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Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
About this Title
Leonid Positselski, Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Bolshoy Karetny per. 19 str. 1, Moscow 127994, Russia
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 212, Number 996
ISBNs: 978-0-8218-5296-5 (print); 978-1-4704-0613-4 (online)
DOI: https://doi.org/10.1090/S0065-9266-2010-00631-8
Published electronically: November 19, 2010
Keywords: DG-algebra,
CDG-coalgebra,
curved A-infinity coalgebra,
comodule,
contramodule,
derived category of the second kind,
triangulated category,
model category,
D-module,
Koszul duality
MSC: Primary 18E30, 18G10, 16T15, 16S37, 14F10; Secondary 18G55, 17B55, 16E65, 18G15, 58J10
Table of Contents
Chapters
- Introduction
- 1. Derived Category of DG-Modules
- 2. Derived Categories of DG-Comodules and DG-Contramodules
- 3. Coderived and Contraderived Categories of CDG-Modules
- 4. Coderived Category of CDG-Comodules and Contraderived Category of CDG-Contramodules
- 5. Comodule-Contramodule Correspondence
- 6. Koszul Duality: Conilpotent and Nonconilpotent Cases
- 7. $\mathrm {A}_\infty$-Algebras and Curved $\mathrm {A}_\infty$-Coalgebras
- 8. Model Categories of DG-Modules, CDG-Comodules, and CDG-Contramodules
- 9. Model Categories of DG-Algebras and CDG-Coalgebras
- Appendix A. Homogeneous Koszul Duality
- Appendix B. $\mathcal {D}$–$\Omega$ Duality
Abstract
The aim of this paper is to construct the derived nonhomogeneous Koszul duality. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived category of CDG-comodules, and the contraderived category of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or “triality” (an equivalence between exotic derived categories corresponding to Koszul dual (C)DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various A-infinity structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and D-$\Omega$ duality are discussed in the appendices.- A. L. Agore. Limits of coalgebras, bialgebras and Hopf algebras. Electronic preprint arXiv:1003.0318 [math.QA].
- Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1–50. MR 1237825
- A. Beilinson, V. Drinfeld. Quantization of Hitchin’s integrable system and Hecke eigensheaves. February 2000. Available from http://www.math.utexas.edu/˜benzvi/Langlands.html .
- A. A. Beĭlinson, V. A. Ginsburg, and V. V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317–350. MR 1048505, DOI 10.1016/0393-0440(88)90028-9
- Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
- I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
- Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
- Jonathan Block, Duality and equivalence of module categories in noncommutative geometry, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, 2010, pp. 311–339. MR 2648899, DOI 10.1090/crmp/050/24
- A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23–42. MR 992977, DOI 10.1070/IM1990v034n01ABEH000583
- 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. MR 1055981, DOI 10.1070/SM1991v070n01ABEH001253
- Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 120260, DOI 10.1090/S0002-9947-1960-0120260-3
- Samuel Eilenberg and John C. Moore, Limits and spectral sequences, Topology 1 (1962), 1–23. MR 148723, DOI 10.1016/0040-9383(62)90093-9
- Samuel Eilenberg and J. C. Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. 55 (1965), 39. MR 178036
- David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. MR 570778, DOI 10.1090/S0002-9947-1980-0570778-7
- Gunnar Fløystad, Koszul duality and equivalences of categories, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2373–2398. MR 2204036, DOI 10.1090/S0002-9947-05-04035-3
- Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
- L. Gruson and C. U. Jensen, Dimensions cohomologiques reliées aux foncteurs $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980) Lecture Notes in Math., vol. 867, Springer, Berlin, 1981, pp. 234–294 (French). MR 633523
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291–3323. MR 1465117, DOI 10.1080/00927879708826055
- Vladimir Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), no. 2-3, 209–250. MR 1843805, DOI 10.1016/S0022-4049(00)00121-3
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- J. Huebschmann. Homological perturbations, equivariant cohomology, and Koszul duality. Electronic preprint arXiv:math/0401160 [math.AT].̇
- Dale Husemoller, John C. Moore, and James Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113–185. MR 365571, DOI 10.1016/0022-4049(74)90045-0
- J. F. Jardine, A closed model structure for differential graded algebras, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995) Fields Inst. Commun., vol. 17, Amer. Math. Soc., Providence, RI, 1997, pp. 55–58. MR 1478701
- D. Kaledin. Derived Mackey functors. Electronic preprint arXiv:0812.2519 [math.KT].̇
- M. M. Kapranov, On DG-modules over the de Rham complex and the vanishing cycles functor, Algebraic geometry (Chicago, IL, 1989) Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 57–86. MR 1181207, DOI 10.1007/BFb0086264
- Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406
- B. Keller. Koszul duality and coderived categories (after K. Lefèvre). October 2003. Available from http://www.math.jussieu.fr/˜keller/publ/index.html .
- B. Keller. Pseudocompact DG algebras and derived categories. Appendix to the paper: B. Keller, D. Yang. Derived equivalences from mutations of quivers with potential. Electronic preprint arXiv:0906.0761 [math.RT].̇
- Bernhard Keller, Wendy Lowen, and Pedro Nicolás, On the (non)vanishing of some “derived” categories of curved dg algebras, J. Pure Appl. Algebra 214 (2010), no. 7, 1271–1284. MR 2587002, DOI 10.1016/j.jpaa.2009.10.011
- Henning Krause, A Brown representability theorem via coherent functors, Topology 41 (2002), no. 4, 853–861. MR 1905842, DOI 10.1016/S0040-9383(01)00010-6
- Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. MR 2157133, DOI 10.1112/S0010437X05001375
- Srikanth Iyengar and Henning Krause, Acyclicity versus total acyclicity for complexes over Noetherian rings, Doc. Math. 11 (2006), 207–240. MR 2262932
- H. Krause. Localization theory for triangulated categories. Electronic preprint arXiv:0806.1324 [math.CT].̇
- K. Lefèvre-Hasegawa. Sur les $\mathrm {A}_\infty$-catégories. Thèse de doctorat, Université Denis Diderot – Paris 7, November 2003. arXiv:math.CT/0310337.̇ Corrections, by B. Keller. Available from http://people.math.jussieu.fr/˜keller/lefevre/publ.html .
- Ivan Mirković and Simon Riche, Linear Koszul duality, Compos. Math. 146 (2010), no. 1, 233–258. MR 2581249, DOI 10.1112/S0010437X09004357
- Amnon Neeman, The derived category of an exact category, J. Algebra 135 (1990), no. 2, 388–394. MR 1080854, DOI 10.1016/0021-8693(90)90296-Z
- Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234. MR 1214458
- Amnon Neeman, 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 (1992), no. 5, 547–566. MR 1191736
- Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
- Pedro Nicolás, The bar derived category of a curved dg algebra, J. Pure Appl. Algebra 212 (2008), no. 12, 2633–2659. MR 2452316, DOI 10.1016/j.jpaa.2008.04.001
- D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240–262 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 227–248. MR 2101296
- David Pauksztello, Compact corigid objects in triangulated categories and co-$t$-structures, Cent. Eur. J. Math. 6 (2008), no. 1, 25–42. MR 2379950, DOI 10.2478/s11533-008-0003-2
- Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MR 2177131, DOI 10.1090/ulect/037
- Leonid Positselski, Koszul property and Bogomolov’s conjecture, Int. Math. Res. Not. 31 (2005), 1901–1936. MR 2171198, DOI 10.1155/IMRN.2005.1901
- L. Positselski. Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. With appendices coauthored by S. Arkhipov and D. Rumynin. Electronic preprint arXiv:0708.3398 [math.CT],̇ to be published in Birkhäuser’s series Monografie Matematyczne, vol. 70 in 2010.
- L. Positselski. Mixed Artin–Tate motives with finite coefficients. Electronic preprint arXiv:1006.4343 [math.KT].̇
- N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
- María José Souto Salorio, On the cogeneration of $t$-structures, Arch. Math. (Basel) 83 (2004), no. 2, 113–122. MR 2104939, DOI 10.1007/s00013-004-1064-5
- J.-L. Verdier. Catégories dérivées, état 0. SGA 4 1/2. Lect. Notes Math. 569, p. 262–311, 1977.
- Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. MR 1453167
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136