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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Betti tables of monomial ideals fixed by permutations of the variables


Author: Satoshi Murai
Journal: Trans. Amer. Math. Soc. 373 (2020), 7087-7107
MSC (2010): Primary 13D02; Secondary 13A50
DOI: https://doi.org/10.1090/tran/8159
Published electronically: August 5, 2020
MathSciNet review: 4155201
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Abstract: Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti tables of $I_m$ for some small integers $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.


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Additional Information

Satoshi Murai
Affiliation: Department of Mathematics, Faculty of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan
MR Author ID: 800440
Email: s-murai@waseda.jp

Received by editor(s): August 5, 2019
Received by editor(s) in revised form: January 7, 2020
Published electronically: August 5, 2020
Additional Notes: The research of the author was partially supported by KAKENHI 16K05102.
Article copyright: © Copyright 2020 American Mathematical Society