Abstract
In this article we prove that, under certain hypotheses, Morita context algebras with zero bimodule morphisms have finite \(\phi \)-dimension. For these algebras we also study the behaviour of the \(\phi \)-dimension for an algebra and its opposite. In particular we show that the \(\phi \)-dimension of an Artin algebra is not symmetric, i.e. there exists an Artin algebra A such that \(\phi \dim (A) \not = \phi \dim (A^{op})\).
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References
Auslander, M.: On the dimension of modules and algebras (iii). Global dimension. Nagoya Mathematical Journal 9, 67–77 (1955)
Assem, I., Simson, D., Skowroński, A.: Elements of the representation Theory of Associative Algebras, vol. 1. Cambridge University Press, Cambridge (2006)
Barrios, M., Mata, G.: On algebras of \(\Omega ^{n}\)-finite and \(\Omega ^{\infty }\)-infinite representation type. https://doi.org/10.1142/S0219498824501834 (2024, to appear)
Barrios, M., Mata, G., Rama, G.: Igusa-Todorov \(\phi \) function for truncated path algebras. Algebr. Represent. Theor. 23(3), 1051–1063 (2020)
Cibils, C., Redondo, M.J., Solotar, A.: Han’s conjecture and Hochschild homology for null-square projective algebras. Indiana Univ. Math. J. 70(2), 639–668 (2021)
Gao, N., Psaroudakis, C.: Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings. Algebr. Represent. Theor. 20, 487–529 (2017)
Green, E.L., Psaroudakis, C.: On Artin Algebras Arising from Morita Contexts. Algebr. Represent. Theor. 17, 1485–1525 (2014)
Huard, F., Lanzilotta, M.: Selfinjective right artinian rings and Igusa-Todorov functions. Algebr. Represent. Theor. 16(3), 765–770 (2012)
Huard, F., Lanzilotta, M., Mendoza, O.: An approach to the finitistic dimension conjecture. J. Algebra 319(9), 3916–3934 (2008)
Haim, M., Lanzilotta, M., Mata, G.: The Igusa-Todorov function for comodules. São Paulo J. Math. Sci. 11, 59–67 (2017)
Igusa, K., Todorov, G.: On finitistic global dimension conjecture for artin algebras. Representations of algebras and related topics. Fields Inst. Commun. 45, American Mathematical Society, pp. 201–204 (2005)
Lanzilotta, M., Mata, G.: Igusa-Todorov functions for Artin algebras. Journal of Pure and Applied Algebra 222(1), 202–212 (2018)
Lanzilotta, M., Marcos, E., Mata, G.: Igusa-Todorov functions for radical square zero algebras. J. Algebra 487, 357–385 (2017)
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The authors M. Barrios and G. Mata have contributed equally.
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Presented by: Christof Geiß.
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Barrios, M., Mata, G. The Igusa-Todorov \(\phi \)-Dimension on Morita Context Algebras. Algebr Represent Theor 26, 3255–3269 (2023). https://doi.org/10.1007/s10468-023-10218-w
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DOI: https://doi.org/10.1007/s10468-023-10218-w