Abstract
We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie’s strongly (co)cartesian n-cubes. As applications we construct abstract Auslander–Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi–Yau property. This is used to offer a new perspective on May’s axioms for monoidal, triangulated categories.
Funding statement: The first-named author was supported by the Dutch Science Foundation (NWO). The second-named author was supported by grant GAČR P201/12/G028 from the Czech Science Foundation.
References
[1]
M. Ando, A. J. Blumberg and D. Gepner,
Twists of K-theory and TMF,
Superstrings, geometry, topology, and
[2] M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins and C. Rezk, Units of ring spectra and Thom spectra, preprint (2009), http://arxiv.org/abs/0810.4535v3. Search in Google Scholar
[3] L. Angeleri Hügel, D. Happel and H. Krause (eds.), Handbook of tilting theory, London Math. Soc. Lecture Note Ser. 332, Cambridge University Press, Cambridge 2007. 10.1017/CBO9780511735134Search in Google Scholar
[4] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. 94, American Mathematical Society, Providence 1969. 10.1090/memo/0094Search in Google Scholar
[5] M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, Cambridge Stud. Adv. Math. 36, Cambridge University Press, Cambridge 1997; corrected reprint of the 1995 original. Search in Google Scholar
[6] J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, Astérisque 315, Société Mathématique de France, Paris 2007. Search in Google Scholar
[7] H. Becker, Models for singularity categories, Adv. Math. 254 (2014), 187–232. 10.1016/j.aim.2013.11.016Search in Google Scholar
[8] D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80. 10.4064/fm-153-1-59-80Search in Google Scholar
[9] D. J. Benson, S. B. Iyengar and H. Krause, Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), no. 3, 1643–1684. 10.4007/annals.2011.174.3.6Search in Google Scholar
[10] I. N. Bernšteĭn, I. M. Gel’fand and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2, 19–33. 10.1070/RM1973v028n02ABEH001526Search in Google Scholar
[11] A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205. Search in Google Scholar
[12] F. Borceux, Handbook of categorical algebra 1, Encyclopedia Math. Appl. 50, Cambridge University Press, Cambridge 1994. 10.1017/CBO9780511525858Search in Google Scholar
[13] R.-O. Buchweitz, Maximal Cohen–Macaulay modules and Tate–cohomology over Gorenstein rings, preprint (1987), http://hdl.handle.net/1807/16682. Search in Google Scholar
[14] D.-C. Cisinski, Images directes cohomologiques dans les catégories de modèles, Ann. Math. Blaise Pascal 10 (2003), 195–244. 10.5802/ambp.174Search in Google Scholar
[15] W. Crawley-Boevey, Rigid integral representations of quivers, Representation theory of algebras (Cocoyoc 1994), CMS Conf. Proc. 18, American Mathematical Society, Providence (1996), 155–163. Search in Google Scholar
[16] D. Dugger and B. Shipley, K-theory and derived equivalences, Duke Math. J. 124 (2004), no. 3, 587–617. 10.1215/S0012-7094-04-12435-2Search in Google Scholar
[17] A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys Monogr. 47, American Mathematical Society, Providence 1997; with an appendix by M. Cole. Search in Google Scholar
[18] E. E. Enochs and O. M. G. Jenda, Relative homological algebra, De Gruyter Exp. Math. 30, De Gruyter, Berlin 2000. 10.1515/9783110803662Search in Google Scholar
[19] J. Franke, Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, preprint (1996), www.math.uiuc.edu/K-theory/0139/. Search in Google Scholar
[20] B. Fresse, Modules over operads and functors, Lecture Notes in Math. 1967, Springer, Berlin 2009. 10.1007/978-3-540-89056-0Search in Google Scholar
[21] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309. 10.1007/BF01298413Search in Google Scholar
[22] P. Gabriel, Auslander–Reiten sequences and representation-finite algebras, Representation theory, I (Ottawa 1979), Lecture Notes in Math. 831, Springer, Berlin (1980), 1–71. 10.1007/BFb0089778Search in Google Scholar
[23] J. Gillespie, Model structures on exact categories, J. Pure Appl. Algebra 215 (2011), no. 12, 2892–2902. 10.1016/j.jpaa.2011.04.010Search in Google Scholar
[24] T. G. Goodwillie, Calculus. II. Analytic functors, K-Theory 5 (1991/92), no. 4, 295–332. 10.1007/BF00535644Search in Google Scholar
[25]
M. Groth,
Short course on
[26] M. Groth, Derivators, pointed derivators, and stable derivators, Algebr. Geom. Topol. 13 (2013), 313–374. 10.2140/agt.2013.13.313Search in Google Scholar
[27] M. Groth, K. Ponto and M. Shulman, The additivity of traces in monoidal derivators, J. K-Theory 14 (2014), no. 3, 422–494. 10.1017/is014005011jkt262Search in Google Scholar
[28] M. Groth, K. Ponto and M. Shulman, Mayer–Vietoris sequences in stable derivators, Homology Homotopy Appl. 16 (2014), no. 1, 265–294. 10.4310/HHA.2014.v16.n1.a15Search in Google Scholar
[29] M. Groth and M. Shulman, Something on enriched derivators and weighted homotopy (co)limits, in preparation. Search in Google Scholar
[30] M. Groth and J. Šťovíček, Abstract representation theory of Dynkin quivers of type A, preprint (2014), http://arxiv.org/abs/1409.5003. 10.1016/j.aim.2016.02.018Search in Google Scholar
[31] M. Groth and J. Šťovíček, Abstract tilting theory for quivers and related categories, preprint (2015), http://arxiv.org/abs/1512.06267. 10.2140/akt.2018.3.71Search in Google Scholar
[32] M. Groth and J. Šťovíček, Tilting theory for trees via stable homotopy theory, J. Pure Appl. Algebra 220 (2016), no. 6, 2324–2363. 10.1016/j.jpaa.2015.11.009Search in Google Scholar
[33] A. Grothendieck, Les dérivateurs, preprint (1990), http://people.math.jussieu.fr/␣maltsin/groth/Derivateursengl.html. Search in Google Scholar
[34] R. Guitart, Relations et carrés exacts, Ann. Sci. Math. Québec 4 (1980), no. 2, 103–125. Search in Google Scholar
[35] R. Guitart, Contractible exact squares, Appl. Categ. Structures 22 (2014), no. 5–6, 873–898. 10.1007/s10485-013-9353-4Search in Google Scholar
[36] D. Happel, Dynkin algebras, Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin, 37ème année (Paris 1985), Lecture Notes in Math. 1220, Springer, Berlin (1986), 1–14. 10.1007/BFb0099501Search in Google Scholar
[37] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. 10.1007/BF02564452Search in Google Scholar
[38] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge University Press, Cambridge 1988. 10.1017/CBO9780511629228Search in Google Scholar
[39] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar
[40]
M. Hazewinkel, W. Hesselink, D. Siersma and F. D. Veldkamp,
The ubiquity of Coxeter-Dynkin diagrams (an introduction to the
[41] A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71, American Mathematical Society, Providence 1988. 10.1090/memo/0383Search in Google Scholar
[42] V. Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291–3323. 10.1080/00927879708826055Search in Google Scholar
[43] H. Holm, Gorenstein homological algebra, Ph.D. thesis, University of Copenhagen, 2004; www.math.ku.dk/~holm/download/PhDthesis.pdf. Search in Google Scholar
[44] M. Hovey, Model categories, Math. Surveys Monogr. 63, American Mathematical Society, Providence 1999. Search in Google Scholar
[45] M. Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2441–2457. 10.1090/S0002-9947-01-02721-0Search in Google Scholar
[46] M. Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. 10.1007/s00209-002-0431-9Search in Google Scholar
[47] M. Hovey, B. Shipley and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. 10.1090/S0894-0347-99-00320-3Search in Google Scholar
[48] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–553. 10.4171/dm/86Search in Google Scholar
[49] A. Joyal, The theory of quasi-categories and its applications, Lectures at CRM Barcelona, 2008. Search in Google Scholar
[50] B. Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. 10.24033/asens.1689Search in Google Scholar
[51] B. Keller, On the construction of triangle equivalences, Derived equivalences for group rings, Lecture Notes in Math. 1685, Springer, Berlin (1998), 155–176. 10.1007/BFb0096374Search in Google Scholar
[52] B. Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. 10.4171/dm/199Search in Google Scholar
[53]
B. Keller and A. Neeman,
The connection between May’s axioms for a triangulated tensor product and Happel’s description of the derived category of the quiver
[54] B. Keller and S. Scherotzke, The integral cluster category, Int. Math. Res. Not. IMRN 2012 (2012), no. 12, 2867–2887. 10.1093/imrn/rnr129Search in Google Scholar
[55] M. Köhler, Universal coefficient theorems in equivariant KK-theory, Ph.D. thesis, Georg-August-Universität Göttingen, 2011. Search in Google Scholar
[56] H. Krause and J. Le, The Auslander–Reiten formula for complexes of modules, Adv. Math. 207 (2006), no. 1, 133–148. 10.1016/j.aim.2005.11.008Search in Google Scholar
[57] S. Ladkani, Universal derived equivalences of posets, preprint (2007), http://arxiv.org/abs/0705.0946. Search in Google Scholar
[58] S. Ladkani, Homological properties of finite partially ordered sets, Ph.D. thesis, The Hebrew University of Jerusalem, 2008. Search in Google Scholar
[59] L. G. Lewis Jr., J. P. May, M. Steinberger and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, Berlin 1986; with contributions by J. E. McClure. 10.1007/BFb0075778Search in Google Scholar
[60] J. Lurie, Higher topos theory, Ann. of Math. Stud. 170, Princeton University Press, Princeton 2009. 10.1515/9781400830558Search in Google Scholar
[61] J. Lurie, Higher algebra, preprint (2014), www.math.harvard.edu/␣lurie/papers/higheralgebra.pdf. Search in Google Scholar
[62] S. Mac Lane, Categories for the working mathematician, 2nd ed., Grad. Texts in Math. 5, Springer, New York 1998. Search in Google Scholar
[63] G. Maltsiniotis, La K-théorie d’un dérivateur triangulé, Categories in algebra, geometry and mathematical physics, Contemp. Math. 431, American Mathematical Society, Providence (2007), 341–368. 10.1090/conm/431/08280Search in Google Scholar
[64] G. Maltsiniotis, Carrés exacts homotopiques, et dérivateurs, preprint (2011), http://arxiv.org/pdf/1101.4144v1. Search in Google Scholar
[65] M. A. Mandell and J. P. May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159, American Mathematical Society, Providence 2002. 10.1090/memo/0755Search in Google Scholar
[66] M. A. Mandell, J. P. May, S. Schwede and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. 10.1112/S0024611501012692Search in Google Scholar
[67] J. P. May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001), no. 1, 34–73.10.1006/aima.2001.1995Search in Google Scholar
[68] J. P. May and J. Sigurdsson, Parametrized homotopy theory, Math. Surveys Monogr. 132, American Mathematical Society, Providence 2006. 10.1090/surv/132Search in Google Scholar
[69]
R. Meyer and R. Nest,
[70] J.-I. Miyachi and A. Yekutieli, Derived Picard groups of finite-dimensional hereditary algebras, Compositio Math. 129 (2001), no. 3, 341–368. 10.1023/A:1012579131516Search in Google Scholar
[71]
F. Morel and V. Voevodsky,
[72] K. Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83–142. Search in Google Scholar
[73] K. Ponto and M. Shulman, The linearity of traces in monoidal categories and bicategories, preprint (2014), http://arxiv.org/abs/1406.7854. Search in Google Scholar
[74] D. G. Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer, Berlin 1967. 10.1007/BFb0097438Search in Google Scholar
[75] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366. 10.1090/S0894-0347-02-00387-9Search in Google Scholar
[76] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. 10.1112/jlms/s2-39.3.436Search in Google Scholar
[77] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. 10.1112/jlms/s2-43.1.37Search in Google Scholar
[78] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), no. 1, 149–170. 10.1112/S0024610797005309Search in Google Scholar
[79] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224. 10.1007/BF02566682Search in Google Scholar
[80] A.-C. van Roosmalen, Abelian hereditary fractionally Calabi–Yau categories, Int. Math. Res. Not. IMRN 2012 (2012), no. 12, 2708–2750. 10.1093/imrn/rnr118Search in Google Scholar
[81] S. Schwede, Morita theory in abelian, derived and stable model categories, Structured ring spectra, London Math. Soc. Lecture Note Ser. 315, Cambridge University Press, Cambridge (2004), 33–86. 10.1017/CBO9780511529955.005Search in Google Scholar
[82] S. Schwede, The p-order of topological triangulated categories, J. Topol. 6 (2013), 868–914. 10.1112/jtopol/jtt018Search in Google Scholar
[83] S. Schwede and B. E. Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491–511. 10.1112/S002461150001220XSearch in Google Scholar
[84] S. Schwede and B. Shipley, Stable model categories are categories of modules, Topology 42 (2003), no. 1, 103–153. 10.1016/S0040-9383(02)00006-XSearch in Google Scholar
[85] J. Šťovíček, Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves, Advances in representation theory of algebras, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2013), 297–367. 10.4171/125-1/10Search in Google Scholar
[86] J. Šťovíček and D. Pospíšil, On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings, Trans. Amer. Math. Soc. (2015), 10.1090/tran/6561. 10.1090/tran/6561Search in Google Scholar
[87]
V. Voevodsky,
[88] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick 1983), Lecture Notes in Math. 1126, Springer, Berlin (1985), 318–419. 10.1007/BFb0074449Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
