Triangulated categories of periodic complexes and orbit categories

Jian Liu

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 3, page 765-792
  • ISSN: 0011-4642

Abstract

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We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if A , B are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of n -periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.

How to cite

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Liu, Jian. "Triangulated categories of periodic complexes and orbit categories." Czechoslovak Mathematical Journal 73.3 (2023): 765-792. <http://eudml.org/doc/299105>.

@article{Liu2023,
abstract = {We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if $A$, $B$ are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of $n$-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.},
author = {Liu, Jian},
journal = {Czechoslovak Mathematical Journal},
keywords = {periodic complex; orbit category; triangulated hull; derived category; derived equivalence; dg category; Koszul duality},
language = {eng},
number = {3},
pages = {765-792},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Triangulated categories of periodic complexes and orbit categories},
url = {http://eudml.org/doc/299105},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Liu, Jian
TI - Triangulated categories of periodic complexes and orbit categories
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 765
EP - 792
AB - We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if $A$, $B$ are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of $n$-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.
LA - eng
KW - periodic complex; orbit category; triangulated hull; derived category; derived equivalence; dg category; Koszul duality
UR - http://eudml.org/doc/299105
ER -

References

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  1. Avramov, L. L., Buchweitz, R.-O., Iyengar, S. B., 10.1007/s00222-007-0041-6, Invent. Math. 169 (2007), 1-35. (2007) Zbl1153.13010MR2308849DOI10.1007/s00222-007-0041-6
  2. Avramov, L. L., Buchweitz, R.-O., Iyengar, S. B., Miller, C., 10.1016/j.aim.2009.10.009, Adv. Math. 223 (2010), 1731-1781. (2010) Zbl1186.13006MR2592508DOI10.1016/j.aim.2009.10.009
  3. Beilinson, A. A., Coherent sheaves on P n and problems of linear algebra, Funkts. Anal. Prilozh. 12 (1978), 68-69 Russian. (1978) Zbl0402.14006MR0509388
  4. Beilinson, A. A., Bernstein, J., Deligne, P., Faisceaux pervers, Analysis and Topology on Singular Spaces. I Astérisque 100. Société mathématique de France, Paris (1982), 5-171 French. (1982) Zbl0536.14011MR0751966
  5. Beilinson, A. A., Ginzburg, V., Soergel, W., 10.1090/S0894-0347-96-00192-0, J. Am. Math. Soc. 9 (1996), 473-527. (1996) Zbl0864.17006MR1322847DOI10.1090/S0894-0347-96-00192-0
  6. Benson, D. J., Iyengar, S. B., Krause, H., 10.4007/annals.2011.174.3.6, Ann. Math. (2) 174 (2011), 1643-1684. (2011) Zbl1261.20057MR2846489DOI10.4007/annals.2011.174.3.6
  7. Bernstein, I. N., Gel'fand, I. M., Gel'fand, S. I., Algebraic vector bundles on P n and problems of linear algebra, Funkts. Anal. Prilozh. 12 (1978), 66-67 Russian. (1978) Zbl0402.14005MR0509387
  8. Bökstedt, M., Neeman, A., Homotopy limits in triangulated categories, Compos. Math. 86 (1993), 209-234. (1993) Zbl0802.18008MR1214458
  9. Bruns, W., Herzog, J., 10.1017/CBO9780511608681, Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1998). (1998) Zbl0909.13005MR1251956DOI10.1017/CBO9780511608681
  10. Buchweitz, R.-O., 10.1090/surv/262, Mathematical Surveys and Monographs 262. AMS, Providence (2021). (2021) Zbl07498869MR4390795DOI10.1090/surv/262
  11. Cartan, H., Eilenberg, S., Homological Algebra, Princeton Mathematical Series 19. Princeton University Press, Princeton (1956). (1956) Zbl0075.24305MR0077480
  12. Chen, X.-W., Liu, J., Wang, R., 10.1007/s10468-021-10104-3, (to appear) in Algebr. Represent. Theory. MR4568856DOI10.1007/s10468-021-10104-3
  13. Drinfeld, V., 10.1016/j.jalgebra.2003.05.001, J. Algebra 272 (2004), 643-691. (2004) Zbl1064.18009MR2028075DOI10.1016/j.jalgebra.2003.05.001
  14. Eisenbud, D., Fløystad, G., Schreyer, F.-O., 10.1090/S0002-9947-03-03291-4, Trans. Am. Math. Soc. 355 (2003), 4397-4426. (2003) Zbl1063.14021MR1990756DOI10.1090/S0002-9947-03-03291-4
  15. Enochs, E. E., Jenda, O. M. G., 10.1515/9783110803662, De Gruyter Expositions in Mathematics 30. Walter De Gruyter, Berlin (2000). (2000) Zbl0952.13001MR1753146DOI10.1515/9783110803662
  16. Happel, D., 10.1007/BF02564452, Comment. Math. Helv. 62 (1987), 339-389. (1987) Zbl0626.16008MR0910167DOI10.1007/BF02564452
  17. Iyengar, S. B., Krause, H., 10.4171/dm/209, Doc. Math. 11 (2006), 207-240. (2006) Zbl1119.13014MR2262932DOI10.4171/dm/209
  18. Iyengar, S. B., Letz, J. C., Liu, J., Pollitz, J., 10.2140/pjm.2022.318.275, Pac. J. Math. 318 (2022), 275-293. (2022) Zbl07578612MR4474363DOI10.2140/pjm.2022.318.275
  19. Kalck, M., Yang, D., 10.1016/j.jpaa.2017.11.011, J. Pure Appl. Algebra 222 (2018), 3005-3035. (2018) Zbl1410.16012MR3795632DOI10.1016/j.jpaa.2017.11.011
  20. Keller, B., 10.24033/asens.1689, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 63-102. (1994) Zbl0799.18007MR1258406DOI10.24033/asens.1689
  21. Keller, B., 10.1016/S0022-4049(97)00152-7, J. Pure Appl. Algebra 136 (1999), 1-56. (1999) Zbl0923.19004MR1667558DOI10.1016/S0022-4049(97)00152-7
  22. Keller, B., 10.4171/dm/199, Doc. Math. 10 (2005), 551-581. (2005) Zbl1086.18006MR2184464DOI10.4171/dm/199
  23. Keller, B., Corrections to `On triangulated orbit categories', Available at (2009), 5 pages. (2009) 
  24. Krause, H., 10.1112/S0010437X05001375, Compos. Math. 141 (2005), 1128-1162. (2005) Zbl1090.18006MR2157133DOI10.1112/S0010437X05001375
  25. Krause, H., 10.1017/CBO9781139107075.005, Triangulated Categories London Mathematical Society Lecture Note Series 375. Cambridge University Press, Cambridge (2010), 161-235. (2010) Zbl1232.18012MR2681709DOI10.1017/CBO9781139107075.005
  26. Neeman, A., 10.24033/asens.1659, Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 547-566. (1992) Zbl0868.19001MR1191736DOI10.24033/asens.1659
  27. Neeman, A., 10.1090/S0894-0347-96-00174-9, J. Am. Math. Soc. 9 (1996), 205-236. (1996) Zbl0864.14008MR1308405DOI10.1090/S0894-0347-96-00174-9
  28. Neeman, A., 10.1515/9781400837212, Annals of Mathematics Studies 148. Princeton University Press, Princeton (2001). (2001) Zbl0974.18008MR1812507DOI10.1515/9781400837212
  29. Orlov, D., 10.1007/978-0-8176-4747-6_16, Algebra, Arithmetic, and Geometry. Volume II Progress in Mathematics 270. Birkhäuser, Boston (2009), 503-531. (2009) Zbl1200.18007MR2641200DOI10.1007/978-0-8176-4747-6_16
  30. Peng, L., Xiao, J., 10.1006/jabr.1997.7152, J. Algebra 198 (1997), 19-56. (1997) Zbl0893.16007MR1482975DOI10.1006/jabr.1997.7152
  31. Peng, L., Xiao, J., 10.1007/s002220000062, Invent. Math. 140 (2000), 563-603. (2000) Zbl0966.16006MR1760751DOI10.1007/s002220000062
  32. Rickard, J., 10.1112/jlms/s2-39.3.436, J. Lond. Math. Soc., II. Ser. 39 (1989), 436-456. (1989) Zbl0642.16034MR1002456DOI10.1112/jlms/s2-39.3.436
  33. Rickard, J., 10.1112/jlms/s2-43.1.37, J. Lond. Math. Soc., II. Ser. 43 (1991), 37-48. (1991) Zbl0683.16030MR1099084DOI10.1112/jlms/s2-43.1.37
  34. Ringel, C. M., Zhang, P., 10.1016/j.jalgebra.2016.12.001, J. Algebra 475 (2017), 327-360. (2017) Zbl1406.16010MR3612474DOI10.1016/j.jalgebra.2016.12.001
  35. Stai, T., 10.4310/MRL.2018.v25.n1.a9, Math. Res. Lett. 25 (2018), 199-236. (2018) Zbl1427.16005MR3818620DOI10.4310/MRL.2018.v25.n1.a9
  36. Tang, X., Huang, Z., 10.1016/j.jalgebra.2019.12.011, J. Algebra 549 (2020), 128-164. (2020) Zbl1432.18009MR4050670DOI10.1016/j.jalgebra.2019.12.011
  37. Zhao, X., 10.1007/s11425-014-4852-9, Sci. China, Math. 57 (2014), 2329-2334. (2014) Zbl1304.18029MR3266493DOI10.1007/s11425-014-4852-9

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