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A counterexample to the \(\phi \)-dimension conjecture

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Abstract

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the \(\phi \)-dimension. The \(\phi \)-dimension conjecture states that this upper bound is always finite, a fact that would imply the finitistic dimension conjecture. In this paper, we present a counterexample to the \(\phi \)-dimension conjecture and explain where it comes from. We also discuss implications for further research and the finitistic dimension conjecture.

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References

  1. Álvarez-Gavela, D., Igusa, K.: A Legendrian Turaev torsion via generating families. J. Éc. Polytech. Math. 8, 57–119 (2021)

    Article  MathSciNet  Google Scholar 

  2. Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., Trnka, J.: Grassmannian geometry of scattering amplitudes. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9781316091548. Available as arXiv:1212.5605 (2016)

  3. Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras, 1: Techniques of representation theory, London Math. Soc. Student Texts, p. 65. Cambridge University Press, New York (2006)

    Book  Google Scholar 

  4. Auslander, M., Reiten, I.: On a generalized version of the Nakayama conjecture. Proc. Am. Math. Soc. 52, 69–75 (1975)

    Article  MathSciNet  Google Scholar 

  5. Barrios, M., Mata, G.: On algebras of \(\Omega ^n\)-finite and \(\Omega ^{\infty }\)-infinite representation type. arXiv:1911.02325 (2019)

  6. Barrios, M., Mata, G., Rama, G.: Igusa–Todorov \(\phi \) function for truncated path algebras. Algebr. Represent Theory 23, 1051–1063 (2019)

  7. Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Am. Math. Soc. 95, 466–488 (1960)

    Article  Google Scholar 

  8. Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 118(883) (2007)

  9. Eilenberg, S., Rosenberg, A., Zelinsky, D.: On the dimension of modules and algebras, VIII. Dimension of tensor products. Nagoya Math. J. 12, 71–93 (1957)

    Article  MathSciNet  Google Scholar 

  10. Elsener, A.G., Schiffler, R.: On syzygies over 2-Calabi–Yau tilted algebras. J. Algebra 470, 91–121 (2017)

    Article  MathSciNet  Google Scholar 

  11. Erdmann, K., Holm, T., Iyama, O., Schröer, J.: Radical embeddings and representation dimension. Adv. Math. 185(1), 159–177 (2004)

    Article  MathSciNet  Google Scholar 

  12. Fernandes, S., Lanzilotta, M., Mendoza, O.: The \(\phi \)-dimension: a new homological measure. Algebr. Represent. Theory 18(2), 463–476 (2015)

    Article  MathSciNet  Google Scholar 

  13. Fock, V.V., Goncharov, A.B.: Cluster \(\cal{X}\)-varieties, amalgamation, and Poisson–Lie groups. In: Ginzburg, G. (ed.) Algebraic Geometry and Number Theory. Progress in Mathematics, vol 253. Birkhäuser, Boston (2006)

  14. Hanson, E.J., Igusa, K.: Resolution quiver and cyclic homology criteria for Nakayama algebras. J. Algebra 553, 138–153 (2020)

    Article  MathSciNet  Google Scholar 

  15. Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimension Algebras. London Math. Soc. Lecture Note Series, p. 119. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  16. Herschend, M.: Tensor products on quiver representations. J. Algebra 212(2), 452–469 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Huard, F., Lanzilotta, M.: Self-injective right Artinian rings and Igusa–Todorov functions. Algebr. Represent. Theory 26(3), 765–770 (2013)

    Article  MathSciNet  Google Scholar 

  18. Igusa, K., Todorov, G.: On the finitistic global dimension conjecture for Artin algebras. In: Buchweitz, R.-O., Lenzing, H., (eds.) Representation theory of Algebras and Related Topics. Fields Institute Communications, vol. 45. American Mathematical Society, Providence, RI (2005)

  19. Igusa, K., Zacharia, D.: On the cyclic homology of monomial relation algebras. J. Algebra 151, 502–521 (1992)

    Article  MathSciNet  Google Scholar 

  20. Kirkman, E., Kuzmanovich, J., Small, L.: Finitistic dimensions of Noetherian rings. J. Algebra 147(2), 350–364 (1992)

    Article  MathSciNet  Google Scholar 

  21. Lanzilotta, M., Mata, G.: Igusa–Todorov functions for Artin algebras. J. Pure Appl. Algebra 222(1), 202–212 (2018)

    Article  MathSciNet  Google Scholar 

  22. Lanzilotta, M., Mendoza, O.: Relative Igusa–Todorov functions and relative homological dimensions. Algebr. Represent. Theory 20(3), 765–802 (2017)

    Article  MathSciNet  Google Scholar 

  23. Ringel, C.M.: The Gorenstein projective modules for the Nakayama algebras I. J. Algebra 385, 241–261 (2013)

    Article  MathSciNet  Google Scholar 

  24. Şen, E.: The \(\varphi \)-dimension of cyclic Nakayama algebras. Commun. Algebra 49(6), 2278–2299 (2021)

  25. Shen, D.: A note on homological properties of Nakayama algebras. Arch. Math. 108(3), 251–261 (2017)

    Article  MathSciNet  Google Scholar 

  26. Wei, J.: Finitistic and representation dimensions. arXiv:0803.3364 (2008)

  27. Wei, J.: Finitistic dimension and Igusa–Todorov algebras. Adv. Math. 222(6), 2215–2226 (2009)

    Article  MathSciNet  Google Scholar 

  28. Xi, C.: On the finitistic dimension conjecture II: related to finite global dimension. Adv. Math. 201, 116–142 (2006)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Both authors are thankful to Gordana Todorov, who is working with them on the larger study of amalgamation, for numerous meaningful conversations and support. They would also like to thank Kaveh Mousavand, who asked them whether they could generalize the result of [14] to arbitrary monomial relation algebras, starting the investigation that led to this paper. The second author is thankful to Daniel Álvarez-Gavela for their collaboration on the use of amalgamation to describe examples and invariants in contact topology (see [1]), which motivated the larger study of amalgamation. Moreover, the second author thanks An Huang for sharing the reference [2] with him, which laid the ground work for the connection between [1, 14], and this paper. Lastly, the authors are thankful to Rene Marczinzik for pointing out that \(\mathsf {fin.dim}\left( A\otimes _K A_3^{CT}\right) = 0\) as a consequence of [9, Thorem 16] and to both Liang Chen and an anonymous referee for pointing out a errors in Lemma 6.7.

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Correspondence to Eric J. Hanson.

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Hanson, E.J., Igusa, K. A counterexample to the \(\phi \)-dimension conjecture. Math. Z. 300, 807–826 (2022). https://doi.org/10.1007/s00209-021-02795-7

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