No abstract

We study the asymptotic behavior of the Castelnuovo–Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.

Let R and S be standard graded algebras over a field k, and I ⊆ R and J ⊆ S homogeneous ideals. Denote by P the sum of the extensions of I and J to R ⊗ k S. We investigate several important homological invariants of powers of P based on the information about I and J, with focus on finding the exact formulas for these invariants. Our investigation exploits certain Tor vanishing property of natural inclusion maps between consecutive powers of I and J. As a consequence, we provide fairly complete information about the depth and regularity of powers of P given that R and S are polynomial rings and either char k = 0 or I and J are generated by monomials.2010 Mathematics Subject Classification. 13D02, 13C05, 13D05, 13H99.