Frobenius-Perron theory for projective schemes

Chen, J.; Gao, Z.; Wicks, E.; Zhang, J.; Zhang, X. H.; Zhu, H.

doi:10.1090/tran/8624

Frobenius-Perron theory for projective schemes
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by J. M. Chen, Z. B. Gao, E. Wicks, J. J. Zhang, X. H. Zhang and H. Zhu;

Trans. Amer. Math. Soc. 376 (2023), 2293-2324

DOI: https://doi.org/10.1090/tran/8624

Published electronically: January 18, 2023

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Abstract:

The Frobenius-Perron theory of an endofunctor of a $\Bbbk$-linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.

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