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GORENSTEIN HOMOLOGICAL PROPERTIES OF TENSOR RINGS

Part of: Homological methods Homological algebra

Published online by Cambridge University Press:  07 June 2018

XIAO-WU CHEN  and
MING LU
XIAO-WU CHEN
Affiliation:
Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, PR China email xwchen@mail.ustc.edu.cn
MING LU
Affiliation:
Department of Mathematics, Sichuan University, Chengdu 610064, PR China email luming@scu.edu.cn
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Abstract

Let $R$ be a two-sided Noetherian ring, and let $M$ be a nilpotent $R$-bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_{R}(M)$. Under certain conditions, the ring $R$ is Gorenstein if and only if so is $T_{R}(M)$. We characterize Gorenstein projective $T_{R}(M)$-modules in terms of $R$-modules.

MSC classification

Primary: 16E10: Homological dimension 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 18G25: Relative homological algebra, projective classes
Type
Article
Information
Nagoya Mathematical Journal , Volume 237 , March 2020 , pp. 188 - 208
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

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References

Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential , Ann. Inst. Fourier (Grenoble) 59(6) (2009), 2525–2590.Google Scholar
Assem, I., Brustle, T. and Schiffler, R., Cluster-tilted algebras as trivial extensions , Bull. Lond. Math. Soc. 40 (2008), 151–162.Google Scholar
Auslander, M., Reiten, I. and Smalø, S. O., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36 , Cambridge University Press, 1995.Google Scholar
Beligiannis, A., Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras , J. Algebra 288 (2005), 137–211.Google Scholar
Chen, X. W., Singularity categories, Schur functors and triangular matrix rings , Algebr. Represent. Theory 12 (2009), 181–191.Google Scholar
Chen, X. W., Three results on Frobenius categories , Math. Z. 270 (2012), 43–58.Google Scholar
Cohn, P. M., Algebra, Vol. 3, 2nd ed., John Wiley Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1991.Google Scholar
Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra, De Gruyter Expositions in Math. 30 , Walter de Gruyter, Berlin, New York, 2000.Google Scholar
Geiss, C., Leclerc, B. and Schroer, J., Quivers with relations for symmetrizable Cartan matrices I: Foundations , Invent. Math. 209 (2017), 61–158.Google Scholar
Geuenich, J., Quiver modulations and potentials, PhD Thesis, University of Bonn, 2017.Google Scholar
Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc., Lecture Notes Ser. 119 , Cambridge University Press, 1988.Google Scholar
Keller, B., Chain complexes and stable categories , Manuscripta Math. 67 (1990), 379–417.Google Scholar
Keller, B. and Reiten, I., Cluster-tilted algebras are Gorenstein and stably Calabi–Yau , Adv. Math. 211 (2007), 123–151.Google Scholar
Mac Lane, S., Categories for the Working Mathematician, Grad. Texts in Math. 5 , Springer-Verlag, New York, 1998.Google Scholar
Li, F. and Ye, C., Representations of Frobenius-type triangular matrix algebras , Acta Math. Sin. (Engl. Ser.) 33(3) (2017), 341–361.Google Scholar
Luo, X. and Zhang, P., Monic representations and Gorenstein-projective modules , Pacific J. Math. 264(1) (2013), 163–194.Google Scholar
Minamoto, H., Ampleness of two-sided tilting complexes , Int. Math. Res. Not. IMRN 1 (2012), 67–101.Google Scholar
Minamoto, H. and Yamaura, K., Homological dimension formulas for trivial extension algebras, preprint, 2017, arXiv:1710.01469v1.Google Scholar
Roganov, Yu. V., The dimension of the tensor algebra of a projective bimodule , Math. Notes 18(5–6) (1975), 1119–1123.Google Scholar
Wang, R., Gorenstein triangular matrix rings and category algebras , J. Pure Appl. Algebra 220(2) (2016), 666–682.Google Scholar
Xiong, B. L. and Zhang, P., Gorenstein-projective modules over triangular matrix Artin algebras , J. Algebra Appl. 11(4) (2012), 1250066.Google Scholar
Zaks, A., Injective dimensions of semiprimary rings , J. Algebra 13 (1969), 73–89.Google Scholar
Zhang, P., Gorenstein-projective modules and symmetric recollements , J. Algebra 388 (2013), 65–80.Google Scholar