ATMW Classical Algebraic K - theory (2013)

Venue: TIFR, Mumbai
Dates: 15th -24 thJuly , 2013


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name Ravi A. Rao

Jean Fasel

Nikolai Vavilov 
Mailing Address

Tata Institute of Fundamental Research (TIFR),
Dr. Homi Bhabha Road,
Navy Nagar, Mumbai 400-005,


Fachbereich Mathematik
Universitt Duisburg-Essen
Altendorfer Strasse 11
45117 Essen


Department of Mathematics and Mechanics,
Saint Petersburg State University,
University prospect 28 Stary Peterhof,
Saint Petersburg. 198504, Russia


Speakers and Syllabus 


Brief description of the workshop:-

  1. Quillen-Suslin-Bak, localization and patching
    Quillen-Suslin-Bak, localization and patchingOne of the most powerful ideas in the study of groups of points of reductivegroups over rings is localization. It allows to reduce many important prob-lems over arbitrary commutative rings, to similar problems for semi-localrings. Localization comes in a number of versions. The two most famil-iar ones are localization and patching, proposed by Daniel Quillen [31] andAndrei Suslin [47], and localization–completion, proposed by Anthony Bak[3].Recently, these methods were further generalized to unitary groups, andisotropic reductive groups, and used to study nilpotent structure of K1 -groupof various classical-like groups by Bak, Basu, Hazrat, Petrov, Rao, Stepanov,Vavilov and others [4, 9, 5, ?, ?, ?, 10, ?, ?, ?, ?, 28, 29, 30, 35, ?]. One part ofthe workshop is talks to present various work done recently in this direction.Talks will be given by Rao, Stepanov and Vavilov. Rao will emphasize theconnection between some of these topics with the next topic.
  2. Study of projective modules over an affine algebra
    Study of projective modules over an affine algebra[23, ?] This relates to recent progress in this subject by Fasel-Rao-Swan on aquestion of Suslin, and also reinterprets the earlier results of Suslin in light1of the modern approach proposed.If X = Spec(R) is a smooth affine threefold over a field k of characteristicdifferent from 2 then the elementary symplectic Witt group WE (R) has a nicecohomological description as H2 (X, K3 ). Using Bloch-Kato for K1 , K2 it is shown that this is a divisible group prime to the characteristic. The followingcancellation theorem is obtained from this result: If X is a non-singular affinevariety of dimension d over an algebraically closed field k, and if char(k) ≥ dthen any stably trivial vector bundle of rank (d − 1) over X is trivial. Thehypothesis that X is non-singular can be weakened to X is normal if d ≥ 4.The general case when X is singular is also discussed.The new idea behind these results is the use of the theory of Grothendieck-Witt groups (aka Hermitian K-theory). Talks will be given by Fasel, Schlicht-ing on this topic. Rao will talk on some applications.There will also be talks on the recent work of J.Fasel and Aravind Asokon classification of projective modules over affine threefolds, and some talksby Aravind Asok on A1 -homotopy.

  3. The Bass-Suslin conjecture
    The Bass-Suslin conjectureRecent progress on this long standing conjecture by Rao-Swan in [25] willbe described by Rao. The connection with the theory of Suslin matrices willalso be be outlined. Talks will be given by Rao.There will be six to seven streams of 4-6 lectures each, one stream oftutorials, and a Conference for the last 2 or 3 days. The number of lecturesper day will not exceed 5.

Possible invited speakers

  1. Localization techniques (N. Vavilov, A. Stepanov )
  2. Stabilization and Prestabilization (N. Vavilov, R.A. Rao)
  3. K-theory of Exceptional groups. (N. Vavilov,  N. Petrov)
  4. Witt and Grothendieck groups (B. Calme`s)
  5. Higher Grothendieck-Witt groups (M. Schlichting)
  6. Recent developments on classification of projective modules over an affine algebra (J. Fasel, Aravind Asok , perhaps R.A. Rao)
  7. A1 -homotopy and classification problems (Aravind Asok )
  8. Suslin matrices, Bass-Suslin conjecture (R.A. Rao)



  • [1] A. Asok, J. Fasel, A cohomological classification of vector bundles on smooth affine threefolds, preprint.
  • [2] A. Asok, J. Fasel, Algebraic vector bundles on spheres, preprint.
  • [3] A. Bak, Nonabelian K-theory: the nilpotent class of K1 and general stability, K-Theory 4 (1991), pp 363–397.
  • [4] A. Bak, R. Basu, R. A. Rao, Local-global principle for transvection groups, accepted in Proc. Amer. Math. Soc. arXiv:0908.3094v2 [math.AC].
  • [5] A. Bak, R. Hazrat and N. Vavilov, Localization completion strikes again: relative K1 is nilpotent by abelian. J. Pure Appl. Algebra 213 (2009), pp 1075–1085.
  • [6] Balmer, Paul Witt groups. Handbook of K-theory. Vol. 1, 2, 539576, Springer, Berlin, 2005.
  • [7] H. Bass, K-theory and stable algebra. Inst. Hautes Etudes Sci., Publ. Math.,22 (1964), 5–60.New York, 1968.
  • [8] H. Bass, J. Milnor, J.-P. Serre, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2). Publ. Math. Inst. Hautes Etudes Sci. 33 (1967), 59–137.
  • [9] R. Basu, Local-global principle for quadratic and hermitian groups and the nilpotency of K1 . (2011), to appear.
  • [10] R. Basu, R. A. Rao, R. Khanna, On Quillen’s local global principle, Commutative algebra and algebraic geometry, 17–30, Contemp. Math., 390, Amer. Math. Soc., Providence, RI, 2005.
  • [11] Calms, Baptiste; Hornbostel, Jens Push-forwards for Witt groups of schemes.Comment. Math. Helv. 86 (2011), no. 2, 437468.
  • [12] nuradha Garge, R.A. Rao, A nice group structure on the orbit space of uni-modular rows, K-theory 38, No. 2, 2008, 113–133.
  • [13] njan Gupta, Decrease in injective stability for symplectic groups, preprint.
  • [14] A. J. Hahn and O. T. O’Meara. The Classical Groups and K-Theory, Springer, 1989.
  • [15] R. Hazrat, A. Stepanov, N. Vavilov, Z. Zhang, The yoga of commutators, J.Math. Sci. 387 (2011), 53–82.
  • [16] R. Hazrat, N. Vavilov, Bak’s work on the K-theory of rings, with an appendix by Max Karoubi, J. K-Theory, 4 (2009), 1–65.
  • [17] R. Hazrat, N. Vavilov, Z. Zhang, Relative unitary commutator calculus, and applications, J. Algebra, to appear.
  • [18] Hornbostel, Jens Constructions and dvissage in Hermitian K-theory. K-Theory 26 (2002), no. 2, 139170.
  • [19] Hornbostel, Jens; Schlichting, Marco Localization in Hermitian K-theory of rings. J. London Math. Soc. (2) 70 (2004), no. 1, 77124.
  • [20] Selby Jose, R.A. Rao, A structure theorem for the elementary unimodular vector group, Trans. Amer. Math. Soc. 358, (2006), no. 7, 3097–3112.
  • [21] S. Jose, R. A. Rao, A fundamental property of Suslin matrices, J. K-Theory 5 (2010), no. 3, 407436.
  • [22] J. Fasel, Mennicke symbols, K=cohomology and a Bass-Kubota theorem, preprint.
  • [23] J. Fasel, R. A. Rao, R. G. Swan, On stably free modules over affine algebras, arXiv:1107.1051.
  • [24] J. Fasel, R. A. Rao, R. G. Swan, The study of Projective modules over an affine algebra, in preparation.
  • [25] J. Fasel, R. A. Rao, R. G. Swan On some actions of stably elementary matrices on alternating matrices, in preparation.
  • [26] M. Karoubi, in Algebraic K-theory, III: Hermitian K-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 301411. Lecture Notes in Math., 343, Springer, Berlin, 1973.
  • [27] M. Nori,; R. A. Rao, R. Swan, Non-self-dual stably free modules, Quadratic forms, linear algebraic groups, and cohomology, 315324, Dev. Math., 18, Springer, New York, 2010.
  • [28] V. A. Petrov, Odd unitary groups, J. Math. Sci. 130 (2003), no. 3, 4752–4766.
  • [29] V. A. Petrov, Overgroups of classical groups, Doktorarbeit Univ. St.Petersburg 2005, 1–129 (in Russian).
  • [30] V. A. Petrov, A. K. Stavrova, Elementary subgroups of isotropic reductive groups, St. Petersburg Math. J. 20 (2008), no. 3, 160–188.
  • [31] D. Quillen, Projective modules over polynomial rings Invent. Math. 36 (1976) 167–171.
  • [32] R. A. Rao, The Bass-Quillen conjecture in dimension three but characteristic = 2, 3 via a question of A. Suslin. Invent. Math. 93 (1988), no. 3, 609–618.
  • [33] R. A. Rao, W. van der Kallen, Improved stability for K1 and WMSd of a non-singular affine algebra, K-theory (Strasbourg, 1992), Asterisque no. 226 (1994), 11, 411420.
  • [34] R.A. Rao, R.G. Swan, A Regenerative Property of a Fibre of Invertible Alternating Matrices, (see Swan’s homepage for a preview).
  • [35] A. K. Stavrova, Structure of isotropic reductive groups, Doktorarbeit Univ. St.-Petersburg 2009, 1–158 (in Russian).
  • [36] M. R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971) no.4, 965–1004.
  • [37] M. R. Stein, Relativising functors on rings and algebraic K-theory, J. Algebra 19 (1971), no. 1, 140–152.
  • [38] M. R. Stein, Stability theorems for K1 , K2 and related functors modeled on Chevalley groups, Japan. J. Math., 4 (1978), no.1, 77–108.
  • [39] R. Steinberg, Lectures on Chevalley groups, Yale University, 1967.
  • [40] A.Stepanov, Universal localisation in algebraic groups. publicat.html (2010) to appear.
  • [41] A. Stepanov, N. Vavilov, Decomposition of transvections: a theme with variations. K-theory, 19 (2000), pp 109–153.
  • [42] Schlichting, Marco A note on K-theory and triangulated categories. Invent.Math. 150 (2002), no. 1, 111116.
  • [43] Schlichting, Marco The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes. Invent. Math. 179 (2010), no. 2, 349433.
  • [44] Schlichting, Marco Hermitian K-theory of exact categories. J. K-Theory 5 (2010), no. 1, 105165.[45] Suslin, A. A. Projective modules over polynomial rings are free. (Russian) Dokl. Akad. Nauk SSSR 229 (1976), no. 5, 10631066.
  • [46] Suslin, A. A. Stably free modules. (Russian) Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537550, 632.
  • [47] A. A. Suslin, On the structure of the special linear group over the ring of polynomials, Izv. Akad. Nauk SSSR, Ser. Mat. 141 (1977) no.2, 235–253.
  • [48] Suslin, A. A.; Tulenbaev, M. S. A theorem on stabilization for Milnor’s K2- functor. (Russian) Rings and modules. Zap. Naun. Sem. Leningrad. Otdel.Mat. Inst. Steklov. (LOMI) 64 (1976), 131152, 162. (Reviewer: A. A. Ranicki) 18F25 (12A65)
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  • [50] Suslin, A. A. The cancellation problem for projective modules, and related questions. (Russian) Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 323330, Acad. Sci. Fennica, Helsinki, 1980.
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  • [53] L. N. Vaserstein, On the normal subgroups of GLn over a ring. Lecture Notes in Math. 854 (1981), pp 456–465.


Selected Applicants

Sr.No Name Affilation
1 Ambily Asokan  ISI, Bangalore
2 Simon Markett University of War-wick, UK
3 Heng Xie University of Warwick, Uk
4 Andrei Smolensky St. Petersburg University, Russia
5 Andrei Lavrenov St. Petersburg University, Russia
6 Anand Sawant TIFR, Mumbai
7 Vineeth Reddy TIFR, Mumbai
8 Anjan Gupta TIFR, Mumbai
9 Bhamidi Sai Somanjana Sreedhar TIFR, Mumbai
10 Charanya Ravi TIFR, Mumbai
11 Mandira Mondal TIFR, Mumbai
12 Vivek Sadhu IIT, Powai
13 Husney Parvez Sarwar IIT, Powai
14 Sumit Kumar Upadhyay MNIIT, Allahabad
15 Bibekananda Mishra University of Warwick, UK
1 Sergey Sinchuk St. Petersburg University, Russia
2 Yong Yang National University of Defense and Techonlogy, Changsha
3 Girija Shankar Tripathi TIFR, Mumbai
4 Umesh K. V. Dubey TIFR, Mumbai
5 Sagar Kolte KIAS, Seoul, Korea
6 Shameek Paul CBS Mumbai
7 Sarang Sane Kansas State University
8 Pratyusha Chattopadhyay ISI, Kolkata
9 Rajneesh Kumar Singh University of  Muenster
1 Manoj Keshari IIT, Powai
2 Mrinal Kanti Das ISI, Kolkata
3 Rabeya Basu IISER, Pune
4 Neena Gupta ISI, Kolkata
5 Alexander Luzgarev St. Petersburg University, Russia
6 Anuradha Garge University of Mumbai
7 Shripad Garge IIT, Powai
8 Selby Jose Institute of Science, Mumbai
9 Vivek Mallick IISER, Pune
List of invited teaching Faculty participants
1 Jean Fasel University of Essen Duisburg
2 Marco Schlichting University of Warwick
3 Baptiste Calmes University D’Artois, France
4 Aravind Asok University of Southern California (to be confirmed)
5 Mathias Wendt Universit ̈t Freiburg, Germany (to be confirmed)
6 Alexei Stepanov St. Petersburg University, Russia
Faculty visitors for the program
1  S.M. Bhatwadekar Bhaskaracharya Prathisthana, Pune
2  V.I. Kopeiko Kalmyckia State University, Russia
3  Wilberd van der Kallen University of Utrecht, Holland
4  Satyagopal Mandal University of Kansas at Lawrence
6  V. Suresh University of Hyderabad, India
6 Parimala Raman Emory University, USA
7 R.C. Cowsik CBS, Mumbai
8 Vikram Mehta  IIT, Powai
List of Distinguished visiting Faculty to T.I.F.R. who will participate
1  Nikolai Vavilov St. Petersburg University, Russia
Several other Faculty members, and academic members of the School of Mathematics are also expected to participate and enjoy this workshop.