Betti tables of monomial ideals fixed by permutations of the variables

Murai, Satoshi

doi:10.1090/tran/8159

Betti tables of monomial ideals fixed by permutations of the variables
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Trans. Amer. Math. Soc. 373 (2020), 7087-7107 Request permission

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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