ATMW Classical Algebraic K  theory (2013)
Venue:  TIFR, Mumbai 
Dates:  15th 24 thJuly , 2013 
Convener(s)  Speakers, Syllabus and Time table  Applicants/Participants 
Name  Ravi A. Rao 
Jean Fasel 
Nikolai Vavilov 
Mailing Address 
Tata Institute of Fundamental Research (TIFR),

Fachbereich Mathematik

Department of Mathematics and Mechanics, Saint Petersburg State University, University prospect 28 Stary Peterhof, Saint Petersburg. 198504, Russia 
Brief description of the workshop:
 QuillenSuslinBak, localization and patching
QuillenSuslinBak, 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 problems over arbitrary commutative rings, to similar problems for semilocalrings. Localization comes in a number of versions. The two most familiar 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 classicallike 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.  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 FaselRaoSwan 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 BlochKato 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 nonsingular 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 nonsingular 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 GrothendieckWitt groups (aka Hermitian Ktheory). Talks will be given by Fasel, Schlichting 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.  The BassSuslin conjecture
The BassSuslin conjectureRecent progress on this long standing conjecture by RaoSwan 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 46 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
 Localization techniques (N. Vavilov, A. Stepanov )
 Stabilization and Prestabilization (N. Vavilov, R.A. Rao)
 Ktheory of Exceptional groups. (N. Vavilov, N. Petrov)
 Witt and Grothendieck groups (B. Calme`s)
 Higher GrothendieckWitt groups (M. Schlichting)
 Recent developments on classification of projective modules over an affine algebra (J. Fasel, Aravind Asok , perhaps R.A. Rao)
 A1 homotopy and classification problems (Aravind Asok )
 Suslin matrices, BassSuslin conjecture (R.A. Rao)
References
 [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 Ktheory: the nilpotent class of K1 and general stability, KTheory 4 (1991), pp 363–397.
 [4] A. Bak, R. Basu, R. A. Rao, Localglobal 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 Ktheory. Vol. 1, 2, 539576, Springer, Berlin, 2005.
 [7] H. Bass, Ktheory 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, Localglobal 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 Pushforwards 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 unimodular rows, Ktheory 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 KTheory, 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 Ktheory of rings, with an appendix by Max Karoubi, J. KTheory, 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 Ktheory. KTheory 26 (2002), no. 2, 139170.
 [19] Hornbostel, Jens; Schlichting, Marco Localization in Hermitian Ktheory 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. KTheory 5 (2010), no. 3, 407436.
 [22] J. Fasel, Mennicke symbols, K=cohomology and a BassKubota 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 Ktheory, III: Hermitian Ktheory 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, Nonselfdual 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 BassQuillen 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 nonsingular affine algebra, Ktheory (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 http://math.uchicago.edu/swan/ 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 Ktheory, 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. http://alexei.stepanov.spb.ru/ publicat.html (2010) to appear.
 [41] A. Stepanov, N. Vavilov, Decomposition of transvections: a theme with variations. Ktheory, 19 (2000), pp 109–153.
 [42] Schlichting, Marco A note on Ktheory and triangulated categories. Invent.Math. 150 (2002), no. 1, 111116.
 [43] Schlichting, Marco The MayerVietoris principle for GrothendieckWitt groups of schemes. Invent. Math. 179 (2010), no. 2, 349433.
 [44] Schlichting, Marco Hermitian Ktheory of exact categories. J. KTheory 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)
 [49] Suslin, A. A. A cancellation theorem for projective modules over algebras.(Russian) Dokl. Akad. Nauk SSSR 236 (1977), no. 4, 808811.
 [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.
 [51] Suslin, A. A. Cancellation for affine varieties. (Russian) Modules and algebraic groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 187195.
 [52] Suslin, A. A. Mennicke symbols and their applications in the Ktheory of fields. Algebraic Ktheory, Part I (Oberwolfach, 1980), pp. 334356, Lecture Notes in Math., 966, Springer, BerlinNew York, 1982.
 [53] L. N. Vaserstein, On the normal subgroups of GLn over a ring. Lecture Notes in Math. 854 (1981), pp 456–465.
Students  
Sr.No  Name  Affilation 
1  Ambily Asokan  ISI, Bangalore 
2  Simon Markett  University of Warwick, 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 
Postdocs  
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 
Faculty  
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. 