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On reduction numbers and Castelnuovo–Mumford regularity of blowup rings and modules

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Abstract

We prove new results on the interplay between reduction numbers and the Castelnuovo–Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff–Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we largely generalize a result of A. Mafi on ideals in two-dimensional Cohen–Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen–Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.

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Acknowledgements

The first author was partially supported by CNPq (grants 301029/2019-9 and 406377/2021-9). The second author was supported by a CAPES Doctoral Scholarship. They are grateful to the anonymous reviewers for a careful reading of the manuscript and helpful suggestions, including an improvement in Corollary 5.1.

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  1. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, PB, 58051-900, Brazil

    Cleto B. Miranda-Neto & Douglas S. Queiroz

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  1. Cleto B. Miranda-Neto
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Correspondence to Cleto B. Miranda-Neto.

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Dedicated to the memory of Professor Wolmer Vasconcelos, mentor of generations of commutative algebraists.

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Miranda-Neto, C.B., Queiroz, D.S. On reduction numbers and Castelnuovo–Mumford regularity of blowup rings and modules. Collect. Math. (2024). https://doi.org/10.1007/s13348-024-00436-w

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