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Rigidity dimension of algebras

Part of: Abelian categories Homological algebra Rings and algebras arising under various constructions 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: [email protected]
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: [email protected]
OTTO KERNER
Affiliation:
Mathematisches Institut, Heinrich–Heine–Universität, 40225, Düsseldorf, Germany. e-mail: [email protected]
STEFFEN KOENIG
Affiliation:
Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. e-mail: [email protected]
KUNIO YAMAGATA
Affiliation:
Institute of Engineering, Tokyo University of Agriculture and Technology, Nakacho 2-24-16, Koganei, Tokyo 184-8588, Japan, e-mail: [email protected]
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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

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