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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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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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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DOI: https://doi.org/10.1007/s00209-021-02795-7