Powers of sums and their homological invariants

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 $\mathrm{char}k=0$ or I and J are generated by monomials.