Higher ideal approximation theory
HTML articles powered by AMS MathViewer
- by Javad Asadollahi and Somayeh Sadeghi PDF
- Trans. Amer. Math. Soc. 375 (2022), 2113-2145 Request permission
Abstract:
Our aim in this paper is to introduce the so-called ideal approximation theory into higher homological algebra. To this end, we introduce some important notions from approximation theory into the theory of $n$-exact categories and prove some results. In particular, the higher version of notions such as ideal cotorsion pairs, phantom ideals, Salce’s Lemma and Wakamatsu’s Lemma for ideals are introduced and studied. Our results motivate the definitions and show that $n$-exact categories are the appropriate context for the study of higher ideal approximation theory.References
- Javad Asadollahi, Rasool Hafezi, and Somayeh Sadeghi, $n\Bbb Z$-Gorenstein cluster tilting subcategories, J. Algebra 580 (2021), 127–157. MR 4243360, DOI 10.1016/j.jalgebra.2021.04.004
- Javad Asadollahi, Azadeh Mehregan, and Somayeh Sadeghi, Cotorsion classes in higher homological algebra, J. Pure Appl. Algebra 226 (2022), no. 2, Paper No. 106839, 14. MR 4287791, DOI 10.1016/j.jpaa.2021.106839
- Maurice Auslander, Representation dimension of artin algebras, Queen Mary College Notes, London, 1971.
- Maurice Auslander and Idun Reiten, Stable equivalence of dualizing $R$-varieties, Advances in Math. 12 (1974), 306–366. MR 342505, DOI 10.1016/S0001-8708(74)80007-1
- Maurice Auslander and Idun Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111–152. MR 1097029, DOI 10.1016/0001-8708(91)90037-8
- M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI 10.1016/0021-8693(80)90113-1
- M. Auslander and Sverre O. Smalø, Addendum to: “Almost split sequences in subcategories”, J. Algebra 71 (1981), no. 2, 592–594. MR 630621, DOI 10.1016/0021-8693(81)90199-X
- M. Auslander and Ø. Solberg, Relative homology and representation theory. I. Relative homology and homologically finite subcategories, Comm. Algebra 21 (1993), no. 9, 2995–3031. MR 1228751, DOI 10.1080/00927879308824717
- Apostolos Beligiannis and Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207. MR 2327478, DOI 10.1090/memo/0883
- D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory. I, Fund. Math. 161 (1999), no. 1-2, 37–91. Algebraic topology (Kazimierz Dolny, 1997). MR 1713200, DOI 10.4064/fm-161-1-2-37-91
- D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory. II, Algebr. Represent. Theory 4 (2001), no. 4, 395–404. MR 1863392, DOI 10.1023/A:1012475019810
- L. Bican, R. El Bashir, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390. MR 1832549, DOI 10.1017/S0024609301008104
- Simion Breaz and George Ciprian Modoi, Ideal cotorsion theories in triangulated categories, J. Algebra 567 (2021), 475–532. MR 4163075, DOI 10.1016/j.jalgebra.2020.09.018
- Theo Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. MR 2606234, DOI 10.1016/j.exmath.2009.04.004
- Peter Dräxler, Idun Reiten, Sverre O. Smalø, and Øyvind Solberg, Exact categories and vector space categories, Trans. Amer. Math. Soc. 351 (1999), no. 2, 647–682. With an appendix by B. Keller. MR 1608305, DOI 10.1090/S0002-9947-99-02322-3
- Ramin Ebrahimi and Alireza Nasr-Isfahani, Pure semisimple $n$-cluster tilting subcategories, J. Algebra 549 (2020), 177–194. MR 4050672, DOI 10.1016/j.jalgebra.2019.11.043
- Edgar E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. MR 636889, DOI 10.1007/BF02760849
- Sergio Estrada, Pedro A. Guil Asensio, and Furuzan Ozbek, Covering ideals of morphisms and module representations of the quiver $\Bbb {A}_2$, J. Pure Appl. Algebra 218 (2014), no. 10, 1953–1963. MR 3195419, DOI 10.1016/j.jpaa.2014.02.016
- Francesca Fedele, $d$-Auslander-Reiten sequences in subcategories, Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 342–373. MR 4089379, DOI 10.1017/s0013091519000312
- Leonhard Frerick and Dennis Sieg, Exact categories in functional analysis, Online Lecture Notes, 2010. https://www.math.uni-trier.de/abteilung/analysis/HomAlg.pdf.
- X. H. Fu, P. A. Guil Asensio, I. Herzog, and B. Torrecillas, Ideal approximation theory, Adv. Math. 244 (2013), 750–790. MR 3077888, DOI 10.1016/j.aim.2013.05.020
- X. H. Fu and I. Herzog, Powers of the phantom ideal, Proc. Lond. Math. Soc. (3) 112 (2016), no. 4, 714–752. MR 3483130, DOI 10.1112/plms/pdw006
- Gilles Ph. Gnacadja, Phantom maps in the stable module category, J. Algebra 201 (1998), no. 2, 686–702. MR 1612359, DOI 10.1006/jabr.1997.7303
- Rüdiger Göbel and Jan Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006. MR 2251271, DOI 10.1515/9783110199727
- Martin Herschend, Peter Jørgensen, and Laertis Vaso, Wide subcategories of $d$-cluster tilting subcategories, Trans. Amer. Math. Soc. 373 (2020), no. 4, 2281–2309. MR 4069219, DOI 10.1090/tran/8051
- Ivo Herzog, The phantom cover of a module, Adv. Math. 215 (2007), no. 1, 220–249. MR 2354989, DOI 10.1016/j.aim.2007.03.010
- Ivo Herzog, Phantom morphisms and Salce’s lemma, Expository lectures on representation theory, Contemp. Math., vol. 607, Amer. Math. Soc., Providence, RI, 2014, pp. 57–83. MR 3204866, DOI 10.1090/conm/607/12079
- P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
- Osamu Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22–50. MR 2298819, DOI 10.1016/j.aim.2006.06.002
- Osamu Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51–82. MR 2298820, DOI 10.1016/j.aim.2006.06.003
- Osamu Iyama, Cluster tilting for higher Auslander algebras, Adv. Math. 226 (2011), no. 1, 1–61. MR 2735750, DOI 10.1016/j.aim.2010.03.004
- Osamu Iyama and Gustavo Jasso, Higher Auslander correspondence for dualizing $R$-varieties, Algebr. Represent. Theory 20 (2017), no. 2, 335–354. MR 3638352, DOI 10.1007/s10468-016-9645-0
- Gustavo Jasso, $n$-abelian and $n$-exact categories, Math. Z. 283 (2016), no. 3-4, 703–759. MR 3519980, DOI 10.1007/s00209-016-1619-8
- Gustavo Jasso and Sondre Kvamme, An introduction to higher Auslander-Reiten theory, Bull. Lond. Math. Soc. 51 (2019), no. 1, 1–24. MR 3919557, DOI 10.1112/blms.12204
- Peter Jørgensen, Torsion classes and t-structures in higher homological algebra, Int. Math. Res. Not. IMRN 13 (2016), 3880–3905. MR 3544623, DOI 10.1093/imrn/rnv265
- Bernhard Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), no. 4, 379–417. MR 1052551, DOI 10.1007/BF02568439
- Sondre Kvamme, $d\Bbb Z$-cluster tilting subcategories of singularity categories, Math. Z. 297 (2021), no. 1-2, 803–825. MR 4204714, DOI 10.1007/s00209-020-02534-4
- C. A. McGibbon, Phantom maps, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 1209–1257. MR 1361910, DOI 10.1016/B978-044481779-2/50026-2
- Amnon Neeman, The Brown representability theorem and phantomless triangulated categories, J. Algebra 151 (1992), no. 1, 118–155. MR 1182018, DOI 10.1016/0021-8693(92)90135-9
- Furuzan Ozbek, Precovering and preenveloping ideals, Comm. Algebra 43 (2015), no. 6, 2568–2584. MR 3344207, DOI 10.1080/00927872.2014.903404
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- Luigi Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 11–32. MR 565595
- Hiroyuki Tachikawa, On dominant dimensions of $\textrm {QF}$-3 algebras, Trans. Amer. Math. Soc. 112 (1964), 249–266. MR 161888, DOI 10.1090/S0002-9947-1964-0161888-8
- Takayoshi Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, J. Algebra 134 (1990), no. 2, 298–325. MR 1074331, DOI 10.1016/0021-8693(90)90055-S
Additional Information
- Javad Asadollahi
- Affiliation: Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, P.O.Box: 81746-73441, Isfahan, Iran
- MR Author ID: 55173
- ORCID: 0000-0002-7330-2558
- Email: asadollahi@sci.ui.ac.ir, asadollahi@ipm.ir
- Somayeh Sadeghi
- Affiliation: Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, P.O.Box: 81746-73441, Isfahan, Iran
- MR Author ID: 1432356
- Email: so.sadeghi@sci.ui.ac.ir
- Received by editor(s): October 27, 2020
- Received by editor(s) in revised form: June 18, 2021, August 17, 2021, and August 26, 2021
- Published electronically: January 12, 2022
- Additional Notes: Javad Asadollahi is the corresponding author.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 2113-2145
- MSC (2020): Primary 18E05, 18G25, 18G15, 18E99, 16E30
- DOI: https://doi.org/10.1090/tran/8562
- MathSciNet review: 4378089