Betti tables of monomial ideals fixed by permutations of the variables

Abstract

Let S n S_n be a polynomial ring with n n variables over a field and { I n } n ≥ 1 \{I_n\}_{n \geq 1} a chain of ideals such that each I n I_n is a monomial ideal of S n 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 I_n for all large intergers n n from the Z m \mathbb Z^m -graded Betti tables of I m I_m for some small integers m m . Our main result shows that the projective dimension and the regularity of I n I_n eventually become linear functions on n n , confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.