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47.CrossRefGoogle Scholar
40.Bak, Anthony; Vavilov, Nikolai; Presenting powers of augmentation ideal and Pfister forms. K-Theory 20 (2000), no. 4, 299–309.CrossRefGoogle Scholar
41.Bak, Anthony; Vavilov, Nikolai, Structure of hyperbolic unitary groups. I. Elementary subgroups. Algebra Colloq. 7 (2000), no. 2, 159–196CrossRefGoogle Scholar
42.Bak, Anthony; Vavilov, Nikolai, Cubic form parameters, preprint.Google Scholar
43.Bak, Anthony; Hazrat, Roozbeh; Vavilov, Nikolai, Structure of hyperbolic unitary groups. II. Normal subgroups, preprint.Google Scholar
44.Bak, Anthony; Hazrat, Roozbeh; Vavilov, Nikolai, Localisation-completion strikes again: Relative K 1 is nilpotent by abelian, J. Pure Appl. Algebra 213, (2009) 1075–1085.CrossRefGoogle Scholar
45.Bass, Hyman, K-theory and stable algebra, Publ. Math. Inst Hautes Etudes Sci. 22 (1964), 5–60.CrossRefGoogle Scholar
46.Bass, Hyman, Algebraic K-theory, Benjamin, New York, 1968.Google Scholar
47.Bass, HymanUnitary algebraic K-theory. Lecture Notes Math. 343 (1973), 57–265.CrossRefGoogle Scholar
48.Bass, Hyman; Pardon, William, Some hybrid asymplectic group phenomena. J. Algebra 53 (1978), no. 2, 327–333.CrossRefGoogle Scholar
49.Bass, Hyman, John, Milnor, the algebraist, Topological Methods in modern mathematics (Stony Brook – 1991) Publish or Perish, Houston, 1993, 45–84.Google Scholar
50.Bass, Hyman, Personal reminiscences of birth of algebraic K-theory. K-theory 30 (2003), 203–209CrossRefGoogle Scholar
51.Bass, Hyman; Milnor, John; Serre, Jean-Pierre, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2). Publ. Math. Inst. Hautes Etudes Sci. 33 (1967), 59–137.CrossRefGoogle Scholar
52.Basu, Rabeya; Rao, Ravi. A.; Khanna, Reema; On Quillen's local global principle. Commutative algebra and algebraic geometry, 17–30, Contemp. Math. 390, Amer. Math. Soc., Providence, RI, 2005.CrossRefGoogle Scholar
53.Bayer-Fluckiger, Eva, Principe de Hasse faible pour les systèmes des formes quadratiques. J. reine angew. Math. 378 (1987), 53–59.Google Scholar
54.Borel, Armand; Serre, Jean-Pierre; Le théoreme de Riemann-Roch, Bull Soc. Math. France 86 (1958), 97–136.CrossRefGoogle Scholar
55.Borewicz, Zenon; Vavilov, Nikolai, The distribution of subgroups in the full linear group over a commutative ring, Proc. Steklov Institute Math 3 (1985), 27–46.Google Scholar
56.Browder, William; Surgery on simply-connected manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972Google Scholar
57.Chevalley, Claude; Theory of Lie groups, vol. I, Princeton Univ. Press, Princeton N.J., 1946; 15th printing, 1999.Google Scholar
58.Chevalley, Claude; Sur certains groupes simples. Tôhoku Math. J. 7 (1955), 14–66.CrossRefGoogle Scholar
59.Chevalley, Claude; Classification des groupes algébriques semi-simples. Collected works, vol.3, Springer-Verlag, Berlin, 2005, 1–276.Google Scholar
60.Costa, Douglas, Keller, Gordon; Radix redux: normal subgroups of symplectic groups. J. reine angew. Math. 427 (1992), 51–105.Google Scholar
61.Dennis, Keith, Stability for K 2. Lect. Notes. Math. 353 (1973), 85–94.CrossRefGoogle Scholar
62.Dickson, Leonard, Linear groups: with an exposition of the Galois field theory. Dover publications, N.Y., 1958.Google Scholar
63.Dickson, Leonard Eugene, Theory of linear groups in an arbitrary field. Trans. Amer. Math. Soc. 2 (1901), no. 4, 363–394.CrossRefGoogle Scholar
64.Dieudonné, Jean, Sur les groupes classiques. 3ème ed., Hermann. Paris, 1973.Google Scholar
65.Dieudonné, Jean, On the automorphism of the classical groups. Mem. Amer. Math. Soc. (1951), no. 2, 1–122.Google Scholar
66.Dieudonné, Jean, La géometrie des groupes classiques. 3ème ed., Springer Verlag, Berlin, 1971.Google Scholar
67.Dieudonné, Jean, Panorama des mathématiques pures. Le Choix bourbachique. Gauthier-Villars, Paris, 1977.Google Scholar
68.Estes, David; Ohm, Jack, Stable range in commutative rings. J. Algebra, 7 (1967), no. 3, 343–362.CrossRefGoogle Scholar
69.Fröhlich, Albrecht, Hermitian and quadratic forms over rings with involution. Quart. J. Math. Oxford Ser.2, 20 (1969), 297–317.CrossRefGoogle Scholar
70.Gerasimov, Viktor, The group of units of a free product of rings, Math. U.S.S.R. Sbornik 62 (1989), no. 141–63.CrossRefGoogle Scholar
71.Golubchik, Igor, On the general linear group over an associative ring. Uspehi Mat. Nauk. 28 (1973), no. 3, 179–180.Google Scholar
72.Golubchik, Igor, On the normal subgroups of orthogonal group over an associative ring with involution. Uspehi Mat. Nauk. 30 (1975), no. 6, 165.Google Scholar
73.Golubchik, Igor, The normal subgroups of linear and unitary groups over rings. Ph. D. Thesis, Moscow State Univ. (1981) 1–117 (in Russian).Google Scholar
74.Golubchik, Igor, On the normal subgroups of the linear and unitary groups over associative rings. Spaces over algebras and some problems in the theory of nets. Ufa, 1985, 122–142 (in Russian).Google Scholar
75.Golubchik, Igor, Mikhalev Alexander, Elementary subgroup of a unitary group over a PI-ring. Vestnik Mosk. Univ., ser.1, Mat., Mekh. (1985), no. 1, 30–36.Google Scholar
76.Golubchik, Igor, Mikhalev, Alexander, On the elementary group over a PI-ring. Studies in Algebra, Tbilisi (1985), 20–24. (in Russian).Google Scholar
77.Habdank, Günter, A classification of subgroups of Λ-quadratic groups normalized by relative elementary subgroups. Adv. Math. 110 (1995), no. 2, p.191–233.CrossRefGoogle Scholar
78.Hahn, Alexander; O'Meara, O. T., The classical groups and K-theory, Springer, Berlin 1989.CrossRefGoogle Scholar
79.Hazrat, Roozbeh, Dimension theory and nonstable K 1 of quadratic modules, K-theory 27, (2002) 293–328.CrossRefGoogle Scholar
80.Hazrat, Roozbeh; Vavilov, Nikolai, K 1 of Chevalley groups are nilpotent. J. Pure Appl. Algebra 179, (2003) 99–116.CrossRefGoogle Scholar
81.Jordan, Camille, Traité des substitutions et des équations algébriques. Reprint of the 1870 original. Editions Jacques Gabay, Sceaux, 1989Google Scholar
82.van der Kallen, Wilberd, Injective stability for K 2. Lect. Notes Math. 551 (1976), 77–156.CrossRefGoogle Scholar
83.van der Kallen, Wilberd, Another presentation for Steinberg groups, Indag. Math. 39, no. 4 (1977), 304–312.CrossRefGoogle Scholar
84.van der Kallen, Wilberd, The K 2 of rings with many units, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), no. 4, 473–515.CrossRefGoogle Scholar
85.van der Kallen, Wilberd, A group structure on certain orbit sets of unimodular rows, J. Algebra 82 (1983), 363–397.CrossRefGoogle Scholar
86.van der Kallen, Wilberd, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra 57 (1989), 281–316.CrossRefGoogle Scholar
87.van der Kallen, Wilberd; Magurn, Bruce; Vaserstein, Leonid; Absolute stable rank and Witt cancellation for non-commutative rings. Invent. Math. 91 (1988) 525–542.Google Scholar
88.Kato, KazuyaSymmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two. Invent. Math. 66 (1982), no. 3, 493–510.CrossRefGoogle Scholar
89.Karoubi, Max; Periodicity theorems and conjectures in hermitian K-theory, an appendix to the current paper, J. K-theory 4 (2009), 67–75.CrossRefGoogle Scholar
90.Kervaire, Michel; Milnor, John, Groups of homotopy spheres. I. Ann. of Math. (2) 77 1963 504–537.CrossRefGoogle Scholar
91.Kervaire, Michel; Multiplicateurs de Schur et K-theory. Essays on Topology and Related Topics, Mém. dédiés à G. de Rham. Springer Verlag: Berlin et al. 1970, 212–225.CrossRefGoogle Scholar
92.Klein, Igor, Mikhalev, Alexander; The orthogonal Steinberg group over a ring with involution Algebra and Logic 9 (1970), 88–103.Google Scholar
93.Klein, Igor, Mikhalev, Alexander; The unitary Steinberg group over a ring with involution Algebra and Logic 9 (1970), 305–312.Google Scholar
94.Knus, Max-Albert, Quadratic and hermitian forms over rings. Springer Verlag, Berlin. 1991.CrossRefGoogle Scholar
95.Knus, Max-Albert; Merkuriev, Alexander; Rost, Marcus; Tignol, Jean-Pierre. The Book of involutions. Maer. Math. Soc. Colloq. Publ. 44, 1998.Google Scholar
96.Kolster, Manfred; On injective stability for K 2. Algebraic K-theory, Part I (Oberwolfach, 1980), 128–168, Lecture Notes in Math. 966, Springer, Berlin-New York, 1982.Google Scholar
97.Laitinen, Erkki; Morimoto, Masaharu; Pawalowski, Krzysztof; Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres. Comment. Math. Helv. 70 (1995), no. 1, 10–38.CrossRefGoogle Scholar
98.Lam, Tsit-Yuen, The algebraic theory of quadratic forms. Benhamin. Reading, 1973.Google Scholar
99.Li, Fuan, The structure of symplectic group over arbitrary commutative rings. Acta Math. Sinica, New Series 3 (1987), no. 3, 247–255.Google Scholar
100.Li, Fuan, The structure of orthogonal groups over arbitrary commutative rings. Chinese Ann. Math. 10B (1989), no. 3, 341–350.Google Scholar
101.Li, Shangzhi, A new type of classical groups over skew-fields of characteristic 2. J.Algebra 138 (1991), no. 2, 399–419.CrossRefGoogle Scholar
102.Milnor, John, Algebraic K-theory and quadratic forms. Invent. Math. 9 (1970), 318–344.CrossRefGoogle Scholar
103.Milnor, John, Introduction to algebraic K-theory, Princeton Univ. Press, Princeton, N. J., 1971.Google Scholar
104.Milnor, John, On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64 (1956), 399–405.CrossRefGoogle Scholar
105.Milnor, John; Husemoller, Dale; Symmetric bilinear forms. Springer Verlag, Heidelberg, 1973.CrossRefGoogle Scholar
106.Moore, Calvin; Group extensions of p-adic and adelic linear groups. Publ.Math. Inst. Hautes Etudes Sci. 35 (1968), 157–222.CrossRefGoogle Scholar
107.Morimoto, Masaharu, Bak groups and equivariant surgery. K-theory 2 (1989), no. 4, 465–483.CrossRefGoogle Scholar
108.Morimoto, Masaharu, Bak groups and equivariant surgery. II. K-theory 3 (1990), no. 6, 505–521CrossRefGoogle Scholar
109.Mundkur, Arun, Dimension theory and nonstable K 1. Algebr. Represent. Theory 5 (2002), no. 1, 3–55.CrossRefGoogle Scholar
110.Novikov, Sergey, Homotopically equivalent smooth manifolds. I. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 365–474Google Scholar
111.O'Meara, Timothy, Introduction to quadratic forms. Reprint of the 1973 ed. Springer Verlag, Berlin. 2000Google Scholar
112.Pender, William; Classical groups over division rings of charactristic two. Bull Austral. Math. Soc. 7 (1972), 191–226. Correction, ibid 319.CrossRefGoogle Scholar
113.Petrie, Ted, One fixed point actions on spheres. I, II. Adv. in Math. 46 (1982), no. 1, 3–14, 15–70.Google Scholar
114.Petrov, Viktor, Overgroups of unitary groups. K-theory 29 (2003), no. 3, 147–174.CrossRefGoogle Scholar
115.Petrov, Viktor, Odd unitary groups. J. Math. Sci. 130 (2003), no. 3, 4752–4766.CrossRefGoogle Scholar
116.Petrov, Viktor, Overgroups of classical groups. Ph.D. Thesis, Saint-Petersburg State Univ. (2005), 1–129 (in Russian).Google Scholar
117.Platonov, Vladimir; Rapinchuk, Andrei, Algebraic groups and number theory, Academic Press.CrossRefGoogle Scholar
118.Plotkin, Eugene, Stability theorems for K-functors for Chevalley groups, Proc. Conf. Nonassociative Algebras and Related Topics (Hiroshima — 1990), World Sci. London et al., 1991, 203–217.Google Scholar
119.Plotkin, Eugene, Surjective stabilization for K 1-functor for some exceptional Chevalley groups. J. Soviet Math. 64, 1993, p.751–767.CrossRefGoogle Scholar
120.Plotkin, Eugene, On the stability of the K 1-functor for Chevalley groups of type E 7, J. Algebra 210 (1998), 67–85.CrossRefGoogle Scholar
121.Plotkin, Eugene; Stein, Michael; Vavilov, Nikolai, Stability of K-functors modeled on Chevalley groups, revisited. Unpublished manuscript (2001), 1–21.Google Scholar
122.Raghunathan, M. S., The congruence subgroup problem. Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 299–308.CrossRefGoogle Scholar
123.Rehmann, Ulf, Zentrale Erweiterungen der speziellen linearen Gruppe eines Schiefkörpers. J. reine angew. Math. 301 (1978), 77–104.Google Scholar
124.Rehmann, Ulf, A survey of the congruence subgroup problem. Algebraic K-theory, Part I (Oberwolfach, 1980), 197–207, Lecture Notes in Math. 966, Springer, Berlin-New York, 1982.Google Scholar
125.Sah, Chih Han; Symmetric bilinear forms and quadratic forms. J. Algebra 20 (1972), 144–160.CrossRefGoogle Scholar
126.Serre, Jean-Pierre; Modules projectifs et espaces fibrés a fibre vectorielle. Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc.2, Exposé 23, Paris, 1958Google Scholar
127.Scharlau, Winfried, Quadratic and Hermitian forms. Springer Verlag, Berlin. 1985.CrossRefGoogle Scholar
128.Scharlau, Winfried, On the history of the algebraic theory of quadratic forms. Quadratic forms and their applications (Dublin, 1999), 229–259, Contemp. Math. 272, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
129.Sharpe, Richard, On the structure of the unitary Steinberg groups. Ann. Math. 96 (1972), 444–479.CrossRefGoogle Scholar
130.Smale, Stephen, Generalized Poincaré's conjecture in dimensions greater than four. Ann. of Math. 74 (1961), no. 2, 391–406.CrossRefGoogle Scholar
131.Stafford, J. Tobias, Absolute stable rank and quadratic forms over noncommutative rings. K-theory 4 (1990), 121–130.CrossRefGoogle Scholar
132.Stein, Michael; Relativizing functors on rings and algebraic K-theory, J. Algebra 19 (1971), 140–152.CrossRefGoogle Scholar
133.Stein, Michael; Stability theorems for K 1, K 2 and related functors modeled on Chevalley groups. Japan. J. Math. 4 (1978), no. 1, 77–108.CrossRefGoogle Scholar
134.Steinberg, Robert; Générateurs, rélations et revêtements des groupes algébriques. Colloque Théorie des Groupes Algébriques (Bruxelles—1962), 113–127.Google Scholar
135.Steinberg, Robert; Lectures on Chevalley groups Yale University, 1967.Google Scholar
136.Stepanov, Alexei; Non-standard subgroups between E(n,R) and GL(n,A). Algebra Colloq. 11 (2004), no. 3, 321–334.Google Scholar
137.Stepanov, Alexei, Subgroups of a Chevalley group between two different rings, Preprint.Google Scholar
138.Stepanov, Alexei; Vavilov, Nikolai; Decomposition of transvections: a theme with variations. K-theory 19 (2000), 109–153.CrossRefGoogle Scholar
139.Stepanov, Alexei; Vavilov, Nikolai; You, Hong Localisation-complition in the description of intermediate subgroups (2007), 1–43.Google Scholar
140.Suslin, Andrei, On the structure of the general linear group over polynomial rings, Soviet Math. Izv. 41 (1977), no. 2, 503–516.Google Scholar
141.Suslin, Andrei, Stability in algebraic K-theory. Lect. Notes. Math., 996 (1980), 304–333.Google Scholar
142.Suslin, Andrei, Algebraic K-theory and norm residue homomorphisms, J. Sov. Math. 30 (1985), 2556–2611.CrossRefGoogle Scholar
143.Suslin, Andrei, Kopeiko, Viacheslav, Quadratic modules and orthogonal groups over polynomial rings. J. Sov. Math. 20 (1982), no. 6, 2665–2691.CrossRefGoogle Scholar
144.Suslin, Andrei; Tulenbaev, Marat, Stabilization theorem for Milnor's K 2-functor. J. Sov. Math. 17 (1981), 1804–1819.CrossRefGoogle Scholar
145.Tang, Guoping, Hermitian groups and K-theory, K-theory 13 (1998), no. 3, 209–267.CrossRefGoogle Scholar
146.Tang, Guoping, Presenting powers of augmentation ideals in elementary p-groups. K-theory 13 (2001) no. 1, 31–39.CrossRefGoogle Scholar
147.Tang, Guoping, Hermitian forms over local rings, Algebra Colloq. 8 (2001), no. 1, 1–10.Google Scholar
148.Tits, Jacques; Buildings of spherical type and finite BN-pairs. Lecture Notes Math. 386 (1973).Google Scholar
149.Tits, Jacques; Formes quadratiques, groupes orthogonaux et algébres de Clifford. Invent.Math. 5 (1968), 19–41.CrossRefGoogle Scholar
150.Tulenbaev, Marat, Schur, multiplier of the group of elementary matrices of finite order, J. Sov. Math. 17 (1981), no. 4, 2062–2067.CrossRefGoogle Scholar
151.Vaserstein, Leonid, On the stabilization of the general linear group over a ring. Math USSR Sbornik 8 (1969), 383–400.CrossRefGoogle Scholar
152.Vaserstein, Leonid, K 1-theory and the congruence subgroup problem, Mat. Zametki 5 (1969), 233–244.Google Scholar
153.Vaserstein, Leonid, Stabilization of unitary and orthogonal groups over a ring with involution. Math USSR Sbornik 10 (1970), 307–326.CrossRefGoogle Scholar
154.Vaserstein, Leonid, Stabilization for Milnor's K 2-functor. (Russian) Uspehi Mat. Nauk 30 (1975), no. 1 (181), 224.Google Scholar
155.Vaserstein, Leonid, On the normal subgroups of GLn over a ring, Lecture Notes Math. 854 (1981), 456–465.CrossRefGoogle Scholar
156.Vaserstein, Leonid; The subnormal structure of general linear groups over rings, Math. Proc. Camb. Phil. Soc, 108 (1990), 219–229.CrossRefGoogle Scholar
157.Vaserstein, Leonid; Normal subgroups of orthogonal groups over commutative rings. Amer. J. Math. 110 (1988), no. 5, 955–973.CrossRefGoogle Scholar
158.Vaserstein, Leonid; Normal subgroups of symplectic groups over rings. K-theory 2 (1989), no. 5, 647–673.CrossRefGoogle Scholar
159.Vaserstein, Leonid; You, Hong; Normal subgroups of classical groups over rings. J. Pure Appl. Algebra 105 (1995), 93–106.CrossRefGoogle Scholar
160.Vavilov, Nikolai; The structure of split classical groups over a commutative ring. Soviet Math. Dokl. 37 (1988), no. 2, 550–553.Google Scholar
161.Vavilov, Nikolai; Structure of Chevalley groups over commutative rings. Proc. Conf. Non-Associative Algebras and Related Topics (Hiroshima – 1990). World Sci., London et al., 1991, 219–335.Google Scholar
162.Vavilov, Nikolai; Intermediate subgroups in Chevalley groups. Proc. Conf. Groups of Lie type and their Geometries (Como – 1993). Cambridge Univ. Press, 1995, 233–280.CrossRefGoogle Scholar
163.Vavilov, Nikolai; Subnormal structure of general linear group, Math. Proc. Camb. Phil. Soc. 107 (1990), 103–196.CrossRefGoogle Scholar
164.Vavilov, Nikolai; A third look at weight diagrams. Rend. Sem. Mat. Univ. Padova. 104 (2000) no. 1, 201–250.Google Scholar
165.Vavilov, Nikolai; An A 3-proof of the main structure theorems for Chevalley groups of types E 6 and E 7, Int. J. Algebra. Comput. 17 (2007), no. 5–6, 1283–1298.CrossRefGoogle Scholar
166.Vavilov, Nikolai; Gavrilovich, Mikhail, A 2-proof of the structure theorems for Chevalley groups of types E 6 and E 7, St. Petersburg Math. J. 16 (2005), 649–672.CrossRefGoogle Scholar
167.Vavilov, Nikolai; Gavrilovich, Mikhail; Nikolenko, Sergei, Structure of Chevalley groups: the Proof from the Book, J. Math. Sci. 330 (2006), 36–76.Google Scholar
168.Vavilov, Nikolai; Nikolenko, Sergei, A 2-proof of the structure theorems for Chevalley groups of types F4, St. Petersburg Math. J. 20 (2008), no. 3.Google Scholar
169.Vavilov, Nikolai; Petrov, Viktor, On overgroups of Ep(2l,R), St. Petersburg Math. J. 15 (2004), 515–543.CrossRefGoogle Scholar
170.Vavilov, Nikolai; Petrov, Viktor, On overgroups of EO(n,R, St. Petersburg Math. J. 19 (2007), no. 2, 10–51.Google Scholar
171.Voevodsky, Vladimir, Motivic cohomology with ℤ/2-coefficients. Publ. IHES. 98 (2003), 59–104.CrossRefGoogle Scholar
172.van der Waerden, Bartels, Gruppen der linearen Transformationen. Chelsea, New York, 1948.Google Scholar
173.Wall, Charles Terence Clehh; On the axiomatic foundation of the theory of Hermitian forms. Proc. Cambridge. Phil. Soc. 67 (1970), 243–250.CrossRefGoogle Scholar
174.Wall, Charles Terence Clehh; Surgery of non-simply-connected manifolds. Ann. of Math. (2) 84 (1966), 217–276.CrossRefGoogle Scholar
175.Wall, Charles Terence Clehh; Surgery on compact manifolds. Second edition. Edited and with a foreword by Ranicki, A. A.. Mathematical Surveys and Monographs 69. American Mathematical Society, Providence, RI, 1999CrossRefGoogle Scholar
176.Weil, Andre, Classical groups and algebras with involution, J. Indian Math. Soc. 24 (1961), 589–623.Google Scholar
177.Weyl, Hermann, The classical groups. Their invariants and representations. 15th printing, Princeton Univ. Press, Princeton. 1998Google Scholar
178.Weyl, Hermann, Review of Dieudonné “Sur les groupes classiques”", MR0024439.Google Scholar
179.Wilson, John, The normal and subnormal structure of general linear groups, Proc. Camb. Phil. Soc. 71 (1972), 163–177.CrossRefGoogle Scholar
180.Witt, Ernst, Theorie der quadratischen Formen in beliebigen Körpern. J. reine angew. Math. 176 (1937), 31–44.CrossRefGoogle Scholar
181.Zhang, Zuhong, Stable sandwich classification theorem for classical-like groups, Math. Proc. Cambridge Phil. Soc. 143 (2007), no. 3, 607–619.CrossRefGoogle Scholar