On regularity and projective dimension of invariant chains of monomial ideals

Ideals in infinite-dimensional polynomial rings that are invariant under the action of the monoid of increasing functions have been extensively studied recently. Of particular interest is the asymptotic behavior of truncations of such an ideal in finite-dimensional polynomial subrings. It has been conjectured that the Castelnuovo--Mumford regularity and projective dimension are eventual linear functions along such truncations. In the present paper we provide evidence for these conjectures. We show that for monomial ideals the projective dimension is eventually linear, while the regularity is asymptotically linear.