Duality and symmetry of complexity over complete intersections via exterior homology
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- by Jian Liu and Josh Pollitz PDF
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Abstract:
We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally complete intersection ring are self-dual under Grothendieck duality. This was proved by Stevenson when the ring is a quotient of a regular ring modulo a regular sequence; we offer two independent proofs in the more general setting. Second, we use these techniques to supply new proofs that complete intersections possess symmetry of complexity.References
- Luchezar L. Avramov, Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 1–118. MR 1648664
- Luchezar L. Avramov and Ragnar-Olaf Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318. MR 1794064, DOI 10.1007/s002220000090
- Luchezar L. Avramov, Ragnar-Olaf Buchweitz, Srikanth B. Iyengar, and Claudia Miller, Homology of perfect complexes, Adv. Math. 223 (2010), no. 5, 1731–1781. MR 2592508, DOI 10.1016/j.aim.2009.10.009
- Luchezar L. Avramov and Srikanth B. Iyengar, Cohomology over complete intersections via exterior algebras, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 52–75. MR 2681707
- D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
- Dave Benson, Srikanth B. Iyengar, and Henning Krause, A local-global principle for small triangulated categories, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 3, 451–476. MR 3335421, DOI 10.1017/S0305004115000067
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Jon F. Carlson and Srikanth B. Iyengar, Thick subcategories of the bounded derived category of a finite group, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2703–2717. MR 3301878, DOI 10.1090/S0002-9947-2014-06121-7
- W. Dwyer, J. P. C. Greenlees, and S. Iyengar, Finiteness in derived categories of local rings, Comment. Math. Helv. 81 (2006), no. 2, 383–432. MR 2225632, DOI 10.4171/CMH/56
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
- Michael J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 73–96. MR 932260
- Janina C Letz, Local to global principles for generation time over noether algebras, arXiv:1906.06104 (2019).
- Amnon Neeman, The chromatic tower for $D(R)$, Topology 31 (1992), no. 3, 519–532. With an appendix by Marcel Bökstedt. MR 1174255, DOI 10.1016/0040-9383(92)90047-L
- Josh Pollitz, Cohomological supports over derived complete intersections and local rings, arXiv:1912.12009 (2019).
- Josh Pollitz, The derived category of a locally complete intersection ring, Adv. Math. 354 (2019), 106752, 18. MR 3988642, DOI 10.1016/j.aim.2019.106752
- Gunnar Sjödin, A set of generators for $\textrm {Ext}_{R}(k,k)$, Math. Scand. 38 (1976), no. 2, 199–210. MR 422248, DOI 10.7146/math.scand.a-11629
- Greg Stevenson, Duality for bounded derived categories of complete intersections, Bull. Lond. Math. Soc. 46 (2014), no. 2, 245–257. MR 3194744, DOI 10.1112/blms/bdt089
- Greg Stevenson, Subcategories of singularity categories via tensor actions, Compos. Math. 150 (2014), no. 2, 229–272. MR 3177268, DOI 10.1112/S0010437X1300746X
- Ryo Takahashi, Classifying thick subcategories of the stable category of Cohen-Macaulay modules, Adv. Math. 225 (2010), no. 4, 2076–2116. MR 2680200, DOI 10.1016/j.aim.2010.04.009
Additional Information
- Jian Liu
- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China
- ORCID: 0000-0001-8360-7024
- Email: liuj231@mail.ustc.edu.cn
- Josh Pollitz
- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- MR Author ID: 1335525
- Email: pollitz@math.utah.edu
- Received by editor(s): June 15, 2020
- Received by editor(s) in revised form: July 11, 2020
- Published electronically: December 16, 2020
- Additional Notes: The first author thanks the China Scholarship Council for financial support to visit Srikanth Iyengar at the University of Utah.
The second author was supported by the National Science Foundation under Grant No. 1840190. - Communicated by: Sarah Witherspoon
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 619-631
- MSC (2020): Primary 13D09; Secondary 13D07, 13H10, 16E45
- DOI: https://doi.org/10.1090/proc/15276
- MathSciNet review: 4198070