Abstract
In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field.
-
[BBD] A. A. Beuilinson, J. Bernstein, and P. Deligne, "Faisceaux pervers," in Analysis and Topology on Singular Spaces, I, Soc. Math. France, Paris, 1982, vol. 100, pp. 5-171.
@INCOLLECTION{BBD,
author = {Beĭlinson, A. A. and Bernstein, J. and Deligne, P.},
title = {Faisceaux pervers},
booktitle = {Analysis and Topology on Singular Spaces, {I}},
venue = {{L}uminy, 1981},
series = {Astérisque},
volume = {100},
pages = {5--171},
publisher = {Soc. Math. France, Paris},
year = {1982},
mrclass = {32C38},
mrnumber = {0751966},
mrreviewer = {Zoghman Mebkhout},
zblnumber = {0536.14011},
} -
[BLM] Y. Bespalov, V. Lyubashenko, and O. Manzyuk, Pretriangulated $A_\infty$-Categories, Nats\=ıonal\cprimena Akadem\=ıya Nauk Ukra"ıni, Īnstitut Matematiki, Kiev, 2008, vol. 76.
@BOOK{BLM,
author = {Bespalov, Yu. and Lyubashenko, V. and Manzyuk, O.},
title = {Pretriangulated {$A\sb \infty$}-Categories},
series = {Proc. Inst. Math. NAS Ukraine Math. Appl.},
volume = {76},
publisher = {Nats\=ıonal\cprime{n}a Akadem\=ıya Nauk Ukraïni, Īnstitut Matematiki, Kiev},
year = {2008},
pages = {599},
isbn = {978-966-02-4861-8},
mrclass = {18E30 (18C20 18D15 18D50 53D37)},
mrnumber = {2513383},
mrreviewer = {Beno\^ıt Fresse},
zblnumber = {1199.18001},
} -
[Bondal5]
A. I. Bondal and M. M. Kapranov, "Enhanced triangulated categories," Math. USSR Sb., vol. 70, pp. 93-107, 1991.
@article {Bondal5,
author = {Bondal, A. I. and Kapranov, M. M.},
title = {Enhanced triangulated categories},
journal = {Math. USSR Sb.},
fjournal = {Mathematics of the USSR-Sbornik},
volume = {70},
year = {1991},
mumber = {1},
pages = {93--107},
issn= {0368-8666},
mrclass = {18E30},
mrnumber = {1055981},
mrreviewer = {L. N. Vaserstein},
doi = {10.1070/SM1991v070n01ABEH001253},
url = {https://doi.org/10.1070/SM1991v070n01ABEH001253},
zblnumber = {0729.18008},
} -
[CanonacoStellari]
A. Canonaco and P. Stellari, "A tour about existence and uniqueness of dg enhancements and lifts," J. Geom. Phys., vol. 122, pp. 28-52, 2017.
@ARTICLE{CanonacoStellari,
author = {Canonaco, Alberto and Stellari, Paolo},
title = {A tour about existence and uniqueness of dg enhancements and lifts},
journal = {J. Geom. Phys.},
fjournal = {Journal of Geometry and Physics},
volume = {122},
year = {2017},
pages = {28--52},
issn = {0393-0440},
mrclass = {18E30 (14F05 18E10)},
mrnumber = {3713870},
mrreviewer = {Jian Min Chen},
doi = {10.1016/j.geomphys.2016.11.030},
url = {https://doi.org/10.1016/j.geomphys.2016.11.030},
zblnumber = {1399.14007},
} -
[ChristensenFrankland]
D. J. Christensen and M. Frankland, "Higher Toda brackets and the Adams spectral sequence in triangulated categories," Algebr. Geom. Topol., vol. 17, iss. 5, pp. 2687-2735, 2017.
@ARTICLE{ChristensenFrankland,
author = {Christensen, J. Daniel and Frankland, Martin},
title = {Higher {T}oda brackets and the {A}dams spectral sequence in triangulated categories},
journal = {Algebr. Geom. Topol.},
fjournal = {Algebraic \& Geometric Topology},
volume = {17},
year = {2017},
number = {5},
pages = {2687--2735},
issn = {1472-2747},
mrclass = {55T15 (18E30)},
mrnumber = {3704239},
mrreviewer = {Marius Thaule},
doi = {10.2140/agt.2017.17.2687},
url = {https://doi.org/10.2140/agt.2017.17.2687},
zblnumber = {1421.55015},
} -
[Keller1]
B. Keller, "Deriving DG categories," Ann. Sci. École Norm. Sup. (4), vol. 27, iss. 1, pp. 63-102, 1994.
@ARTICLE{Keller1,
author = {Keller, Bernhard},
title = {Deriving {DG} categories},
journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {27},
year = {1994},
number = {1},
pages = {63--102},
issn = {0012-9593},
mrclass = {18E30 (16D90)},
mrnumber = {1258406},
mrreviewer = {Jeremy Rickard},
doi = {10.24033/asens.1689},
url = {https://doi.org/10.24033/asens.1689},
zblnumber = {0799.18007},
} -
[Lefevre] K. Lefèvre-Hasegawa, Sur les $A_\infty$-catégories, 2003.
@MISC{Lefevre,
author = {Lefèvre-Hasegawa, K.},
title = {Sur les {$A_\infty$}-catégories},
note = {Ph.D. thesis, Université Paris 7},
year = {2003},
zblnumber = {},
arxiv = {math/0310337},
} -
[OrlovLunts]
V. A. Lunts and D. O. Orlov, "Uniqueness of enhancement for triangulated categories," J. Amer. Math. Soc., vol. 23, iss. 3, pp. 853-908, 2010.
@ARTICLE{OrlovLunts,
author = {Lunts, Valery A. and Orlov, Dmitri O.},
title = {Uniqueness of enhancement for triangulated categories},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {23},
year = {2010},
number = {3},
pages = {853--908},
issn = {0894-0347},
mrclass = {14F05 (18E30)},
mrnumber = {2629991},
mrreviewer = {Andrei D. Halanay},
doi = {10.1090/S0894-0347-10-00664-8},
url = {https://doi.org/10.1090/S0894-0347-10-00664-8},
zblnumber = {1197.14014},
} -
[muro2]
F. Muro, S. Schwede, and N. Strickland, "Triangulated categories without models," Invent. Math., vol. 170, iss. 2, pp. 231-241, 2007.
@ARTICLE{muro2,
author = {Muro, Fernando and Schwede, Stefan and Strickland, Neil},
title = {Triangulated categories without models},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {170},
year = {2007},
number = {2},
pages = {231--241},
issn = {0020-9910},
mrclass = {18E30 (18G55 55P42)},
mrnumber = {2342636},
mrreviewer = {Jens Hornbostel},
doi = {10.1007/s00222-007-0061-2},
url = {https://doi.org/10.1007/s00222-007-0061-2},
zblnumber = {1125.18009},
} -
[Orlov4]
D. O. Orlov, "Equivalences of derived categories and $K3$ surfaces," in Algebraic Geometry, Vol. 7, Springer-Verlag, New York, 1997, vol. 84, pp. 1361-1381.
@INCOLLECTION{Orlov4,
author = {Orlov, D. O.},
title = {Equivalences of derived categories and {$K3$} surfaces},
booktitle = {Algebraic Geometry, Vol. 7},
series = {J. Math. Sci.},
publisher={Springer-Verlag, New York},
volume = {84},
year = {1997},
number = {5},
pages = {1361--1381},
issn = {1072-3374},
mrclass = {14J28 (14F05 18E30)},
mrnumber = {1465519},
mrreviewer = {Jong Hae Keum},
doi = {10.1007/BF02399195},
url = {https://doi.org/10.1007/BF02399195},
zblnumber = {0938.14019},
} -
[porta]
M. Porta, "The Popescu-Gabriel theorem for triangulated categories," Adv. Math., vol. 225, iss. 3, pp. 1669-1715, 2010.
@ARTICLE{porta,
author = {Porta, Marco},
title = {The {P}opescu-{G}abriel theorem for triangulated categories},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {225},
year = {2010},
number = {3},
pages = {1669--1715},
issn = {0001-8708},
mrclass = {18E30 (16D90 16E45)},
mrnumber = {2673743},
mrreviewer = {Mike Prest},
doi = {10.1016/j.aim.2010.04.002},
url = {https://doi.org/10.1016/j.aim.2010.04.002},
zblnumber = {1227.18011},
} -
[RizzardoVdB2]
A. Rizzardo, M. Van den Bergh, and A. Neeman, "An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves," Invent. Math., vol. 216, iss. 3, pp. 927-1004, 2019.
@article {RizzardoVdB2,
author = {Rizzardo, Alice and Van den Bergh, Michel and Neeman, Amnon},
TITLE = {An example of a non-{F}ourier-{M}ukai functor between derived categories of coherent sheaves},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {216},
YEAR = {2019},
NUMBER = {3},
PAGES = {927--1004},
ISSN = {0020-9910},
MRCLASS = {14F05 (13D09 14A22 18E30)},
MRNUMBER = {3955712},
DOI = {10.1007/s00222-019-00862-9},
URL = {https://doi.org/10.1007/s00222-019-00862-9},
zblnumber={07066471},
} -
[Rouquier]
R. Rouquier, "Dimensions of triangulated categories," J. K-Theory, vol. 1, iss. 2, pp. 193-256, 2008.
@ARTICLE{Rouquier,
author = {Rouquier, Raphaël},
title = {Dimensions of triangulated categories},
journal = {J. K-Theory},
fjournal = {Journal of K-Theory. K-Theory and its Applications in Algebra, Geometry, Analysis \& Topology},
volume = {1},
year = {2008},
number = {2},
pages = {193--256},
issn = {1865-2433},
mrclass = {18E30 (14F05)},
mrnumber = {2434186},
mrreviewer = {Ioannis Emmanouil},
doi = {10.1017/is007011012jkt010},
url = {https://doi.org/10.1017/is007011012jkt010},
zblnumber = {1165.18008},
} -
[Verdier]
J. Verdier, "Des catégories dérivées des catégories abéliennes," Astérisque, iss. 239, p. xii, 1996.
@ARTICLE{Verdier,
author = {Verdier, Jean-Louis},
title = {Des catégories dérivées des catégories abéliennes},
note = {With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis},
journal = {Astérisque},
fjournal = {Astérisque},
number = {239},
year = {1996},
pages = {xii+253 pp. (1997)},
issn = {0303-1179},
mrclass = {18E30 (18-03 18E35)},
mrnumber = {1453167},
mrreviewer = {Amnon Neeman},
zblnumber = {0882.18010},
url = {https://smf.emath.fr/publications/des-categories-derivees-des-categories-abeliennes},
}