Powers of sums and their homological invariants

Abstract

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⊗kS. 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 chark=0 or I and J are generated by monomials.