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On the structure of the Sally module and the second normal Hilbert coefficient


Authors: Shreedevi K. Masuti, Kazuho Ozeki, Maria Evelina Rossi and Hoang Le Truong
Journal: Proc. Amer. Math. Soc. 148 (2020), 2757-2771
MSC (2010): Primary 13D40, 13A30, 13H10
DOI: https://doi.org/10.1090/proc/14839
Published electronically: April 14, 2020
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Abstract: The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the Hilbert function of the associated graded ring of the normal filtration. A deep analysis of the vanishing of the third normal Hilbert coefficient has been necessary. This study is related to a longstanding conjecture stated by S. Itoh.


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Additional Information

Shreedevi K. Masuti
Affiliation: IIT Dharwad, WALMI Campus, PB Road, near High Court, Dharwad - 580011, Karnataka, India
Email: shreedevi@iitdh.ac.in

Kazuho Ozeki
Affiliation: Department of Mathematical Sciences, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan
Email: ozeki@yamaguchi-u.ac.jp

Maria Evelina Rossi
Affiliation: Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: rossim@dima.unige.it

Hoang Le Truong
Affiliation: Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany; Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, 10307, Hanoi, Viet Nam; and Thang Long Institute of Mathematics and Applied Sciences, Hanoi, Vietnam
Email: hoang@math.uni-sb.de, hltruong@math.ac.vn, truonghoangle@gmail.com

DOI: https://doi.org/10.1090/proc/14839
Keywords: Cohen-Macaulay local ring, associated graded ring, normal Hilbert coefficients, Sally module, Rees algebra
Received by editor(s): March 7, 2019
Published electronically: April 14, 2020
Additional Notes: The first author was supported by the INSPIRE faculty award funded by the Department of Science and Technology, Govt. of India. She was also partially supported by a grant from the Infosys Foundation. She was supported by the INdAM COFUND Fellowships cofunded by Marie Curie actions, Italy, for her research in Genova during which this work was started
The second author was partially supported by Grant-in-Aid for Scientific Researches (C) in Japan (18K03241)
The third author was partially supported by PRIN 2015EYPTSB-008 Geometry of Algebraic Varieties
The fourth author was partially supported by the Alexander von Humboldt Foundation and the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.309
Communicated by: Claudia Polini
Article copyright: © Copyright 2020 American Mathematical Society