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Ideals in infinite-dimensional polynomial rings that are invariant under theaction of the monoid of increasing functions have been extensively studiedrecently. Of particular interest is the asymptotic behavior of truncations ofsuch an ideal in finite-dimensional polynomial subrings. It has beenconjectured that the Castelnuovo--Mumford regularity and projective dimensionare eventual linear functions along such truncations. In the present paper weprovide evidence for these conjectures. We show that for monomial ideals theprojective dimension is eventually linear, while the regularity isasymptotically linear.

On regularity and projective dimension of invariant chains of monomial ideals