Hostname: page-component-5c6d5d7d68-vt8vv Total loading time: 0.001 Render date: 2024-08-18T20:22:09.389Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigidity dimension of algebras

Part of: Abelian categories Rings and algebras arising under various constructions Homological algebra Representation theory of rings and algebras

Published online by Cambridge University Press:  26 November 2019

HONGXING CHEN ,
MING FANG ,
OTTO KERNER ,
STEFFEN KOENIG  and
KUNIO YAMAGATA
HONGXING CHEN
Affiliation:
School of Mathematical Sciences & Academy for Multidisciplinary Studies, Capital Normal University, 105 West Third Ring Road North, Haidian District 100048 Beijing, P.R.China. e-mail: chx19830818@163.com
MING FANG
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, & School of Mathematical Sciences, University of Chinese Academy of Sciences East Zhongguancun Road 100190 Beijing, P.R.China. e-mail: fming@amss.ac.cn
OTTO KERNER
Affiliation:
Mathematisches Institut, Heinrich–Heine–Universität, 40225, Düsseldorf, Germany. e-mail: kerner@math.uni-duesseldorf.de
STEFFEN KOENIG
Affiliation:
Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. e-mail: skoenig@mathematik.uni-stuttgart.de
KUNIO YAMAGATA
Affiliation:
Institute of Engineering, Tokyo University of Agriculture and Technology, Nakacho 2-24-16, Koganei, Tokyo 184-8588, Japan, e-mail: yamagata@cc.tuat.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 18G20: Homological dimension 18E30: Derived categories, triangulated categories 16S50: Endomorphism rings; matrix rings
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 417 - 443
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Auslander, M.. Representation dimension of Artin algebras. Queen Mary College Mathematics Notes (Queen Mary College, London, 1971).Google Scholar
Auslander, M. and Reiten, I.. Representation theory of Artin algebras VI: A functorial approach to almost split sequences. Comm. Algebra 6 (1978), 257–300.CrossRefGoogle Scholar
Auslander, M., Reiten, I. and Smalo, S.. Representation theory of Artin algebras. Cambridge Stud. Adv. Math. 36 (Cambridge University Press, 1995).CrossRefGoogle Scholar
Benson, D.J.. Representations and Cohomology II: Cohomology of Groups and Modules. Cambridge Stud. Adv. Math. 31 (Cambridge University Press, 1991).Google Scholar
Buchweitz, R.–O. Morita contexts, idempotents, and Hochschild cohomology, with application to invariant rings. Contemp. Math. 331 (2003), 25–53.Google Scholar
Buchweitz, R.–O., Green, E., Madsen, D. and Solberg, O.. Finite Hochschild cohomology without finite global dimension. Math. Res. Letters 12 (2005), 805–816.CrossRefGoogle Scholar
Chen, H.X., Fang, M., Kerner, O., Koenig, S. and Yamagata, K.. Rigidity dimension of algebras, II: methods and examples, in preparation.Google Scholar
Chen, H.X., Pan, S.Y. and Xi, C.C.. Inductions and restrictions for stable equivalences of Morita type. J. Pure Appl. Algebra 216 (2012), 643–661.CrossRefGoogle Scholar
Chen, H.X. and Xi, C.C.. Dominant dimensions, derived equivalences and tilting modules. Israel J. Math. 215 (2016), 349–395.CrossRefGoogle Scholar
Crawley–Boevey, W.. On the exceptional fibres of Kleinian singularities. Amer. J. Math. 122 (2000), 1027–1037.CrossRefGoogle Scholar
Erdmann, K. and Skowroński, A.. The stable Calabi–Yau dimension of tame symmetric algebras. J. Math. Soc. Japan 58 (2006), 97–123.CrossRefGoogle Scholar
Fang, M.. Permanents, Doty coalgebras and dominant dimension of Schur algebras. Adv. Math. 264 (2014), 155–182.CrossRefGoogle Scholar
Fang, M. and Miyachi, H.. Hochschild cohomology and dominant dimension. Trans. Amer. Math. Soc. 371 (2019), 5267–5292.CrossRefGoogle Scholar
Fang, M. and Koenig, S.. Schur functors and dominant dimension. Trans. Amer. Math. Soc. 363 (2011), 1555–1576.CrossRefGoogle Scholar
Fang, M. and Koenig, S.. Endomorphism algebras of generators over symmetric algebras. J. Algebra 332 (2011), 428–433.CrossRefGoogle Scholar
Fang, M. and Koenig, S.. Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension. Trans. Amer. Math. Soc. 368 (2016), 5037–5055.CrossRefGoogle Scholar
Geiss, C., Leclerc, B. and Schröer, J.. Rigid modules over preprojective algebras. Invent. Math. 165 (2006) 589–632.CrossRefGoogle Scholar
Guo, X.Q.. Representation dimension: An invariant under stable equivalence. Trans. Amer. Math. Soc. 357 (2005), 3255–3263.CrossRefGoogle Scholar
Happel, D.. Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Note Ser. 119 (Cambridge University Press, 1988).CrossRefGoogle Scholar
Hoshino, M.. Modules without self-extensions and Nakayama conjecture. Arch. Math. 43 (1984), 493–500.CrossRefGoogle Scholar
Hu, W. and Xi, C.C.. Derived equivalences and stable equivalences of Morita type, I. Nagoya Math. J. 200 (2010), 107–152.CrossRefGoogle Scholar
Hu, W. and Xi, C.C.. Derived equivalences for ϕ-Auslander-Yoneda algebras. Trans. Amer. Math. Soc. 365 (2013), 5681–5711.CrossRefGoogle Scholar
Iyama, O.. Finiteness of representation dimension. Proc. Amer. Math. Soc. 131 (2003), 1011–1014.Google Scholar
Iyama, O.. Auslander correspondence. Adv. Math. 210 (2007), 51–82.CrossRefGoogle Scholar
Iyama, O.. Cluster tilting for higher Auslander algebras. Adv. Math. 226 (2011), 1–61.CrossRefGoogle Scholar
Kerner, O. and Yamagata, K.. Morita algebras. J. Algebra 382 (2013), 185–202.CrossRefGoogle Scholar
Linckelmann, M.. Transfer in Hochschild cohomology of blocks of finite groups. Algebr. Represent. Theory 2 (1999), 107–135.CrossRefGoogle Scholar
Liu, Y.M. and Xi, C.C.. Constructions of stable equivalences of Morita type for finite-dimensional algebras, II. Math. Zeit. 251 (2005), 21–39.CrossRefGoogle Scholar
Liu, Y.M. and Xi, C.C.. Constructions of stable equivalences of Morita type for finite-dimensional algebras, III. J. London Math. Soc. 76 (2007), 567–585.CrossRefGoogle Scholar
Martinez–Villa, R.. Properties that are left invariant under stable equivalence. Comm. Algebra 18 (1990), 4141–4169.CrossRefGoogle Scholar
Müller, B.. The classification of algebras by dominant dimension. Canad. J. Math. 20 (1968), 398–409.CrossRefGoogle Scholar
Rickard, J.. Derived categories and stable equivalences. J. Pure Appl. Algebra 61 (1989), 303–317.CrossRefGoogle Scholar
Rouquier, R.. q-Schur algebras and complex reflection groups. Moscow Math. J. 8 (2008), 119–158.CrossRefGoogle Scholar
Snashall, N.. Support varieties and the Hochschild cohomology ring modulo nilpotence, in: Proceedings of the 41st Symposium on Ring Theory and Representation Theory, 68–82, Symp. Ring Theory Represent. Theory Organ. Comm., Tsukuba (2009).Google Scholar
Symonds, P.. On the Castelnuovo-Mumford regularity of the cohomology ring of a group. J. Amer. Math. Soc. 23 (2010), 1159–1173.CrossRefGoogle Scholar
Tachikawa, H.. Quasi-Frobenius rings and generalizations. Lecture Notes in Math. 351 (Berlin–Heidelberg, New York, 1973).CrossRefGoogle Scholar
Xu, F.. Hochschild and ordinary cohomology rings of small categories. Adv. Math. 219 (2008), 1872–1893.CrossRefGoogle Scholar
Yamagata, K.. Frobenius algebras. In Handbook of algebra, vol. 1 (North–Holland, Amsterdam, 1996), pp. 841–887.CrossRefGoogle Scholar
Yamagata, K. and Kerner, O.. Morita theory, revisited. Contemporary Mathematics, A.M.S. 607 (2014), 85–96.CrossRefGoogle Scholar