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On the homotopy finiteness of DG categories. (English. Russian original) Zbl 1445.18006

Russ. Math. Surv. 74, No. 3, 431-460 (2019); translation from Usp. Mat. Nauk 74, No. 3, 63-94 (2019).
Summary: This paper gives a short overview of results related to homotopy finiteness of DG categories. A general plan is explained for proving homotopy finiteness of derived categories of coherent sheaves and coherent matrix factorizations on separated schemes of finite type over a field of characteristic zero.

MSC:

18G80 Derived categories, triangulated categories
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
18G35 Chain complexes (category-theoretic aspects), dg categories
14E05 Rational and birational maps
18-02 Research exposition (monographs, survey articles) pertaining to category theory
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References:

[1] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov 2016 Resolutions in factorization categories Adv. Math.295 195-249 · Zbl 1353.13016 · doi:10.1016/j.aim.2016.02.008
[2] E. Bierstone and P. D. Milman 1997 Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant Invent. Math.128 2 207-302 · Zbl 0896.14006 · doi:10.1007/s002220050141
[3] А. И. Бондал, М. М. Капранов 1990 Оснащенные триангулированные категории Матем. сб.181 5 669-683 · Zbl 0719.18005
[4] English transl. A. I. Bondal and M. M. Kapranov 1991 Enhanced triangulated categories Math. USSR-Sb.70 1 93-107 · Zbl 0729.18008 · doi:10.1070/SM1991v070n01ABEH001253
[5] A. Bondal and D. Orlov 2002 Derived categories of coherent sheaves Proceedings of the International Congress of MathematiciansBeijing, 2002 II Higher Ed. Press, Beijing 47-56 · Zbl 0996.18007
[6] A. Bondal and M. van den Bergh 2003 Generators and representability of functors in commutative and noncommutative geometry Mosc. Math. J.3 1 1-36 · Zbl 1135.18302 · doi:10.17323/1609-4514-2003-3-1-1-36
[7] V. Drinfeld 2004 DG quotients of DG categories J. Algebra272 2 643-691 · Zbl 1064.18009 · doi:10.1016/j.jalgebra.2003.05.001
[8] V. Drinfeld 2004 On the notion of geometric realization Mosc. Math. J.4 3 619-626 · Zbl 1073.55010 · doi:10.17323/1609-4514-2004-4-3-619-626
[9] A. I. Efimov Homotopy finiteness of some DG categories from algebraic geometry J. Eur. Math. Soc. (JEMS) 1308.0135 · Zbl 1478.14010
[10] A. I. Efimov 2018 Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration 1805.09283 21 pp.
[11] A. Efimov Homotopy finiteness of derived categories of coherent D-modules in preparation
[12] A. I. Efimov and L. Positselski 2015 Coherent analogues of matrix factorizations and relative singularity categories Algebra Number Theory9 5 1159-1292 · Zbl 1333.14018 · doi:10.2140/ant.2015.9.1159
[13] P. Gabriel 1962 Sur les catégories abéliennes localement noethériennes et leurs applications aux algèbres étudiées par Dieudonné Séminaire J.-P. Serre 1959/1960, mimeographed notes Collège de France, Paris
[14] P. Gabriel 1962 Des catégories abéliennes Bull. Soc. Math. France90 323-448 · Zbl 0201.35602 · doi:10.24033/bsmf.1583
[15] W. Geigle and H. Lenzing 1991 Perpendicular categories with applications to representations and sheaves J. Algebra144 2 273-343 · Zbl 0748.18007 · doi:10.1016/0021-8693(91)90107-J
[16] M. Hovey 1999 Model categories Math. Surveys Monogr. 63 Amer. Math. Soc., Providence, RI xii+209 pp. · Zbl 0909.55001
[17] J. F. Jardine 1997 A closed model structure for differential graded algebras Cyclic cohomology and noncommutative geometryWaterloo, ON, 1995 Fields Inst. Commun. 17 Amer. Math. Soc., Providence, RI 55-58 · Zbl 0887.55016
[18] B. Keller 1994 Deriving DG categories Ann. Sci. École Norm. Sup. (4)27 1 63-102 · Zbl 0799.18007 · doi:10.24033/asens.1689
[19] B. Keller 1999 On the cyclic homology category of exact categories J. Pure Appl. Algebra136 1 1-56 · Zbl 0923.19004 · doi:10.1016/S0022-4049(97)00152-7
[20] M. Kontsevich and Y. Soibelman 2009 Notes on \(A_{\infty}\)-algebras, \(A_{\infty}\)-categories and non-commutative geometry Homological mirror symmetry Lecture Notes in Phys. 757 Springer, Berlin 153-219 · Zbl 1202.81120 · doi:10.1007/978-3-540-68030-7_6
[21] H. Krause 2005 The stable derived category of a noetherian scheme Compos. Math.141 5 1128-1162 · Zbl 1090.18006 · doi:10.1112/S0010437X05001375
[22] A. Kuznetsov and V. A. Lunts 2015 Categorical resolutions of irrational singularities Int. Math. Res. Not. IMRN2015 13 4536-4625 · Zbl 1338.14020 · doi:10.1093/imrn/rnu072
[23] V. A. Lunts 2010 Categorical resolution of singularities J. Algebra323 10 2977-3003 · Zbl 1202.18006 · doi:10.1016/j.jalgebra.2009.12.023
[24] V. A. Lunts and D. O. Orlov 2010 Uniqueness of enhancement for triangulated categories J. Amer. Math. Soc.23 3 853-908 · Zbl 1197.14014 · doi:10.1090/S0894-0347-10-00664-8
[25] M. Nagata 1963 A generalization of the imbedding problem of an abstract variety in a complete variety J. Math. Kyoto Univ.3 89-102 · Zbl 0223.14011 · doi:10.1215/kjm/1250524859
[26] A. Neeman 1992 The connection between the \(K\)-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel Ann. Sci. École Norm. Sup. (4)25 5 547-566 · Zbl 0868.19001 · doi:10.24033/asens.1659
[27] A. Neeman 1996 The Grothendieck duality theorem via Bousfield’s techniques and Brown representability J. Amer. Math. Soc.9 1 205-236 · Zbl 0864.14008 · doi:10.1090/S0894-0347-96-00174-9
[28] D. Orlov 2016 Smooth and proper noncommutative schemes and gluing of DG categories Adv. Math.302 59-105 · Zbl 1368.14031 · doi:10.1016/j.aim.2016.07.014
[29] D. Pauksztello 2009 Homological epimorphisms of differential graded algebras Comm. Algebra37 7 2337-2350 · Zbl 1189.18005 · doi:10.1080/00927870802623344
[30] L. Positselski 2010 Homological algebra of semimodules and semicontramodules. Semi-infinite homological algebra of associative algebraic structures IMPAN Monogr. Mat. (N. S.) 70 Birkhäuser/Springer Basel AG, Basel xxiv+349 pp. · Zbl 1202.18001 · doi:10.1007/978-3-0346-0436-9
[31] L. Positselski 2011 Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence Mem. Amer. Math. Soc. 212 Amer. Math. Soc., Providence, RI 996 vi+133 pp. · Zbl 1275.18002 · doi:10.1090/S0065-9266-2010-00631-8
[32] L. Positselski 2017 (v1 - 2012) Contraherent cosheaves 1209.2995 257 pp.
[33] D. C. Ravenel 1984 Localization with respect to certain periodic homology theories Amer. J. Math.106 2 351-414 · Zbl 0586.55003 · doi:10.2307/2374308
[34] R. Rouquier 2008 Dimensions of triangulated categories J. K-Theory1 2 193-256 · Zbl 1165.18008 · doi:10.1017/is007011012jkt010
[35] G. Tabuada 2005 Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories C. R. Acad. Sci. Paris Ser. I Math.340 1 15-19 · Zbl 1060.18010 · doi:10.1016/j.crma.2004.11.007
[36] G. N. Tabuada 2007 Théorie homotopique des DG-catégories PhD thesis Univ. Paris Diderot - Paris 7, Paris 0710.4303 178 pp.
[37] B. Toën and M. Vaquié 2007 Moduli of objects in dg-categories Ann. Sci. École Norm. Sup. (4)40 3 387-444 · Zbl 1140.18005 · doi:10.1016/j.ansens.2007.05.001
[38] C. T. C. Wall 1965 Finiteness conditions for CW-complexes Ann. of Math. (2)81 1 56-69 · Zbl 0152.21902 · doi:10.2307/1970382
[39] J. H. C. Whitehead 1949 Combinatorial homotopy. I Bull. Amer. Math. Soc.55 213-245 · Zbl 0040.38704 · doi:10.1090/S0002-9904-1949-09175-9
[40] J. H. C. Whitehead 1950 A certain exact sequence Ann. of Math. (2)52 51-110 · Zbl 0037.26101 · doi:10.2307/1969511
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