St. Petersburg School of Linear Groups: II. Early Works by Suslin

N. A. Vavilov, “St. Petersburg school of linear groups. I. Prehistoric period,” Vestn. St. Petersburg Univ.: Math. 56, 273–288 (2023). https://doi.org/10.1134/S106345412303010X

A. A. Suslin, “On projective modules over polynomial rings,” Math. USSR-Sb. 22, 595–602 (1974).

L. N. Vaserstein and A. A. Suslin, “Serre’s problem on projective modules over polynomial rings and algebraic K-theory,” Funct. Anal. Appl. 8, 148–150 (1974).

A. A. Suslin, “One theorem of Cohn,” J. Sov. Math. 17, 1801–1803 (1981).

A. A. Suslin and M. S. Tulenbaev, “Stabilization theorem for the Milnor K2-functor,” J. Sov. Math. 17, 1804–1819 (1981).

L. N. Vaserstein and A. A. Suslin, “Serre’s problem on projective modules over polynomial rings, and algebraic K-theory,” Math. USSR-Izv. 10, 937–1001 (1976).

A. A. Suslin, “Projective modules over a polynomial ring are free,” Sov. Math. Dokl. 17, 1160–1164 (1977).

A. A. Suslin and V. I. Kopeiko, “Quadratic modules and orthogonal group over polynomial rings,” J. Sov. Math. 20, 2665–2691 (1982).

A. A. Suslin, “On stably free modules,” Math. USSR-Sb. 31, 479–491 (1977).

A. A. Suslin, “On the structure of the special linear group over polynomial rings,” Math. USSR-Izv. 11, 221–238 (1977).

A. A. Suslin, “Locally polynomial rings and symmetric algebras,” Math. USSR-Izv. 11, 472–484 (1977).

A. A. Suslin, “A cancellation theorem for projective modules over algebras,” Sov. Math. Dokl. 18, 1281–1284 (1978).

A. A. Suslin, “Structure of projective modules over rings of polynomials in the case of a noncommutative ring of coefficients,” Proc. Steklov Inst. Math. 148, 245–267 (1980).

V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring,” Math. USSR-Sb. 34, 655–669 (1978).

V. I. Kopeiko and A. A. Suslin, “Quadratic modules over polynomial rings,” J. Sov. Math. 17, 2024–2031 (1981)].

A. A. Suslin, “Reciprocity laws and the stable rank of polynomial rings,” Math. USSR-Izv. 15, 589–623 (1980).

M. S. Tulenbaev, “Schur multiplier of a group of elementary matrices of finite order,” J. Sov. Math. 17, 2062–2067 (1981).

A. A. Suslin, “Cancellation problem for projective modules and similar questions,” In: Proc. Int. Congr. of Mathematicians, Helsinki, Finland, 1978 (Academia Scientiarum Fennica, Helsinki, 1980), Vol. 1, pp. 323–330.

M. S. Tulenbaev, “The Steinberg group of a polynomial ring,” Math. USSR-Sb. 45, 139–154 (1983).

A. A. Suslin, “Stability in algebraic K-theory,” in Algebraic K-Theory: Proc. Conf., Oberwolfach, June 1980 (Springer-Verlag, Berlin, 1982), Vol. 1, in Ser.: Lecture Notes in Mathematics, Vol. 966, 304–333.

A. A. Suslin, “Mennicke symbols and their applications in the K-theory of fields,” Algebraic K-Theory: Proc. Conf., Oberwolfach, June 1980 (Springer-Verlag, Berlin, 1982), Vol. 1, in Ser.: Lecture Notes in Mathematics, Vol. 966, pp. 334–356 (1982).

A. A. Suslin, “Algebraic K-theory,” J. Sov. Math. 28, 870–923 (1985).

A. A. Suslin, “Algebraic K-theory,” Proc. Steklov Inst. Math. 168, 161–177 (1986).

A. A. Suslin, “Algebraic K-theory and the norm residue homomorphism,” J. Sov. Math. 30, 2556–2611 (1985).

R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings (with an appendix by M. Karoubi),” J. K-Theory 4 (1), 1–65 (2009).

N. A. Vavilov, “Simple Lie algebras, simple algebraic groups and simple finite groups,” in Mathematics of the XX Century. View from St. Petersburg, Ed. by A. M. Vershik (Mosk. Tsentr. Nepreryvnogo. Mat. Obraz., Moscow, 2010), pp. 8–46 [in Russian].

E. Artin, Geometric Algebra (Interscience, 1957; Nauka, Moscow, 1969), in Ser.: Interscience Tracts in Pure and Applied Mathematics, Vol. 3.

J. A. Dieudonné, La Géométrie des Groupes Classiques (Springer Verlag, Berlin, 1971

Mir, Moscow, 1974), in Ser.: Ergebnisse der Mathematik und Ihrer Grenzgebiete, Vol. 5.

H. Bass, “K-theory and stable algebra,” Publ. Math., Inst. Hautes Etud. Sci. 22, 489–544 (1964).

H. Bass, Algebraic K-Theory (W. A. Benjamin, New York, 1968; Mir, Moscow, 1973), in Ser.: Mathematics Lecture Note Series.

R. Steinberg, Lectures on Chevalley Groups (American Mathematical Society, Providence, R.I., 2016; Mir, Moscow, 1975), in Ser.: University Lecture Series, Vol. 66.

J. W. Milnor, Introduction to Algebraic K-Theory (Princeton Univ. Press and Univ. of Tokyo Press, Princeton, 1971

Mir, Moscow, 1974), in Ser.: Annals of Mathematics Studies, Vol. 72.

H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2),” Inst. Hautes Etud. Sci. Publ. Math. 33, 59–133 (1967).

A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq. 7, 159–196 (2000).

J. Wilson, “The normal and subnormal structure of general linear groups,” Proc. Cambridge Philos. Soc. 71, 163–177 (1972).

I. Z. Golubchik, “On the general linear group over an associative ring,” Usp. Mat. Nauk 28, 179–180 (1973).

T. Y. Lam, Serre’s Problem on Projective Modules (Springer-Vevrlag, Berlin, 2006), in Ser.: Springer Monographs in Mathematics.

A. S. Merkurjev, “On the norm residue symbol of degree 2,” Sov. Math., Dokl. 24, 546–551 (1981).

A. A. Suslin, “The quaternion homomorphism for the function field on a conic,” Sov. Math., Dokl. 26, 72–77 (1982).

A. S. Merkurjev and A. A. Suslin, “K-cohomology of Severi–Brauer varieties and the norm residue homomorphism,” Sov. Math., Dokl. 25, 690–693 (1982).

A. S. Merkurjev and A. A. Suslin, “K-cohomology of Severi–Brauer varieties and the norm residue homomorphism,” Math. USSR-Izv. 21, 307–340 (1983).

E. M. Friedlander and A. S. Merkurjev, “The mathematics of Andrei Suslin,” Bull. Am. Math. Soc., New Ser. 57, 1–22 (2020).

S. K. Gupta and M. P. Murthy, Suslin’s Work on Linear Groups over Polynomial Rings and Serre Problem (Macmillan, New Delhi, 1980), in Ser.: ISI Lecture Notes, Vol. 8.

A. Hahn and O. T. O’Meara, The Classical Groups and K-Theory (Springer-Verlag, Berlin, 1989), in Ser.: Grundlehren der Mathematischen Wissenschaften, Vol. 291.

N. A. Vavilov and A. V. Stepanov, “Linear groups over general rings. I. Generalities,” J. Math. Sci. (N. Y.) 188, 490–550 (2013)].

L. N. Vaserstein, “On the stabilization of the general linear group over a ring,” Math. USSR-Sb. 8, 383–400 (1969).

L. N. Vaserstein, “Stable rank of rings and dimensionality of topological spaces,” Funct. Anal. Appl. 5, 102–110 (1971).

N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, “Computations in Chevalley groups over commutative rings,” Sov. Math., Dokl. 40, 145–147 (1990)].

A. Stepanov and N. Vavilov, “Decomposition of transvections: A theme with variations,” K-Theory 19, 109–153 (2000).

P. M. Sohn, “On the structure of the GL2 of a ring,” Publ. Math. Inst. Hautes Etud. Sci. 30, 365–413 (1966).

V. N. Gerasimov, “The group of units of a free product of rings,” Math. USSR-Sb. 62, 41–63 (1989)].

I. Z. Golubchik and V. T. Markov, “Localization dimension of P I-rings,” Tr. Semin. im. I. G. Petrovskogo 6, 39–46 (1981).

I. Z. Golubchik and A. V. Mikhalev, “Elementary subgroup of a unitary group over a P I-ring,” Mosc. Univ. Math. Bull. 40, 44–54 (1985).

A. Bak, “Nonabelian K-theory: The nilpotent class of K1 and general stability,” K-Theory 4, 363–397 (1991).

L. N. Vaserstein, On the Normal Subgroups of the GL n of a Ring (Springer-Verlag, Berlin, 1981), in Ser.: Lecture Notes in Mathematics, Vol. 854, pp. 454–465.

Z. I. Borewicz and N. A. Vavilov, “Arrangement of subgroups in the general linear group over a commutative ring,” Proc. Steklov Inst. Math. 165, 27–46 (1985)].

N. A. Vavilov and A. V. Stepanov, “Standard commutator formula,” Vestn. St. Petersburg Univ.: Math. 41, 5–8 (2008).

R. Hazrat and Zuhong Zhang, “Generalized commutator formula,” Commun. Algebra 39, 1441–1454 (2011).

N. A. Vavilov and A. V. Stepanov, “Standard commutator formulae, revisited,” Vestn. St. Petersburg Univ.: Math. 43, 12–17 (2010).

R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “The yoga of commutators,” J. Math. Sci. (N. Y.) 179, 662–678 (2011).

R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “Commutator width in Chevalley groups,” Note Mat. 33, 139–170 (2013).

R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “The yoga of commutators: Further applications,” J. Math. Sci. (N. Y.) 200, 742–768 (2014).

R. Hazrat, N. Vavilov, and Z. Zhang, “The commutators of classical groups,” J. Math. Sci. (N. Y.) 222, 466–515 (2017)].

A. Stepanov, “Structure of Chevalley groups over rings via universal localization,” J. Algebra 450, 522–548 (2016).

G. Taddei, “Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau,” Contemp. Math. 55, 693–710 (1986).

V. A. Petrov and A. K. Stavrova, “Elementary subgroups of isotropic reductive groups,” St. Petersburg Math. J. 20, 625–644 (2009).

R. Preusser, “Structure of hyperbolic unitary groups. II: Classification of E-normal subgroups,” Algebra Colloq. 24, 195–232 (2017).

N. Vavilov, “Structure of Chevalley groups over commutative rings,” in Proc. Int. Symp. on Nonassociative Algebras and Related Topics, Hiroshima, Japan, Aug. 30 – Sept. 1, 1990 (World Scientific, London, 1991), pp. 219–335.

N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “Structure of Chevalley groups: The proof from the book,” J. Math. Sci. (N. Y.) 140, 626–645 (2007).

D. L. Costa and G. E. Keller, “Radix redux: Normal subgroups of symplectic groups,” J. Reine Angew. Math. 427, 51–105 (1992).

D. L. Costa and G. E. Keller, “On the normal subgroups of G2(A),” Trans. Am. Math. Soc. 351, 5051–5088 (1999).

R. Hazrat, V. Petrov, and N. Vavilov, “Relative subgroups in Chevalley groups,” J. K-Theory. 5, 603–618 (2010).

A. Stavrova and A. Stepanov, “Normal structure of isotropic reductive groups over rings,” J. Algebra (2022) (in press).

V. M. Petechuk, “Automorphisms of matrix groups over commutative ring,” Math. USSR-Sb. 45, 527–542 (1982).

I. Z. Golubchik and A. V. Mikhalev, “Isomorphisms of a complete linear group over an associative ring,” Mosc. Univ. Math. Bull. 38, 73–85 (1983).

I. Z. Golubchik and A. V. Mikhalev, “Isomorphisms of unitary groups over associative rings,” J. Sov. Math. 30, 1863–1871 (1985).

E. I. Zel’manov, “Isomorphisms of linear groups over an associative ring,” Sib. Math. J. 26, 515–530 (1985).

F. Keune, “The relativization of K2,” J. Algebra 54, 159–177 (1978).

J. L. Loday, “Cohomologie et groupe de Steinberg relatifs,” J. Algebra 54, 178–202 (1978).

E. Voronetsky, “An explicit presentation of relative Steinberg groups,” J. Algebra 602, 278–299 (2022).

W. van der Kallen, “Another presentation for Steinberg groups,” Nederl. Akad. Wet. Proc. Ser. A 80, 304–312 (1977).

V. Wendt, “On homotopy invariance for homology of rank two groups,” J. Pure Appl. Algebra 216, 2291–2301 (2012).

A. Lavrenov, “Another presentation for symplectic Steinberg groups,” J. Pure Appl. Algebra 219, 3755–3780 (2015).

S. Sinchuk, “On centrality of K2 for Chevalley groups of type El,” J. Pure Appl. Algebra 220, 857–875 (2016).

A. Lavrenov and S. Sinchuk, “On centrality of even orthogonal K2,” J. Pure Appl. Algebra 221, 1134–1145 (2017).

E. Voronetsky, “Centrality of K2-functor revisited,” J. Pure Appl. Algebra 225, 106547 (2021).

E. Voronetsky, “Centrality of odd unitary K2-functor,” preprint (2020). arXiv 2005.02926v2 [math.GR]

A. Lavrenov, S. Sinchuk, and E. Voronetsky, “Centrality of K2 for Chevalley groups: A pro-group approach,” preprint (2020). arXiv 2009.03999v1 [math.GR]

R. K. Dennis, “Stability for K2,” in Proc. Conf. on Orders, Group Rings and Related Topics, Columbus, Ohio, May 1972 (Springer-Verlag, Berlin, 1973), in Ser.: Lecture Notes in Mathematics, Vol. 353, pp. 85–94.

M. R. Stein, “Stability theorems for K1, K2 and related functors modeled on Chevalley groups,” Jpn. J. Math. 4 (1), 77–108 (1978).

N. A. Vavilov and S. S. Sinchuk, “Dennis–Vaserstein type decompositions,” J. Math. Sci. (N. Y.) 171, 331–337 (2010).

N. A. Vavilov and S. S. Sinchuk, “Parabolic factorizations of split classical groups,” St. Petersburg Math. J. 23, 637–657 (2012).

A. Bak and G. Tang, “Stability for Hermitian K1,” J. Pure Appl. Algebra 150, 107–121 (2000).

V. A. Petrov, “Odd unitary groups,” J. Math. Sci. (N. Y.) 30, 4752–4766 (2005).

A. Bak, V. Petrov, and G. Tang, “Stability for quadratic K1,” J. K-Theory 30 (1), 1–11 (2003).

S. Sinchuk, “Injective stability for unitary K1,” J. K-Theory 11, 233–242 (2013).

W. Yu, “Injective stability for odd-dimensional unitary K1,” J. Group Theory 23, 313–325 (2020). https://doi.org/10.1515/jgth-2019-0099

E. Voronetsky, “Injective stability for odd unitary K1,” J. Group Theory 23, 781–800 (2020).

E. Plotkin, “On the stability of the K1-functor for Chevalley groups of type E7,” J. Algebra 210, 67–85 (1998).

W. van der Kallen, “Injective stability for K2,” in Algebraic K-Theory: Proc. Conf., Evanston, Jan. 12–16, 1976 (Springer Verlag, Berlin, 1976), in Ser.: Lecture Notes in Mathematics, Vol. 551, pp. 77–154.

K. Kolster, “On injective stability for K2,” in Algebraic K-Theory: Proc. Conf., Oberwolfach, June 1980 (Sprin-ger-Verlag, Berlin, 1982), Vol. 1, in Ser.: Lecture Notes in Mathematics, Vol. 966, pp. 128–168.

K. Kolster, “Improvement of K2-stability under transitive actions of elementary groups,” J. Pure Appl. Algebra 24, 277–282 (1982).

W. van der Kallen, “Homology stability for linear groups,” Invent. Math. 60, 269–295 (1980).

A. A. Suslin, “Homology of GLn, characteristic classes and Milnor K-theory,” Proc. Steklov Inst. Math. 165, 207–226 (1985)].

Yu. P. Nesterenko and A. A. Suslin, “Homology of the full linear group over a local ring, and Milnor’s K-theory,” Math. USSR-Izv. 34, 121–145 (1990)].

I. A. Panin, “Stabilization for orthogonal and symplectic algebraic K-theories,” Leningr. Math. J. 1, 741–764 (1990).

H. Bass, A. Heller, and R. G. Swan, “The Whitehead group of a polynomial extension,” Publ. Math., Inst. Hautes Etud. Sci. 22, 545–564 (1964).

T. Vorst, “The general linear group of polynomial rings over regular rings,” Commun. Algebra 9, 499–509 (1981).

E. Abe, “Whitehead groups of Chevalley groups over polynomial rings,” Commun. Algebra 11, 1271–1307 (1983).

F. Grunewald, J. Mennicke, and L. Vaserstein, “On symplectic groups over polynomial rings,” Math. Z. 206, 35–56 (1991).

A. Stavrova, “Homotopy invariance of non-stable K1-functors,” J. K-Theory 13, 199–248 (2014).

A. Stavrova, “Non-stable K1-functors of multiloop groups,” Can. J. Math. 68, 150–178 (2016).

A. Stavrova, “Chevalley groups of polynomial rings over Dedekind domains,” J. Group Theory 23, 121–132 (2020).

A. Lavrenov, “A local-global principle for symplectic K2,” Doc. Math. 23, 653–675 (2018).

A. Lavrenov and S. Sinchuk, “A Horrocks-type theorem for even orthogonal K2,” Doc. Math. 25, 767–809 (2020).

D. Quillen, “Projective modules over polynomial rings,” Invent. Math. 36, 167–171 (1976).

A. Suslin, “Quillen’s solution of Serre’s problem,” J. K-Theory 11, 549–552 (2013).

R. Hazrat, “Dimension theory and nonstable K1 of quadratic modules,” J. K-Theory 27, 293–327 (2002).

R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra 179, 99–116 (2003).

A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes Again: relative K1 is nilpotent by Abelian,” J. Pure Appl. Algebra 213, 1075–1085 (2009).

R. Hazrat, N. Vavilov, and Z. Zhang, “Relative unitary commutator calculus, and applications,” J. Algebra 343, 107–137 (2011).

R. Hazrat, N. Vavilov, and Z. Zhang, “Relative commutator calculus in Chevalley groups,” J. Algebra 385, 262–293 (2013).

R. Basu, R. A. Rao, and R. Khanna, “On Quillen’s local global principle,” in Commutative Algebra and Algebraic Geometry: Proc. Joint Int. Meeting of the American Mathematical Society and the Indian Mathematical Society, Bangalore, India, Dec. 17–20, 2003, Ed. by S. Ghorpade (American Mathematical Society, Providence, R.I., 2003), in Ser.: Contemporary Mathematics, Vol. 390, pp. 17–30 (2005).

A. Bak, R. Basu, and R. A. Rao, “Local-global principle for transvection groups,” Proc. Am. Math. Soc. 138, 1191–1204 (2010).

H. Apte and A. Stepanov, “Local-global principle for congruence subgroups of Chevalley groups,” Cent. Eur. J. Math. 12, 801–812 (2014).

R. Basu, “Local-global principle for the general quadratic and general Hermitian groups and the nilpotency of KH1,” J. Math. Sci. (N. Y.) 232, 591–609 (2018).

R. Basu, R. Khanna, and R. A. Rao, “The pillars of relative Quillen—Suslin theory,” in Leavitt Path Algebras and Classical K-theory: Based on the Int. Workshop on Leavitt Path Algebras and K-Theory, Kerala, India, July 1–3, 2017, Ed. by A. A. Ambily, et al. (Springer-Verlag, Singapore, 2020), in Ser.: Indian Statistical Institute Series, pp. 211–223.

R. Basu, “A note on general quadratic groups,” J. Algebra Appl. 17, 1850217 (2018).

A. A. Ambily, “Yoga of commutators in DSER elementary orthogonal group,” J. Homotopy Relat. Struct. 14, 595–610 (2019).

R. Basu and M. K. Singh, “Quillen–Suslin theory for classical groups: Revisited over graded rings,” in Categorical, Homological and Combinatorial Methods in Algebra: Proc. AMS Special Meeting in Honor of S. K. Jain’s 80th Birthday, Columbus, Ohio, Mar. 16–18, 2018, Ed. by K. A. Srivastava, et al. (American Mathematical Society, Providence, R.I., 2020), in Ser: Contemporary Mathematics, Vol. 751, pp. 5–18.

R. G. Swan and J. Towber, “A class of projective modules which are nearly free,” J. Algebra 36, 427–434 (1975).

M. Krusemeyer, “Skewly completable rows and a theorem of Swan and Towber,” Commun. Algebra 4, 657–663 (1976).

S. Mandal, Projective Modules and Complete Intersections (Springer-Verlag, Berlin, 1997), in Ser.: Lecture Notes in Mathematics, Vol. 1672.

N. M. Kumar, “A note on the cancellation of reflexive modules,” J. Ramanujan Math. Soc. 17, 93–100 (2002).

R. Sridharan and S. K. Yadav, “On a theorem of Suslin,” in Leavitt Path Algebras and Classical K-Theory (Springer-Verlag, Singapore, 2020), in Ser.: Indian Statistical Institute Series, pp. 241–260.

S. Jose and R. A. Rao, “A fundamental property of Suslin matrices,” J. K-Theory 5, 407–436 (2010).

R. A. Rao and S. Jose, “A study of Suslin matrices: Their properties and uses,” in Proc. Int. Congr. on Algebra and Its Applications, Aligarh, India, Dec. 15–17, 2014, Ed. by S. T. Rizvi (Springer-Verlag, Singapore, 2016), in Ser.: Mathematics & Statistics, Vol. 174, pp. 89–121.

N. A. Vavilov, A. V. Smolensky, and B. Sury, “Unitriangular factorisations of Chevalley groups,” J. Math. Sci. (N. Y.) 183, 584–599 (2012).