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Essential dimension: a survey

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Abstract

In the paper we survey research on the essential dimension. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. We present self-contained proofs of these cases and give applications in the theory of simple algebras and quadratic forms.

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References

  1. S. Amitsur, L. Rowen, J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), no. 2, 133–148.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Baek, Essential dimension of simple algebras with involutions, Bull. Lond. Math. Soc. 44 (2012), no. 3, 578–590.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Baek, Essential dimension of simple algebras in positive characteristic, C. R. Math. Acad. Sci. Paris 349 (2011), no. 7–8, 375–378.

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Baek, A. Merkurjev, Essential dimension of central simple algebras, Acta Mathematica 209 (2012), no. 1, 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Beauville, On finite simple groups of essential dimension 3, arXiv:1101.1372v2, 18 Jan. 2011.

  6. G. Berhuy, Z. Reichstein, On the notion of canonical dimension for algebraic groups, Adv. Math. 198 (2005), no. 1, 128–171.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330 (electronic).

    MathSciNet  MATH  Google Scholar 

  8. P. Brosnan, The essential dimension of a g-dimensional complex abelian variety is 2g, Transform. Groups 12 (2007), no. 3, 437–441.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), no. 1, 533–544.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension of moduli of curves and other algebraic stacks, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1079–1112, With an appendix by Najmuddin Fakhruddin.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Brosnan, R. Sreekantan, Essential dimension of abelian varieties over number fields, C. R. Math. Acad. Sci. Paris 346 (2008), no. 7–8, 417–420.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159–179.

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Chernousov, S. Gille, A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (2005), no. 1, 137–159.

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Chernousov, A. Merkurjev, Essential p-dimension of split simple groups of type A n , Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 429, Apr. 15, 2011, to appear in Math. Annalen.

  15. V. Chernousov, A. Merkurjev, Essential dimension of spinor and Clifford groups, Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 455, Jan. 2, 2012.

  16. V. Chernousov, J.-P. Serre, Lower bounds for essential dimensions via orthogonal representations, J. Algebra 305 (2006), no. 2, 1055–1070.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Chu, S.-J. Hu, M.-C. Kang, J. Zhang, Groups with essential dimension one, Asian J. Math. 12 (2008), no. 2, 177–191.

    MathSciNet  MATH  Google Scholar 

  18. Ж.-Л. Кольо-Телен, Н. А. Карпенко, А. C. Меркурьев, Рациональные поверхносми и каноническая размерносмь группы PGL 6, Алгебра и Анализ 19 (2007), no. 5, 159–178. Engl. transl.: J.-L. Colliot-Thélǹe, N. Karpenko, A. Merkurjev, Rational surfaces and the canonical dimension of the group PGL 6, St. Petersburg Math. J. 19 (2008), no. 5, 793–804.

  19. M. Demazure, A. Grothendieck, Schémas en Groupes, SGA 3, Vol. 1, Springer-Verlag, Berlin, 1970.

    Google Scholar 

  20. A. Duncan, Z. Reichstein, Versality of algebraic group actions and rational points on twisted varieties, preprint, 2012.

  21. A. Duncan, Finite groups of essential dimension 2, to appear in Comment. Math. Helv.

  22. A. Duncan, Essential dimensions of A 7 and S 7, Math. Res. Lett. 17 (2010), no. 2, 263–266.

    MathSciNet  MATH  Google Scholar 

  23. R. Elman, N. Karpenko, A. Merkurjev, The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications, Vol. 56, American Mathematical Society, Providence, RI, 2008.

    MATH  Google Scholar 

  24. M. Florence, On the essential dimension of cyclic p-groups, Invent. Math. 171 (2008), no. 1, 175–189.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984.

    MATH  Google Scholar 

  26. R. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological Invariants in Galois Cohomology, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  27. R. Garibaldi, Cohomological Invariants: Exceptional Groups and Spin Groups, Mem. Amer. Math. Soc. 200 (2009), no. 937, xii+81, With an appendix by D. W. Hoffmann.

    MathSciNet  Google Scholar 

  28. Ph. Gille, Z. Reichstein, A lower bound on the essential dimension of a connected linear group, Comment. Math. Helv. 84 (2009), no. 1, 189–212.

    Article  MathSciNet  MATH  Google Scholar 

  29. I. Herstein, Noncommutative Rings, Mathematical Association of America, Washington, DC, 1994, Reprint of the 1968 original, with an afterword by L. W. Small.

    Google Scholar 

  30. В. В. Ишханов, Б. Б. Лурье, Д. К. Фаддеев, Задача noгрyженuя в меopии Галуа, Наука, M., 1990. Engl. transl.: V. Ishkhanov, B. Lur’e, D. Faddeev, The Embedding Problem in Galois Theory, Translations of Mathematical Monographs, Vol. 165, American Mathematical Society, Providence, RI, 1997.

  31. В. А. Исковских, Минимальные модели рациональных поверхностей над произвольными nолями, Изв. Акад. Наук CCCP, cep. мат 43 (1979), no. 1, 19–43, 237. Engl. transl.: V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary fields, Math. USSR, Izv. 14 (1980), 17–39.

    Google Scholar 

  32. B. Kahn, Comparison of some field invariants, J. Algebra 232 (2000), no. 2, 485–492.

    Article  MathSciNet  MATH  Google Scholar 

  33. M.-C. Kang, A central extension theorem for essential dimensions, Proc. Amer. Math. Soc. 136 (2008), no. 3, 809–813 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  34. Н. A. Карпенко, Чжoy-мотивы Гpoмeндикa мнoгooбpaзий Севери–Брауэра, АБлгeбра и анализ 7 (1995), no. 4, 196–213. Engl. transl.: N. Karpenko, Grothen-dieck Chow motives of Severi−Brauer varieties, St. Petersburg Math. J. 7 (1996), no. 4, 649–661.

  35. N. Karpenko, On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15 (2000), no. 1, 1–22.

    MathSciNet  MATH  Google Scholar 

  36. N. Karpenko, Canonical dimension of orthogonal groups, Transform. Groups 10 (2005), no. 2, 211–215.

    Article  MathSciNet  MATH  Google Scholar 

  37. N. Karpenko, A bound for canonical dimension of the (semi)spinor groups, Duke Math. J. 133 (2006), no. 2, 391–404.

    Article  MathSciNet  MATH  Google Scholar 

  38. N. Karpenko, Canonical dimension of (semi-)spinor groups of small ranks, Pure Appl. Math. Q. 4 (2008), no. 4, part 1, 1033–1039.

    MathSciNet  MATH  Google Scholar 

  39. N. Karpenko, Canonical dimension, in: Proceedings of the International Congress of Mathematicians, Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 146–161.

    Google Scholar 

  40. N. Karpenko, Incompressibility of orthogonal grassmannians, C. R. Math. Acad. Sci. Paris 349 (2011), no. 21–22, 1131–1134.

    Article  MathSciNet  MATH  Google Scholar 

  41. N. Karpenko, Incompressibility of quadratic Weil transfer of generalized Severi− Brauer varieties, J. Inst. Math. Jussieu 11 (2012), no. 1, 119–131.

    Article  MathSciNet  MATH  Google Scholar 

  42. N. Karpenko, Upper motives of algebraic groups and incompressibility of Severi− Brauer varieties, to appear in J. Reine Angew. Math.

  43. N. Karpenko, Unitary grassmannians, Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 424, Mar. 17, 2011, J. Pure Appl. Algebra 216 (2012), no. 12, 2586–2600.

  44. N. Karpenko, Incompressibility of generic torsors of norm tori, Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 475, Aug. 8, 2012.

  45. N. Karpenko, A. Merkurjev, Essential dimension of quadrics, Invent. Math. 153 (2003), no. 2, 361–372.

    Article  MathSciNet  MATH  Google Scholar 

  46. N. Karpenko, A. Merkurjev, Canonical p-dimension of algebraic groups, Adv. Math. 205 (2006), no. 2, 410–433.

    Article  MathSciNet  MATH  Google Scholar 

  47. N. Karpenko, A. Merkurjev, Essential dimension of finite p-groups, Invent. Math. 172 (2008), no. 3, 491–508.

    Article  MathSciNet  MATH  Google Scholar 

  48. M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, American Mathematical Society, Providence, RI, 1998, with a preface in French by J. Tits.

    MATH  Google Scholar 

  49. В. Э. Kopдoнский, О существенной размерности и гипотезе Серра II для исключuмельных групп, Мат. заметки 68 (2000), no. 4, 539–547. Engl. transl.: V. È. Kordonskiĭ, On the essential dimension and Serre’s conjecture II for exceptional groups, Mathematical Notes 68 (2000), no. 4, 464–470.

  50. H. Kraft, R. Lötscher, G. Schwarz, Compression of finite group actions and covariant dimension. II, J. Algebra 322 (2009), no. 1, 94–107.

    Article  MathSciNet  MATH  Google Scholar 

  51. H. Kraft, G. Schwarz, Compression of finite group actions and covariant dimension, J. Algebra 313 (2007), no. 1, 268–291.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Ledet, On the essential dimension of some semi-direct products, Canad. Math. Bull. 45 (2002), no. 3, 422–427.

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Ledet, On p-groups in characteristic p, in: Algebra, Arithmetic and Geometry with Applications, (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 591–600.

    Google Scholar 

  54. A. Ledet, On the essential dimension of p-groups, in: Galois Theory and Modular Forms, Dev. Math., Vol. 11, Kluwer, Boston, MA, 2004, pp. 159–172.

    Chapter  Google Scholar 

  55. A. Ledet, Finite groups of essential dimension one, J. Algebra 311 (2007), no. 1, 31–37.

    Article  MathSciNet  MATH  Google Scholar 

  56. N. Lemire, Essential dimension of algebraic groups and integral representations of Weyl groups, Transform. Groups 9 (2004), no. 4, 337–379.

    Article  MathSciNet  MATH  Google Scholar 

  57. M. Lorenz, Z. Reichstein, L. Rowen, D. Saltman, Fields of definition for division algebras, J. London Math. Soc. (2) 68 (2003), no. 3, 651–670.

    Article  MathSciNet  MATH  Google Scholar 

  58. R. Lötscher, Application of multihomogeneous covariants to the essential dimension of finite groups, Transform. Groups 15 (2010), no. 3, 611–623.

    Article  MathSciNet  MATH  Google Scholar 

  59. R. Lötscher, M. MacDonald, A. Meyer, Z. Reichstein, Essential p-dimension of algebraic tori, to appear in J. Reine Angew. Math.

  60. R. Lötshcer, A fiber dimension theorem for essential and canonical dimension, Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 454, Dec. 16, 2011.

  61. M. MacDonald, Essential p-dimension of the normalizer of a maximal torus, Transform. Groups 16 (2011), no. 4, 1143–1171.

    Article  MathSciNet  MATH  Google Scholar 

  62. M. MacDonald, Upper bounds for the essential dimension of E 7, Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 449, Nov. 16, 2011.

  63. Ю. Манин, Рациональные поверхности над сoвершнными полями. II, Мaт. cб 72 (114) (1967), 161–192. Engl. transl.: Yu. Manin, Rational surfaces over perfect fields. II, Math. of the USSR-Sbornik 1 (1967), no. 2, 141–168.

  64. B. Mathews, Canonical dimension of projective PGL1(A)-homogeneous varieties, Linear Algebraic Groups and Related Structures (preprint server), http://www.math.uni-bielefeld.de/LAG/, preprint no. 332, Mar. 30, 2009.

  65. B. Mathews, Incompressibility of orthogonal Grassmannians of rank 2, J. Algebra 349 (2012), no. 1, 353–363.

    Article  MathSciNet  MATH  Google Scholar 

  66. B. Mathews, Canonical dimension of projective homogeneous varieties of inner type A and type B, ProQuest LLC, Ann Arbor, MI, 2009, Thesis (Ph.D.), University of California, Los Angeles.

  67. A. Merkurjev, Maximal indexes of Tits algebras, Doc. Math. 1 (1996), no. 12, 229–243 (electronic).

    MathSciNet  MATH  Google Scholar 

  68. A. Merkurjev, Steenrod operations and degree formulas, J. Reine Angew. Math. 565 (2003), 13–26.

    MathSciNet  MATH  Google Scholar 

  69. A. Merkurjev, R-equivalence on three-dimensional tori and zero-cycles, Algebra Number Theory 2 (2008), no. 1, 69–89.

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Merkurjev, Essential dimension, in: Quadratic Forms|Algebra, Arithmetic, and Geometry, Contemp. Math., Vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 299–325.

    Chapter  Google Scholar 

  71. A. Merkurjev, Essential p-dimension of PGL(p 2), J. Amer. Math. Soc. 23 (2010), no. 3, 693–712.

    Article  MathSciNet  MATH  Google Scholar 

  72. A. Merkurjev, A lower bound on the essential dimension of simple algebras, Algebra Number Theory 4 (2010), no. 8, 1055–1076.

    Article  MathSciNet  MATH  Google Scholar 

  73. А. С. Меркурьев, А. А. Суслин, K-когомологии многообpaзий Севери–Брауэра и гомоморфизм норменного вычета, Изв. АН СССР, cep. мат 46 (1982), no. 5, 1011–1046. Engl. transl.: A. Merkurjev, A. Suslin, K-cohomology of Severi−Brauer varieties and the norm residue homomorphism, Math. of the USSR-Izvestiya 21 (1983), 2, 307–340.

  74. A. Meyer, Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra Number Theory 3 (2009), no. 4, 467–487.

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Meyer, Z. Reichstein, Some consequences of the Karpenko−Merkurjev theorem, Doc. Math. (2010), no. Extra volume: Andrei A. Suslin sixtieth birthday, 445–457.

  76. A. Meyer, Z. Reichstein, An upper bound on the essential dimension of a central simple algebra, J. Algebra 329 (2011), 213–221.

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Milne, Étale Cohomology, Princeton University Press, Princeton, N.J., 1980. Russian transl.: Дж. Милн, Эмальныe когомологии, Миp, M., 1983.

  78. J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.

    Article  MathSciNet  MATH  Google Scholar 

  79. А. М. Попов, Конечные стационарные подгруппы общего положения простых линейных групп Ли, Тpyды Mocк. Мат. О-ва 48 (1985), 7–59. Engl. transl.: A. M. Popov, Finite isotropy subgroups in general position of simple linear Lie groups, Trans. Mosc. Math. Soc. 1986, 3–63.

  80. Yu. Prokhorov, Simple finite subgroups of the Cremona group of rank 3, J. Algebraic Geom. 21 (2012), 563–600.

    Article  MathSciNet  MATH  Google Scholar 

  81. D. Quillen, Higher algebraic K-theory. I, in: Algebraic K-theory, I: Higher K- theories (Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147.

    Google Scholar 

  82. Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.

    Article  MathSciNet  MATH  Google Scholar 

  83. Z. Reichstein, Essential dimension, Proceedings of the International Congress of Mathematicians, Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188.

    Google Scholar 

  84. Z. Reichstein, A. Vistoli, A genericity theorem for algebraic stacks and essential dimension of hypersurfaces, to appear in JEMS.

  85. Z. Reichstein, B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, Canad. J. Math. 52 (2000), no. 5, 1018–1056, with an appendix by J. Kollár and E. Szabó.

    Article  MathSciNet  MATH  Google Scholar 

  86. A. Ruozzi, Algebraic tori and essential dimension, Thesis (Ph.D.), University of California, Los Angeles.

  87. A. Ruozzi, Essential p-dimension of PGL n , J. Algebra 328 (2011), 488–494.

    Article  MathSciNet  MATH  Google Scholar 

  88. N. Semenov, K. Zainoulline, Essential dimension of Hermitian spaces, Math. Ann. 346 (2010), no. 2, 499–503.

    Article  MathSciNet  MATH  Google Scholar 

  89. J.-P. Serre, Cohomologie galoisienne: progrès et problèmes, Séminaire Bourbaki, Vol. 1993/94, Astérisque no. 227 (1995), Exp. no. 783, 4, 229–257.

  90. J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York, 1977.

    Book  MATH  Google Scholar 

  91. J.-P. Serre, Local Fields, Springer-Verlag, New York, 1979.

    MATH  Google Scholar 

  92. J.-P. Serre, Galois Cohomology, Springer-Verlag, Berlin, 1997.

    Book  MATH  Google Scholar 

  93. J.-P. Serre, Le groupe de Cremona et ses sous-groupes finis, Séminaire Bourbaki, Vol. 2008/2009, Astérisque, no. 332 (2010), Exp. no. 1000, 75–100.

  94. A. Suslin, S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1–2, 245–276.

    Article  MathSciNet  MATH  Google Scholar 

  95. R. Thomason, Algebraic K-theory of group scheme actions, in: Algebraic Topology and Algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., Vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563.

    Google Scholar 

  96. J.-P. Tignol, Algèbres indécomposables d’exposant premier, Adv. in Math. 65 (1987), no. 3, 205–228.

    Article  MathSciNet  MATH  Google Scholar 

  97. D. Tossici, A. Vistoli, On the essential dimension of infinitesimal group schemes, arXiv:1101.3988v3, 25 Oct. 2010.

  98. B. Totaro, Birational geometry of quadrics in characteristic 2, J. Algebraic Geom. 17 (2008), no. 3, 577–597.

    Article  MathSciNet  MATH  Google Scholar 

  99. A. Vishik, On the Chow groups of quadratic Grassmannians, Doc. Math. 10 (2005), 111–130 (electronic).

    MathSciNet  MATH  Google Scholar 

  100. A. Vishik, Direct summands in the motives of quadrics, preprint, 1999, available at http://www.maths.nott.ac.uk/personal/av/papers.html.

  101. W. Wong, On the essential dimension of cyclic groups, J. Algebra 334 (2011), 285–294.

    Article  MathSciNet  MATH  Google Scholar 

  102. K. Zainoulline, Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants, Bull. Lond. Math. Soc. 39 (2007), no. 2, 301–304.

    Article  MathSciNet  MATH  Google Scholar 

  103. O. Zariski, P. Samuel, Commutative Algebra, Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York, 1975, reprint of the 1960 edition. Russian transl.: O. Зарисский, П. Самюэль, Коммутативная алгебра, т. 2, ИЛ. M., 1963.

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  1. Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA

    Alexander S. Merkurjev

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*The work has been supported by the NSF grant DMS #1160206.

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Merkurjev, A.S. Essential dimension: a survey. Transformation Groups 18, 415–481 (2013). https://doi.org/10.1007/s00031-013-9216-y

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