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From Ohkawa to Strong Generation via Approximable Triangulated Categories—A Variation on the Theme of Amnon Neeman’s Nagoya Lecture Series

  • Norihiko MinamiEmail author
Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 309)

Abstract

This survey stems from Amnon Neeman’s lecture series at Ohakawa’s memorial workshop. Starting with Ohakawa’s theorem, this survey intends to supply enough motivation, background and technical details to read Neeman’s recent papers on his “approximable triangulated categories” and his \({{\,\mathrm{\mathbf {D}}\,}}^{b}_{\mathrm {coh}}(X)\) strong generation sufficient criterion via de Jong’s regular alteration, even for non-experts.

Keywords

Research exposition Derived categories Triangulated categories Stable homotopy theory Bouseld class Motivic homotopy theory 

2010 Mathematics Subject Classification

14-02 18-02 55-02 14F05 14F42 18E30 18G55 55P42 55N20 55U35 

References

  1. 1.
    Alonso Tarrío, L., Jeremías López, A., Souto Salorio, M.J.: Construction of t-structures and equivalences of derived categories. Trans. Am. Math. Society. 355(6), 2523–2543 (2003) MR1974001 (2004c:18020)Google Scholar
  2. 2.
    Alonso Tarrío, L., Jeremías López, A., Souto Salorio, M.J.: Bousfield localization on formal schemes. J. Algebra 278(2), 585–610 (2004). MR2071654 (2005g:14037)Google Scholar
  3. 3.
    Balmer, P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149–168 (2005). MR2196732Google Scholar
  4. 4.
    Balmer, P.: Supports and filtrations in algebraic geometry and modular representation theory. Am. J. Math. 129(5), 1227–1250 (2007). MR2354319 (2009d:18017)Google Scholar
  5. 5.
    Balmer, P.: Tensor triangular geometry. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 85–112. Hindustan Book Agency, New Delhi (2010). MR2827786 (2012j:18016)Google Scholar
  6. 6.
    Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J. Algebra 236(2), 819–834 (2001). MR1813503 (2002a:18013)Google Scholar
  7. 7.
    Balmer, P., Favi, G.: Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3) 102(6), 1161–1185 (2011). MR2806103 (2012d:18010)Google Scholar
  8. 8.
    Balmer, P., Sanders B.: The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math. 208(1), 283–326 (2017). MR3621837Google Scholar
  9. 9.
    Barthel, T.: A short introduction to the telescope and chromatic splitting conjectures, in this proceedingsGoogle Scholar
  10. 10.
    Barthel, T., Heard, D., Valenzuela, G.: The algebraic chromatic splitting conjecture for Noetherian ring spectra. Math. Z. 290(3–4), 1359–1375 (2018). MR3856857Google Scholar
  11. 11.
    Beaudry, A.: The chromatic splitting conjecture at \(n = p = 2\), Geom. Topol. 21(6), 3213–3230 (2017). MR3692966Google Scholar
  12. 12.
    Beaudry, A., Goerss, P.G., Henn, H.-W.: Chromatic splitting for the \(K(2)\)-local sphere at \(p=2\). arXiv:1712.08182
  13. 13.
    Behrens, M., Rezk, C.: Spectral algebra models of unstable \(v_n\)-periodic homotopy theory, in this proceedingsGoogle Scholar
  14. 14.
    Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and Topology on Singular Spaces, I (Luminy, 1981). Astérisque, vol. 100, pp. 5–171. Société Mathématique de France, Paris (1982). MR0751966 (86g:32015)Google Scholar
  15. 15.
    Benson, D. J., Carlson, J.F., Rickard, J., Complexity and varieties for infinitely generated modules. II, Math. Proc. Cambridge Philos. Soc. 120(4), 597–615 (1996). MR1401950 (97f:20008)Google Scholar
  16. 16.
    Benson, D., Iyengar, S.B., Krause, H., Pevtsova, J.: Stratification for module categories of finite group schemes. J. Am. Math. Society. 31(1), 265–302 (2018). MR3718455Google Scholar
  17. 17.
    Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86(2), 209–234 (1993). MR1214458 (94f:18008)Google Scholar
  18. 18.
    Bondal, A., Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125(3), 327–344 (2001). MR1818984 (2001m:18014)Google Scholar
  19. 19.
    Bondal, A., Orlov, D.: Derived categories of coherent sheaves. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. II, pp. 47–56. Higher Education Press, Beijing (2002). MR1818984Google Scholar
  20. 20.
    Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36, 258 (2003). MR1996800 (2004h:18009)Google Scholar
  21. 21.
    Bousfield, A.K.: The localization of spectra with respect to homology. Topology, 18(4), 257–281 (1979). MR551009 55N20 (55N15 55P60)Google Scholar
  22. 22.
    Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007). MR2373143 (2009c:14026)Google Scholar
  23. 23.
    Buan, A.B., Krause, H., Solberg, Ø.: Support varieties: an ideal approach. Homol. Homotopy Appl. 9(1), 45–74 (2007). MR2280286 (2008i:18007)Google Scholar
  24. 24.
    Burke, J., Neeman, A., Pauwels, B.: Gluing approximable triangulated categories. arxiv.1806.05342
  25. 25.
    Calabrese, J., Groechenig, M.: Moduli problems in abelian categories and the reconstruction theorem. Algebr. Geom. 2(1), 1–18 (2015). MR3322195Google Scholar
  26. 26.
    Casacuberta, C.: Depth and simplicity of Ohkawa’s argument, in this proceedingsGoogle Scholar
  27. 27.
    Casacuberta, C., Rosický, J.: Combinatorial homotopy categories, in this proceedingsGoogle Scholar
  28. 28.
    Casacuberta, C., Gutiérrez, J.J., Rosický, J.: A generalization of Ohkawa’s theorem. Compos. Math. 150(5), 893–902 (2014). MR3209799 55N20 18G55 55P42 55U40Google Scholar
  29. 29.
    Christensen, J.D.: Ideals in triangulated categories: phantoms, ghosts and skeleta. Adv. Math. 136(2), 284–339 (1998). MR1626856 (99g:18007)Google Scholar
  30. 30.
    Dao, H., Takahashi, R.: The dimension of a subcategory of modules. Forum Math. Sigma 3, e19 (2015) 31 pp. MR3482266Google Scholar
  31. 31.
    Dao, H., Takahashi, R.: Upper bounds for dimensions of singularity categories. Comptes Rendus Math. Acad. Sci. (Paris) 353(4), 297–301 (2015). MR3319124Google Scholar
  32. 32.
    de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Ht. Études Sci. Publ. Math. (83), 51–93 (1996). MR1423020 (98e:14011)Google Scholar
  33. 33.
    de Jong, A.J.: Families of curves and alterations. Ann. Inst. Fourier (Grenoble) 47(2), 599–621 (1997). MR1450427 (98f:14019)Google Scholar
  34. 34.
    Dell’Ambrogio, I., Stevenson, G.: On the derived category of a graded commutative Noetherian ring. J. Algebra 373, 356–376 (2013). MR2995031Google Scholar
  35. 35.
    Devinatz, E.S., Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. I. Ann. Math. (2) 128(2), 207–241 (1988). MR0960945 (89m:55009)Google Scholar
  36. 36.
    Douglas, M.R.: Dirichlet branes, homological mirror symmetry, and stability. In: Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. III, pp. 395–408. Higher Education Press, Beijing (2002). MR1957548Google Scholar
  37. 37.
    Dwyer, W.G., Palmieri, J.H., Ohkawa’s theorem: there is a set of Bousfield classes, Proc. Amer. Math. Soc. 129(3), 881-886 (2001). MR1712921 (2001f:55015)Google Scholar
  38. 38.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part II. AMS/IP Studies in Advanced Mathematics, vol. 46.2. American Mathematical Society, Providence; International Press, Somerville (2009). xii+396 pp. MR2548482 (2011c:53218)Google Scholar
  39. 39.
    Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I. AMS/IP Studies in Advanced Mathematics, vol. 46.1. American Mathematical Society, Providence; International Press, Somerville (2009). xii+396 pp. MR2553465 (2011c:53217)Google Scholar
  40. 40.
    Gabber, O.: Finiteness theorems for tale cohomology of excellent schemes. In: Conference in honor of P. Deligne on the occasion of his 61st birthday, IAS, Princeton, October 2005, p. 45 (2005)Google Scholar
  41. 41.
    Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. Fr. 90, 323–448 (1962). MR0232821 (38 #1144)Google Scholar
  42. 42.
    Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer, New York (1967) x+168 pp. MR0210125 (35 # 1019)Google Scholar
  43. 43.
    Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2003). xx+372 pp. MR1950475 (2003m:18001)Google Scholar
  44. 44.
    Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Ins. Ht. Études Sci. Publ. Math. 11, 5–167 (1961)Google Scholar
  45. 45.
    Grothendieck, A., Raynaud, M.: Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique. I.H.E.S (1963)Google Scholar
  46. 46.
    Hall, J.: GAGA theorems. arXiv:1804.01976
  47. 47.
    Hall, J., Rydh, D.: The telescope conjecture for algebraic stacks. J. Topol. 10(3), 776–794 (2017). MR3797596Google Scholar
  48. 48.
    Hopkins, M.: Global methods in homotopy theory. Homotopy Theory (Durham, 1985). London Mathematical Society Lecture Note Series, vol. 117, pp. 73–96. Cambridge University Press, Cambridge (1987). MR0932260 (89g:55022)Google Scholar
  49. 49.
    Hopkins, M.J., Gross, B.H.: The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory. Bull. Am. Math. Soc. (N.S.) 30(1), 7686 (1994). MR1217353 (94k:55009)Google Scholar
  50. 50.
    Hopkins, M.J., Smith, J.H.: Nilpotence and stable homotopy theory. II. Ann. Math. (2) 148(1), 1–49 (1998). MR1652975 (99h:55009)Google Scholar
  51. 51.
    Hovey, M.: Bousfield localization functors and Hopkins’ chromatic splitting conjecture. The Čech centennial, Boston, MA, 1993. Contemporary Mathematics, vol. 181, pp. 225–250. American Mathematical Society, Providence, RI (1995). MR1320994 (96m:55010)Google Scholar
  52. 52.
    Hovey, M.: Cohomological Bousfield classes. J. Pure Appl. Algebra 103(1), 45–59 (1995). MR1354066 (96g:55008)Google Scholar
  53. 53.
    Hovey, M., Palmieri, J.H.: The structure of the Bousfield lattice. Homotopy Invariant Algebraic Structures, Baltimore, MD, 1998, pp. 175–196. Contemporary Mathematics, vol. 239, 1999. MR1718080 (2000j:55033)Google Scholar
  54. 54.
    Hovey, M., Palmieri, J.H., Strickland, N.P.: Axiomatic Stable Homotopy Theory. Memoirs of the American Mathematical Society, vol. 128 (610), x+114 pp. (1997). MR1388895 (98a:55017)Google Scholar
  55. 55.
    Hoyois, M., Kelly, S., Østvær, P.A.: The motivic Steenrod algebra in positive characteristic. J. Eur. Math. Soc. (JEMS) 19(12), 3813–3849 (2017). MR3730515Google Scholar
  56. 56.
    Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006). viii+307 pp. MR2244106 (2007f:14013)Google Scholar
  57. 57.
    Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2010). xviii+325 pp. MR2665168 (2011e:14017)Google Scholar
  58. 58.
    Iyengar, S.B., Krause, H.: The Bousfield lattice of a triangulated category and stratification. Math. Z. 273(3–4), 1215–1241 (2013). MR3030697Google Scholar
  59. 59.
    Iyengar, S.B., Takahashi, R.: Annihilation of cohomology and decompositions of derived categories. Homol. Homotopy Appl. 16(2), 231–237 (2014). MR326389Google Scholar
  60. 60.
    Iyengar, S.B., Takahashi, R.: Annihilation of cohomology and strong generation of module categories. Int. Math. Res. Not. IMRN 2016(2), 499–535 (2016). MR3493424Google Scholar
  61. 61.
    Iyengar, S.B., Lipman, J., Neeman, A.: Relation between two twisted inverse image pseudofunctors in duality theory. Compos. Math. 151(4), 735–764 (2015). MR3334894Google Scholar
  62. 62.
    Joachimi, R.: Thick ideals in equivariant and motivic stable homotopy categories, in this proceedingsGoogle Scholar
  63. 63.
    Jørgensen, P.: A new recollement for schemes. Houst. J. Math. 35(4), 1071–1077 (2009). MR2577142 (2011c:14048)Google Scholar
  64. 64.
    Kapustin, A.N., Li, Y.: Topological correlators in Landau-Ginzburg models with boundaries. Adv. Theor. Math. Phys. 7(4), 727–749 (2003). MR2039036 (2005b:81179a)Google Scholar
  65. 65.
    Kashiwara, M., Schapira, P.: Sheaves on manifolds. With a chapter in French by Christian Houzel. Corrected reprint of the 1990 original. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292. Springer, Berlin (1994). x+512 pp. MR1299726 (95g:58222)Google Scholar
  66. 66.
    Kato, R., Okajima, H., Shimomura, K.: Notes on an algebraic stable homotopy category, in this proceedingsGoogle Scholar
  67. 67.
    Kawamata, Y.: \(D\)-equivalence and \(K\)-equivalence. J. Differ. Geom. 61(1), 147–171 (2002). MR1949787 (2004m:14025)Google Scholar
  68. 68.
    Kawamata, Y.: Birational geometry and derived categories. arXiv:1710.07370
  69. 69.
    Keller, B.: A remark on the generalized smashing conjecture. Manuscr. Math. 84(2) (1994). 193198 MR1285956 (95h:18014)Google Scholar
  70. 70.
    Kelly, G.M.: Chain maps inducing zero homology maps. Proc. Camb. Philos. Soc. 61, 847–854 (1965). MR0188273 (32 # 5712)Google Scholar
  71. 71.
    Kelly, S.: Triangulated categories of motives in positive characteristic. Ph.D. thesis, University of Paris 13; Australian National University (2013). arXiv:1305.5349v2
  72. 72.
    Kelly, S.: Some observations about motivic tensor triangulated geometry over a finite field, in this proceedingsGoogle Scholar
  73. 73.
    Kollár, J., Mori, S.: Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge, 1998, viii+254pp, MR1658959 (2000b:14018)Google Scholar
  74. 74.
    Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, Zurich, 1994, vol. 1, 2, pp. 120–139. Birkhauser, Basel (1995). MR1403918 (97f:32040)Google Scholar
  75. 75.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008). arXiv:0811.2435
  76. 76.
    Krause, H.: Smashing subcategories and the telescope conjecture–an algebraic approach. Invent. Math. 139(1), 99–133 (2000). MR1728877 (2000k:55016)Google Scholar
  77. 77.
    Krause, H.: A Brown representability theorem via coherent functors. Topology 41(4), 853–861 (2002). MR1905842 (2003c:18011)Google Scholar
  78. 78.
    Krause, H.: Localization theory for triangulated categories. Triangulated Categories. London Mathematical Society Lecture Note Series, vol. 375, pp. 161–235 (2010). MR2681709 (2012e:18026)Google Scholar
  79. 79.
    Krause, H.: Completing perfect complexes. arXiv:1805.10751
  80. 80.
    Krause, H., Šťovíček, J.: The telescope conjecture for hereditary rings via Ext-orthogonal pairs. Adv. Math. 225(5), 2341–2364 (2010). MR2680168 (2011j:16013)Google Scholar
  81. 81.
    Lewis Jr., L.G., May, J.P., Steinberger, M.: Equivariant stable homotopy theory. With contributions by J. E. McClure. Lecture Notes in Mathematics, vol. 1213. Springer, Berlin (1986). x+538 pp. MR0866482 (88e:55002)Google Scholar
  82. 82.
    Lipman, J.: Notes on derived functors and Grothendieck duality. Foundations of Grothendieck duality for diagrams of schemes. Lecture Notes in Mathematics, vol. 1960, pp. 1–259. Springer, Berlin (2009). MR2490557 (2011d:14029)Google Scholar
  83. 83.
    Lipman, J., Neeman, A.: Quasi-of the twisted inverse image functor perfect scheme-maps and boundedness. Ill. J. Math. 51(1), 209–236 (2007). MR2346195 (2008m:14004)Google Scholar
  84. 84.
    Lurie, J.: Tannaka duality for geometric stacks. arXiv:math/0412266v2
  85. 85.
    Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009). MR2522659 (2010j:18001)Google Scholar
  86. 86.
    Lurie, J.: Higher algebra. www.math.harvard.edu/~lurie/ (2016)
  87. 87.
    Mahowald, M., Ravenel, D., Shick, P.: The triple loop space approach to the telescope conjecture. Homotopy Methods in Algebraic Topology, Boulder, CO, 1999. Contemporary Mathematics, vol. 271, pp. 217–284. American Mathematical Society, Providence, RI (2001). MR1831355 (2002g:55014)Google Scholar
  88. 88.
    Matsuki, K.: Introduction to the Mori Program. Universitext. Springer, New York (2002). xxiv+478, MR1875410 (2002m:14011)Google Scholar
  89. 89.
    Matsuoka, T.: Koszul duality for \(E_n\)-algebras in a filtered category, in this proceedingsGoogle Scholar
  90. 90.
    Matsuoka, T.: Some technical aspects of factorization algebras on manifolds, in this proceedingsGoogle Scholar
  91. 91.
    Matumoto, T.: Memories on Ohkawa’s mathematical life in Hiroshima, in this proceedingsGoogle Scholar
  92. 92.
    May, J.P.: The additivity of traces in triangulated categories. Adv. Math. 163(1), 3473 (2001). MR1867203 (2002k:18019)Google Scholar
  93. 93.
    Miller, H.: Finite localizations. Papers in honor of Jos Adem Bol. Soc. Mat. Mex. (2) 37(1–2), 383–389 (1992). MR1317588 (96h:55009)Google Scholar
  94. 94.
    Minami, N.: A topologist’s introduction to the motivic homotopy theory for transformation group theorists–1, Geometry of transformation groups and combinatorics, RIMS Kôkyûroku Bessatsu. Res. Inst. Math. Sci. (RIMS), Kyoto, B39, 63–107 (2013). MR3156820Google Scholar
  95. 94.
    Miyaoka, Y., Peternell, T.: Geometry of Higher-Dimensional Algebraic Varieties. DMV Seminar, vol. 26. Birkhuser Verlag, Basel, 1997. vi+217 pp. MR1468476 (98g:14001)Google Scholar
  96. 95.
    Morava, J.: Noetherian localisations of categories cobordism comodules. Ann. Math. (2) 121, 1–39 (1985). MR0782555 (86g:55004)Google Scholar
  97. 96.
    Morava, J.: A remark on Hopkins’ chromatic splitting conjecture. arXiv:1406.3286,
  98. 97.
    Morava, J.: Operations on integral lifts of \(K(n)\), in this proceedingsGoogle Scholar
  99. 98.
    Morava, J.: Toward a fundamental groupoid for the stable homotopy category. In: Proceedings of the Nishida Fest, Kinosaki, 2003. Geometry and Topology Monographs, vol. 10, pp. 293–318. Geometry & Topology Publications, Coventry (2007). MR2402791 (2009e:55018)Google Scholar
  100. 99.
    Morel, F., Voevodsky, V.: \({\bf A}^1\)-homotopy theory of schemes. Inst. Ht. Études Sci. Publ. Math. 90, 1999, 45–143 (2001), MR1813224 (2002f:14029)Google Scholar
  101. 100.
    Mukai, S.: Duality between \(D(X)\) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981). MR0607081 (82f:14036)Google Scholar
  102. 101.
    Nayak, S.: Compactification for essentially finite-type maps. Adv. Math. 222(2), 527–546 (2009)Google Scholar
  103. 102.
    Neeman, A.: The chromatic tower for \(D(R)\). Topology 31(3), 519–532 (1992). With an appendix by Marcel Bökstedt. MR1174255 (93h:18018)Google Scholar
  104. 103.
    Neeman, A.: Triangulated categories with a single compact generator and a Brown representability theorem. arXiv:1804.02240
  105. 104.
    Neeman, A.: The connection between the \(K\)-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. l’Ecole Norm. Supér. (4) 25(5), 547–566 (1992). MR1191736 (93k:18015)Google Scholar
  106. 105.
    Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996). MRMR1308405Google Scholar
  107. 106.
    Neeman, A.: Oddball Bousfield classes. Topology 39(5), 931–935 (2000). MR1763956 (2001c:18007)Google Scholar
  108. 107.
    Neeman, A.: Approximable triangulated categories. arXiv:1806.06995
  109. 108.
    Neeman, A.: Strong generators in \(D^{perf}(X)\) and \(D^b_{coh}(X)\). arXiv:1703.04484
  110. 109.
    Neeman, A.: The categories \({\cal{T}}^c\) and \({\cal{T}}^b_c\) determine each other. arXiv:1806.06471
  111. 110.
    Neeman, A.: Triangulated Categories. Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton (2001). MR1812507 (2001k:18010)Google Scholar
  112. 111.
    Neeman, A.: The K-theory of triangulated categories. Handbook of K-theory, vol. 1, 2, pp. 1011–1078 (2005). MR2181838 (2006g:19004)Google Scholar
  113. 112.
    Noguchi, J.: Analytic Function Theory of Several Variables. Elements of Oka’s Coherence. Springer, Singapore (2016). xvi+397 pp. MR3526579Google Scholar
  114. 113.
    Ohsawa, T.: A role of the \(L^2\) method in the study of analytic families, in this proceedingsGoogle Scholar
  115. 114.
    Ohkawa, T.: The injective hull of homotopy types with respect to generalized homology functors. Hiroshima Math. J. 19(3), 631–639 (1989). MR1035147Google Scholar
  116. 115.
    Ohsawa, T.: \(L^2\) Approaches in Several Complex Variables. Development of Oka-Cartan Theory by \(L^2\) Estimates for the \(\overline{\partial }\) Operator. Springer Monographs in Mathematics. Springer, Tokyo (2015). MR3443603Google Scholar
  117. 116.
    Oort, F.: Alterations can remove singularities. Bull. Am. Math. Soc. (N.S.) 35(4), 319–331 (1998). MR1638306 (99i:14021)Google Scholar
  118. 117.
    Orlov, D.O.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. (Russian. Russian summary) Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh, 240–262; translation in Proc. Steklov Inst. Math. 2004, no. 3(246), 227–248. MR2101296 (2006i:81173)Google Scholar
  119. 118.
    Orlov, D.O.: Matrix factorizations for nonaffine LG-models. Math. Ann. 353(1), 95–108 (2012). MR2910782 14F05 18E30Google Scholar
  120. 119.
    Orlov, D.: Smooth and proper noncommutative schemes and gluing of DG categories. Adv. Math. 302, 59–105 (2016). MR3545926 14F05 (16E45 18E30)Google Scholar
  121. 120.
    Perego, A.: A Gabriel theorem for coherent twisted sheaves. Math. Z. 262(3), 571–583 (2009). MR2506308 (2011a:14032)Google Scholar
  122. 121.
    Ravenel, D.C.: Localization with respect to certain periodic homology theories. Am. J. Math. 106(2), 351–414 (1984). MR0737778 (85k:55009)Google Scholar
  123. 122.
    Ravenel, D.C.: Nilpotence and periodicity in stable homotopy theory. Ann. Math. Stud. 128, Appendix C by Jeff Smith (1992) xiv+209 MR1192553 (94b:55015)Google Scholar
  124. 123.
    Rosenberg, A.: Spectra of ‘Spaces’ Represented by Abelian Categories. MPIM Preprints, 2004-115Google Scholar
  125. 124.
    Rouquier, R.: Dimensions of triangulated categories. J. K-Theory 1(2), 193–256 (2008). MR2434186Google Scholar
  126. 125.
    Rouquier, R.: Derived categories and algebraic geometry. Triangulated Categories. London Mathematical Society Lecture Note Series, vol. 375, pp. 351–370. Cambridge University Press, Cambridge (2010). MR2681712 (2011h:14022)Google Scholar
  127. 126.
    Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001). MR1831820 (2002e:14030)Google Scholar
  128. 127.
    Serre, J.P.: Géométrie algébrique et géométrie analytique. Ann. l’institut Fourier (Grenoble) 6, 1–42 (1955–1956). MR0082175 (18,511a)Google Scholar
  129. 128.
    The Stacks Project, Part 1: Preliminaries, Chapter 17: Sheaves of modules, Section 17.9: Modules of finite typeGoogle Scholar
  130. 129.
    Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Volume III. Progress in Mathematics, vol. 88, pp. 247–435. Birkhuser Boston, Boston (1990). MR1106918 (92f:19001)Google Scholar
  131. 130.
    Thomason, R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997). MR1436741 (98b:18017)Google Scholar
  132. 131.
    Toda, Y.: Limit stable objects on Calabi-Yau 3-folds. Duke Math. J. 149(1), 157–208 (2009). MR2541209 (2011b:14043)Google Scholar
  133. 132.
    Torii, T.: On quasi-categories of comodules and Landweber exactness, in this proceedingsGoogle Scholar
  134. 133.
    Uehara, H.: An example of Fourier-Mukai partners of minimal elliptic surfaces. Math. Res. Lett. 11(2–3), 371–375 (2004). MR2067481 (2005g:14073)Google Scholar
  135. 134.
    Verdier, J.-L.: Catégories dérivées: quelques résultats (état 0). Cohomologie étale. Lecture Notes in Mathematics, vol. 569, pp. 262–311. Springer, Berlin (1977). MR3727440Google Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Nagoya Institute of TechnologyNagoyaJapan

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