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Uniqueness of enhancement for triangulated categories. (English) Zbl 1197.14014

It has been known for a long time that the definition of a triangulated category is not entirely satisfactory, one of the main problems being the non-uniqueness of the cone. A remedy is to use enhancements. By definition, an enhancement of \(\mathcal{T}\) is a pair \((\mathcal{B},\epsilon)\), where \(\mathcal{B}\) is a pretriangulated DG category and \(\epsilon: H^0(\mathcal{B})\rightarrow \mathcal{T}\) is an equivalence of triangulated categories. Recall that a DG category \(\mathcal{A}\) is a category where the space of morphisms between any two objects forms a complex in a natural way. Its homotopy category \(H^0(\mathcal{A})\) is the category having the same objects as \(\mathcal{A}\) and the space of morphisms between any two objects is the zeroth cohomology of the associated complex. To say that a DG category \(\mathcal{A}\) is pretriangulated means, in particular, that the cone is functorial in \(\mathcal{A}\) and that the homotopy category \(H^0(\mathcal{A})\) is triangulated.
Clearly, the question of uniqueness of enhancement is a very important one. The category \(\mathcal{T}\) is said to have a unique enhancement if it has one and for any two enhancements \((\mathcal{B},\epsilon)\) and \((\mathcal{B'},\epsilon')\) there exists a quasi-functor \(\mathcal{B}\rightarrow \mathcal{B'}\) inducing an equivalence \(H^0(\mathcal{B})\rightarrow H^0(\mathcal{B'})\).
In the paper under review, the authors prove general results on the uniqueness of enhancement for triangulated categories. In particular, they show that the following categories associated to a quasi-projective scheme over some field \(k\) have a unique enhancement: the unbounded derived category of quasi-coherent sheaves, the triangulated category of perfect complexes and the bounded derived category of coherent sheaves. If the scheme is projective, then a stronger conclusion, that of strong uniqueness, holds for the category of perfect complexes and for the bounded derived category. These results can, in particular, be used to show that a fully faithful functor between bounded derived categories of coherent sheaves on projective schemes can be represented by an object on the product.
Reviewer: Pawel Sosna (Bonn)

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
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[1] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963 – 1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
[2] M. R. Ballard, Equivalences of derived categories of sheaves on quasi-projective schemes, arXiv:0905.3148.
[3] Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209 – 234. · Zbl 0802.18008
[4] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1 – 36, 258 (English, with English and Russian summaries). · Zbl 1135.18302
[5] A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669 – 683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93 – 107. · Zbl 0719.18005
[6] Alexey I. Bondal, Michael Larsen, and Valery A. Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 29 (2004), 1461 – 1495. · Zbl 1079.18008
[7] Vladimir Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004), no. 2, 643 – 691. · Zbl 1064.18009
[8] Alexander I. Efimov, Valery A. Lunts, and Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories. I. General theory, Adv. Math. 222 (2009), no. 2, 359 – 401. · Zbl 1180.18006
[9] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323 – 448 (French). · Zbl 0201.35602
[10] A. Grotendik and Ž. A. D\(^{\prime}\)edonne, Elements of algebraic topology, Uspehi Mat. Nauk 27 (1972), no. 2(164), 135 – 148 (Russian). Translated from the French (Éléments de géométrie algébrique, I: Le langage des schemas, second edition, pp. 4 – 18, Springer, Berlin, 1971) by F. V. Širokov.
[11] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167.
[12] Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. · Zbl 0212.26101
[13] Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291 – 3323. · Zbl 0894.18008
[14] F. Castaño Iglesias, P. Enache, C. Năstăsescu, and B. Torrecillas, Un analogue du théorème de Gabriel-Popescu et applications, Bull. Sci. Math. 128 (2004), no. 4, 323 – 332 (French, with English and French summaries). · Zbl 1074.18002
[15] Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. · Zbl 1118.18001
[16] Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63 – 102. · Zbl 0799.18007
[17] Bernhard Keller, Derived categories and their uses, Handbook of algebra, Vol. 1, Handb. Algebr., vol. 1, Elsevier/North-Holland, Amsterdam, 1996, pp. 671 – 701. · Zbl 0862.18001
[18] Joseph Lipman, Lectures on local cohomology and duality, Local cohomology and its applications (Guanajuato, 1999) Lecture Notes in Pure and Appl. Math., vol. 226, Dekker, New York, 2002, pp. 39 – 89. · Zbl 1011.13010
[19] C. Menini, Gabriel-Popescu type theorems and graded modules, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 239 – 251. · Zbl 0685.16002
[20] Amnon Neeman, The connection between the \?-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547 – 566. · Zbl 0868.19001
[21] 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
[22] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. · Zbl 0974.18008
[23] Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. · Zbl 0438.32016
[24] D. O. Orlov, Equivalences of derived categories and \?3 surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361 – 1381. Algebraic geometry, 7. · Zbl 0938.14019
[25] D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89 – 172 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 3, 511 – 591. · Zbl 1118.14021
[26] N. Popescu, Abelian categories with applications to rings and modules, Academic Press, London-New York, 1973. London Mathematical Society Monographs, No. 3. · Zbl 0271.18006
[27] Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197 – 278 (French). · Zbl 0067.16201
[28] N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121 – 154. · Zbl 0636.18006
[29] G. Tabuada, Théorie homotopique des DG-catégories, Thèse de l’Univ. Paris 7 (2007).
[30] Bertrand Toën, The homotopy theory of \?\?-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615 – 667. · Zbl 1118.18010
[31] 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
[32] R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1 – 27. · Zbl 0873.18003
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