Geometry & Topology Volume 28, issue 6 (2024)

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Cluster categories and rational curves

Zheng Hua and Bernhard Keller

Geometry & Topology 28 (2024) 2569–2634

DOI: 10.2140/gt.2024.28.2569

Bibliography

1	T Adachi, O Iyama, I Reiten, τ–tilting theory, Compos. Math. 150 (2014) 415 MR3187626
2	T Aihara, O Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. 85 (2012) 633 MR2927802
3	C Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009) 2525 MR2640929
4	J August, On the finiteness of the derived equivalence classes of some stable endomorphism rings, Math. Z. 296 (2020) 1157 MR4159827
5	J August, The tilting theory of contraction algebras, Adv. Math. 374 (2020) 107372 MR4157576
6	D Ben-Zvi, J Francis, D Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010) 909 MR2669705
7	A Blanc, M Robalo, B Toën, G Vezzosi, Motivic realizations of singularity categories and vanishing cycles, J. Éc. polytech. Math. 5 (2018) 651 MR3877165
8	A I Bondal, M M Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1183 MR1039961
9	M Booth, Noncommutative deformation theory, the derived quotient, and DG singularity categories, preprint (2018) arXiv:1810.10060
10	C Braun, J Chuang, A Lazarev, Derived localisation of algebras and modules, Adv. Math. 328 (2018) 555 MR3771137
11	T Brüstle, D Yang, Ordered exchange graphs, from: "Advances in representation theory of algebras" (editors D J Benson, H Krause, A Skowroński), Eur. Math. Soc. (2013) 135 MR3220536
12	R O Buchweitz, Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, preprint (1986)
13	H Clemens, The infinitesimal Abel–Jacobi mapping and moving the 𝒪(2) + 𝒪(−4) curve, Duke Math. J. 59 (1989) 233 MR1016885
14	W Donovan, M Wemyss, Noncommutative deformations and flops, Duke Math. J. 165 (2016) 1397 MR3504176
15	W Donovan, M Wemyss, Contractions and deformations, Amer. J. Math. 141 (2019) 563 MR3956515
16	V Drinfeld, DG quotients of DG categories, J. Algebra 272 (2004) 643 MR2028075
17	A Dugas, A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015) 2281 MR3326012
18	T Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011) 223 MR2824483
19	A I Efimov, V A Lunts, D O Orlov, Deformation theory of objects in homotopy and derived categories, I : General theory, Adv. Math. 222 (2009) 359 MR2538013
20	A I Efimov, V A Lunts, D O Orlov, Deformation theory of objects in homotopy and derived categories, II : Pro-representability of the deformation functor, Adv. Math. 224 (2010) 45 MR2600992
21	D Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980) 35 MR0570778
22	A Elagin, V A Lunts, O M Schnürer, Smoothness of derived categories of algebras, Mosc. Math. J. 20 (2020) 277 MR4088795
23	D Gaitsgory, ind-coherent sheaves, Mosc. Math. J. 13 (2013) 399 MR3136100
24	D Gaitsgory, N Rozenblyum, A study in derived algebraic geometry, I : Correspondences and duality, 221, Amer. Math. Soc. (2017) MR3701352
25	G M Greuel, T H Pham, Mather–Yau theorem in positive characteristic, J. Algebraic Geom. 26 (2017) 347 MR3606998
26	J A Guccione, J J Guccione, M J Redondo, O E Villamayor, Hochschild and cyclic homology of hypersurfaces, Adv. Math. 95 (1992) 18 MR1176152
27	Z Hua, Contraction algebra and singularity of three-dimensional flopping contraction, Math. Z. 290 (2018) 431 MR3848439
28	Z Hua, Y Toda, Contraction algebra and invariants of singularities, Int. Math. Res. Not. 2018 (2018) 3173 MR3805201
29	Z Hua, G Zhou, Quasi-homogeneity of potentials, J. Noncommut. Geom. 15 (2021) 399 MR4325712
30	Z Hua, G Zhou, Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras, Math. Ann. 386 (2023) 1605 MR4612402
31	A Ikeda, Y Qiu, q–stability conditions on Calabi–Yau-𝕏 categories, Compos. Math. 159 (2023) 1347 MR4599210
32	O Iyama, S Oppermann, Stable categories of higher preprojective algebras, Adv. Math. 244 (2013) 23 MR3077865
33	O Iyama, Y Yoshino, Mutation in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (2008) 117 MR2385669
34	G Jasso, F Muro, The derived Auslander–Iyama correspondence, preprint (2022) arXiv:2208.14413
35	G Jasso, B Keller, F Muro, The Donovan-Wemyss conjecture via the derived Auslander–Iyama correspondence, from: "Triangulated categories in representation theory and beyond" (editors P A Bergh, S Oppermann, Ø Solberg), Abel Symp., Springer (2024) 105
36	M Kalck, D Yang, Relative singularity categories, II: DG models, preprint (2018) arXiv:1803.08192
37	S Katz, D R Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom. 1 (1992) 449 MR1158626
38	Y Kawamata, On multi-pointed non-commutative deformations and Calabi–Yau threefolds, Compos. Math. 154 (2018) 1815 MR3867285
39	Y Kawamata, Non-commutative deformations of simple objects in a category of perverse coherent sheaves, Selecta Math. 26 (2020) 43 MR4117994
40	B Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998) 223 MR1492902
41	B Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999) 1 MR1667558
42	B Keller, Bimodule complexes via strong homotopy actions, Algebr. Represent. Theory 3 (2000) 357 MR1808132
43	B Keller, Koszul duality and coderived categories (after K Lefèvre), preprint (2003)
44	B Keller, On differential graded categories, from: "International Congress of Mathematicians, II" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc. (2006) 151 MR2275593
45	B Keller, Calabi–Yau triangulated categories, from: "Trends in representation theory of algebras and related topics", Eur. Math. Soc. (2008) 467 MR2484733
46	B Keller, Deformed Calabi–Yau completions, J. Reine Angew. Math. 654 (2011) 125 MR2795754
47	B Keller, Singular Hochschild cohomology via the singularity category, C. R. Math. Acad. Sci. Paris 356 (2018) 1106 MR3907577
48	B Keller, P Nicolás, Weight structures and simple DG modules for positive DG algebras, Int. Math. Res. Not. 2013 (2013) 1028 MR3031826
49	B Keller, D Vossieck, Sous les catégories dérivées, C. R. Math. Acad. Sci. Paris 305 (1987) 225 MR0907948
50	B Keller, D Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988) 239 MR0976638
51	B Keller, D Yang, Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011) 2118 MR2739775
52	T Kinjo, N Masuda, Global critical chart for local Calabi–Yau threefolds, Int. Math. Res. Not. (2023)
53	J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) MR1658959
54	M Kontsevich, Y Soibelman, Notes on A_∞–algebras, A_∞–categories and non-commutative geometry, from: "Homological mirror symmetry" (editors A Kapustin, M Kreuzer, K G Schlesinger), Lecture Notes in Phys. 757, Springer (2009) 153 MR2596638
55	K Lefèvre-Hasegawa, Sur les A_∞–catégories, PhD thesis, Université Paris 7 (2003) arXiv:math/0310337
56	J Lurie, Higher algebra, preprint (2017)
57	J N Mather, S S T Yau, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982) 243 MR0674404
58	D O Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004) 240 MR2101296
59	Y Palu, Grothendieck group and generalized mutation rule for 2–Calabi–Yau triangulated categories, J. Pure Appl. Algebra 213 (2009) 1438 MR2497588
60	H C Pinkham, Factorization of birational maps in dimension 3, from: "Singularities, II", Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 343 MR0713260
61	P G Plamondon, Cluster characters for cluster categories with infinite-dimensional morphism spaces, Adv. Math. 227 (2011) 1 MR2782186
62	A Polishchuk, A_∞–structures associated with pairs of 1–spherical objects and noncommutative orders over curves, Trans. Amer. Math. Soc. 373 (2020) 6029 MR4155172
63	D Quillen, Cyclic cohomology and algebra extensions, K–Theory 3 (1989) 205 MR1040400
64	M Reid, Minimal models of canonical 3–folds, from: "Algebraic varieties and analytic varieties" (editor S Iitaka), Adv. Stud. Pure Math. 1, North-Holland (1983) 131 MR0715649
65	I Reiten, 2–Calabi–Yau tilted algebras, São Paulo J. Math. Sci. 4 (2010) 529 MR2856198
66	E Segal, The A_∞ deformation theory of a point and the derived categories of local Calabi–Yaus, J. Algebra 320 (2008) 3232 MR2450725
67	B Toën, The homotopy theory of DG-categories and derived Morita theory, Invent. Math. 167 (2007) 615 MR2276263
68	M Tousi, S Yassemi, Tensor products of some special rings, J. Algebra 268 (2003) 672 MR2009326
69	M Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004) 423 MR2057015
70	M Van den Bergh, The signs of Serre duality, (2008)
71	M Van den Bergh, Calabi–Yau algebras and superpotentials, Selecta Math. 21 (2015) 555 MR3338683
72	Z Wang, Gerstenhaber algebra and Deligne’s conjecture on the Tate–Hochschild cohomology, Trans. Amer. Math. Soc. 374 (2021) 4537 MR4273171
73	M Wemyss, Lectures on noncommutative resolutions, lecture notes (2012) arXiv:1210.2564
74	M Wemyss, Flops and clusters in the homological minimal model programme, Invent. Math. 211 (2018) 435 MR3748312
75	M Wemyss, A lockdown survey on cDV singularities, from: "McKay correspondence, mutation and related topics" (editors Y Ito, A Ishii, O Iyama), Adv. Stud. Pure Math. 88, Math. Soc. Japan (2023) 47 MR4603584