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Noncommutative curves and noncommutative surfaces. (English) Zbl 1042.16016
Summary: In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of Abelian categories.
Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic, growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has led to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

MSC:
16S38 Rings arising from noncommutative algebraic geometry
14A22 Noncommutative algebraic geometry
16P90 Growth rate, Gelfand-Kirillov dimension
16W50 Graded rings and modules (associative rings and algebras)
18E15 Grothendieck categories (MSC2010)
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References:
[1] K. Ajitabh, Residue complex for Sklyanin algebras of dimension three, Adv. Math., 144 (1999), 137-160. CMP 99:14 · Zbl 1006.16033
[2] M. Artin, Some problems on three-dimensional graded domains, Representation theory and algebraic geometry (Waltham, MA, 1995) London Math. Soc. Lecture Note Ser., vol. 238, Cambridge Univ. Press, Cambridge, 1997, pp. 1 – 19. · Zbl 0888.16025
[3] Michael Artin and William F. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), no. 2, 171 – 216. · Zbl 0633.16001 · doi:10.1016/0001-8708(87)90034-X · doi.org
[4] M. Artin and J. T. Stafford, Noncommutative graded domains with quadratic growth, Invent. Math. 122 (1995), no. 2, 231 – 276. · Zbl 0849.16022 · doi:10.1007/BF01231444 · doi.org
[5] -, Semiprime graded algebras of dimension two, J. Algebra, 277 (2000). 68-123. CMP 2000:12 · Zbl 0967.16021
[6] M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33 – 85. · Zbl 0744.14024
[7] M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), no. 2, 335 – 388. · Zbl 0763.14001 · doi:10.1007/BF01243916 · doi.org
[8] M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), no. 2, 249 – 271. · Zbl 0717.14001 · doi:10.1016/0021-8693(90)90269-T · doi.org
[9] M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228 – 287. · Zbl 0833.14002 · doi:10.1006/aima.1994.1087 · doi.org
[10] -, Abstract Hilbert schemes, Algebr. Represent. Theory, to appear. · Zbl 1030.14003
[11] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601
[12] Sergey Barannikov and Maxim Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 (1998), 201 – 215. · Zbl 0914.58004 · doi:10.1155/S1073792898000166 · doi.org
[13] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023
[14] Arnaud Beauville, Surfaces algébriques complexes, Société Mathématique de France, Paris, 1978 (French). Avec une sommaire en anglais; Astérisque, No. 54. · Zbl 0394.14014
[15] Alexandre Beĭlinson and Joseph Bernstein, Localisation de \?-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15 – 18 (French, with English summary). · Zbl 0476.14019
[16] A. I. Bondal, Noncommutative deformations and Poisson brackets on projective spaces, MPI-preprint, 1993.
[17] A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183 – 1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519 – 541. · Zbl 0703.14011
[18] A. I. Bondal and D. O. Orlov, Semi-orthogonal decompositions for algebraic varieties, MPI preprint, 1996.
[19] A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras: the method of helices, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3 – 50 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 2, 219 – 260. · Zbl 0847.16010 · doi:10.1070/IM1994v042n02ABEH001536 · doi.org
[20] A. I. Bondal and M. Van den Bergh, in preparation.
[21] P. M. Cohn, Algebra. Vol. 1, 2nd ed., John Wiley & Sons, Ltd., Chichester, 1982. · Zbl 0481.00001
[22] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323 – 448 (French). · Zbl 0201.35602
[23] I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 5 – 19 (French). · Zbl 0144.02104
[24] K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. · Zbl 0679.16001
[25] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119 – 221 (French). · Zbl 0118.26104
[26] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[27] Timothy J. Hodges, Morita equivalence of primitive factors of \?(\?\?(2)), Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 175 – 179. · Zbl 0814.17008 · doi:10.1090/conm/139/1197835 · doi.org
[29] Peter Jørgensen, Non-commutative graded homological identities, J. London Math. Soc. (2) 57 (1998), no. 2, 336 – 350. · doi:10.1112/S0024610798006164 · doi.org
[30] -, Intersection theory on noncommutative surfaces, Trans. Amer. Math. Soc., 352 (2000), 5817-5854. CMP 99:14
[31] P. Jorgensen and J. J. Zhang, Gourmet Guide to Gorensteinness, Adv. Math., 151 (2000), 313-345. CMP 2000:12
[32] M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998), 73 – 118. · Zbl 0918.14001 · doi:10.1515/crll.1998.122 · doi.org
[33] A. Kapustin, A. Kuznetsov, D. Orlov, Noncommutative Instantons and Twistor Transform, hep-th/0002193, 2000. · Zbl 0989.81127
[34] D. S. Keeler, Criteria for \(\sigma\)-ampleness, J. Amer. Math. Soc., 13 (2000), 517-532. CMP 2000:12 · Zbl 0952.14002
[35] Bernhard Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994), no. 2, 193 – 198. · Zbl 0826.18004 · doi:10.1007/BF02567453 · doi.org
[36] -, Introduction to \(A_\infty\) algebras and modules, to appear; math.RA/9910179, 1999.
[37] G. M. Kelly and Ross Street, Review of the elements of 2-categories, Category Seminar (Proc. Sem., Sydney, 1972/1973) Springer, Berlin, 1974, pp. 75 – 103. Lecture Notes in Math., Vol. 420.
[38] Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293 – 344. · Zbl 0146.17001 · doi:10.2307/1970447 · doi.org
[39] M. Kontsevich and A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996-1999, 85-108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000. CMP 2000:17
[40] G. R. Krause and T. H. Lenagan, Growth of algebras and Gel\(^{\prime}\)fand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0564.16001
[41] L. Le Bruyn, Noncommutative geometry at \(n\), math.AG/9904171, 1999.
[42] -, Non-commutative compact manifolds constructed from quivers., AMA Algebra Mpntp. Announc., 1999, No.1. CMP 2000:07
[43] Lieven Le Bruyn, S. P. Smith, and Michel Van den Bergh, Central extensions of three-dimensional Artin-Schelter regular algebras, Math. Z. 222 (1996), no. 2, 171 – 212. · Zbl 0876.17019 · doi:10.1007/PL00004532 · doi.org
[44] Thierry Levasseur and S. Paul Smith, Modules over the 4-dimensional Sklyanin algebra, Bull. Soc. Math. France 121 (1993), no. 1, 35 – 90 (English, with English and French summaries). · Zbl 0823.17020
[45] Yu. I. Manin and M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974. Translated from the Russian by M. Hazewinkel; North-Holland Mathematical Library, Vol. 4.
[46] Yu. I. Manin, Quantum groups and noncommutative geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988. · Zbl 0724.17006
[47] Yuri I. Manin, Topics in noncommutative geometry, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991. · Zbl 0724.17007
[48] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. · Zbl 0644.16008
[49] Barry Mitchell, Rings with several objects, Advances in Math. 8 (1972), 1 – 161. · Zbl 0232.18009 · doi:10.1016/0001-8708(72)90002-3 · doi.org
[50] C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. · Zbl 0494.16001
[51] Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205 – 236. · Zbl 0864.14008
[52] A. V. Odesskii and B. L. Feigin, Sklyanin algebras associated with an elliptic curve, preprint, Institute for Theoretical Physics, Kiev, 1989.
[53] A. V. Odesskiĭ and B. L. Feĭgin, Sklyanin’s elliptic algebras, Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, 45 – 54, 96 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 3, 207 – 214 (1990). · Zbl 0713.17009 · doi:10.1007/BF01079526 · doi.org
[54] D. M. Patrick, Noncommmutative Ruled Surfaces, Ph.D. Thesis, MIT, June 1997; www.math.washington.edu/ patrick/prof.html.
[55] -, Noncommutative symmetric algebras of two-sided vector spaces, J. Algebra 223 (2000), 16-36. · Zbl 1004.16022
[56] I. Reiner, Maximal orders, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1975. London Mathematical Society Monographs, No. 5. · Zbl 0305.16001
[57] I. Reiten and M. Van den Bergh, Noetherian hereditary categories satisfying Serre duality, to appear; math.RT/9911242, 1999. · Zbl 0991.18009
[58] Richard Resco, A dimension theorem for division rings, Israel J. Math. 35 (1980), no. 3, 215 – 221. · Zbl 0437.16014 · doi:10.1007/BF02761192 · doi.org
[59] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. · Zbl 0546.16013
[60] Alexander L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its Applications, vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. · Zbl 0839.16002
[61] Alexander L. Rosenberg, The spectrum of abelian categories and reconstruction of schemes, Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996) Lecture Notes in Pure and Appl. Math., vol. 197, Dekker, New York, 1998, pp. 257 – 274. · Zbl 0898.18005
[62] Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0461.16001
[63] A. H. Schofield, Stratiform simple Artinian rings, Proc. London Math. Soc. (3) 53 (1986), no. 2, 267 – 287. · Zbl 0605.16010 · doi:10.1112/plms/s3-53.2.267 · doi.org
[64] J. P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197-278. · Zbl 0067.16201
[65] I. R. Shafarevitch, Basic Algebraic Geometry I, Springer Verlag, Berlin, 1994.
[66] B. Shelton and M. Vancliff, Embedding a quantum rank three quadric in a quantum \(\mathbb{ P} ^3\), Comm. Algebra 27 (1999), 2877-2904. CMP 99:12 · Zbl 0936.16023
[67] E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 4, 27 – 34, 96 (Russian).
[68] E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, 34 – 48 (Russian). · Zbl 0536.58007
[69] L. W. Small, J. T. Stafford, and R. B. Warfield Jr., Affine algebras of Gel\(^{\prime}\)fand-Kirillov dimension one are PI, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 3, 407 – 414. · Zbl 0561.16005 · doi:10.1017/S0305004100062976 · doi.org
[70] L. W. Small and R. B. Warfield Jr., Prime affine algebras of Gel\(^{\prime}\)fand-Kirillov dimension one, J. Algebra 91 (1984), no. 2, 386 – 389. · Zbl 0545.16011 · doi:10.1016/0021-8693(84)90110-8 · doi.org
[71] S. P. Smith, The four-dimensional Sklyanin algebras, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I (Antwerp, 1992), 1994, pp. 65 – 80. · Zbl 0809.16051 · doi:10.1007/BF00962090 · doi.org
[72] S. P. Smith and J. T. Stafford, Regularity of the four-dimensional Sklyanin algebra, Compositio Math. 83 (1992), no. 3, 259 – 289. · Zbl 0758.16001
[73] S. Paul Smith and J. M. Staniszkis, Irreducible representations of the 4-dimensional Sklyanin algebra at points of infinite order, J. Algebra 160 (1993), no. 1, 57 – 86. · Zbl 0809.16052 · doi:10.1006/jabr.1993.1178 · doi.org
[74] S. P. Smith and J. J. Zhang, Curves on noncommutative schemes, Algebr. Represent. Theory 1 (1998), 311-351. CMP 99:11
[75] J. T. Stafford, Regularity of algebras related to the Sklyanin algebra, Trans. Amer. Math. Soc. 341 (1994), no. 2, 895 – 916. · Zbl 0823.17018
[76] J. T. Stafford and J. J. Zhang, Examples in non-commutative projective geometry, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 415 – 433. · Zbl 0821.16026 · doi:10.1017/S0305004100072716 · doi.org
[77] J. T. Stafford and J. J. Zhang, Homological properties of (graded) Noetherian \?\? rings, J. Algebra 168 (1994), no. 3, 988 – 1026. · Zbl 0812.16046 · doi:10.1006/jabr.1994.1267 · doi.org
[78] Bo Stenström, Rings of quotients, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217; An introduction to methods of ring theory. · Zbl 0296.16001
[79] D. R. Stephenson, Artin-Schelter regular algebras of global dimension three, J. Algebra 183 (1996), no. 1, 55 – 73. · Zbl 0868.16027 · doi:10.1006/jabr.1996.0207 · doi.org
[80] Darin R. Stephenson, Algebras associated to elliptic curves, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2317 – 2340. · Zbl 0868.16028
[81] D. R. Stephenson and J. J. Zhang, Noetherian connected graded algebras of global dimension \(3\), J. Algebra, 230 (2000), 474-495. CMP 2000:16
[82] John Tate and Michel van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), no. 1-3, 619 – 647. · Zbl 0876.17010 · doi:10.1007/s002220050065 · doi.org
[83] R. W. Thomason and Thomas Trobaugh, Higher algebraic \?-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247 – 435. · Zbl 0731.14001 · doi:10.1007/978-0-8176-4576-2_10 · doi.org
[84] Michaela Vancliff and Kristel Van Rompay, Embedding a quantum nonsingular quadric in a quantum \?³, J. Algebra 195 (1997), no. 1, 93 – 129. · Zbl 0910.16013 · doi:10.1006/jabr.1997.7077 · doi.org
[85] M. Vancliff, K. Van Rompay, and L. Willaert, Some quantum \?³s with finitely many points, Comm. Algebra 26 (1998), no. 4, 1193 – 1208. · Zbl 0915.16035 · doi:10.1080/00927879808826193 · doi.org
[86] Michel Van den Bergh, A translation principle for the four-dimensional Sklyanin algebras, J. Algebra 184 (1996), no. 2, 435 – 490. · Zbl 0876.17011 · doi:10.1006/jabr.1996.0269 · doi.org
[87] -, Blowing-up of noncommutative smooth surfaces, Mem. Amer. Math. Soc., to appear; math.QA/980911, 1998.
[88] -, Noncommutative quadrics, in preparation. · Zbl 1311.14003
[89] -, Some noncommutative birational transformations, in preparation. · Zbl 1371.14005
[90] Martine Van Gastel and Michel Van den Bergh, Graded modules of Gelfand-Kirillov dimension one over three-dimensional Artin-Schelter regular algebras, J. Algebra 196 (1997), no. 1, 251 – 282. · Zbl 0902.16035 · doi:10.1006/jabr.1997.7072 · doi.org
[91] Freddy M. J. Van Oystaeyen and Alain H. M. J. Verschoren, Noncommutative algebraic geometry, Lecture Notes in Mathematics, vol. 887, Springer-Verlag, Berlin, 1981. An introduction. · Zbl 0477.16001
[92] Fred Van Oystaeyen and Luc Willaert, Examples and quantum sections of schematic algebras, J. Pure Appl. Algebra 120 (1997), no. 2, 195 – 211. · Zbl 0892.16020 · doi:10.1016/S0022-4049(96)00065-5 · doi.org
[93] A. B. Verëvkin, On a noncommutative analogue of the category of coherent sheaves on a projective scheme, Algebra and analysis (Tomsk, 1989) Amer. Math. Soc. Transl. Ser. 2, vol. 151, Amer. Math. Soc., Providence, RI, 1992, pp. 41 – 53. · Zbl 0920.14001 · doi:10.1090/trans2/151/02 · doi.org
[94] Q. S. Wu and J. J. Zhang, Some homological invariants of local PI algebras, J. Algebra 225 (2000), 904-935. CMP 2000:09
[95] Amnon Yekutieli and James J. Zhang, Serre duality for noncommutative projective schemes, Proc. Amer. Math. Soc. 125 (1997), no. 3, 697 – 707. · Zbl 0860.14001
[96] Amnon Yekutieli and James J. Zhang, Rings with Auslander dualizing complexes, J. Algebra 213 (1999), no. 1, 1 – 51. · Zbl 0948.16006 · doi:10.1006/jabr.1998.7657 · doi.org
[97] James J. Zhang, On lower transcendence degree, Adv. Math. 139 (1998), no. 2, 157 – 193. · Zbl 0924.16015 · doi:10.1006/aima.1998.1749 · doi.org
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