Betti splittings for powers of sums of ideals

Let $A$ and $B$ be standard graded polynomial rings over a field $k$ and $I$ and $J$ be non-zero, proper homogeneous ideals contained in $A$ and $B$, respectively. Denote by $P$ the sum of $I$ and $J$ in $R=A\otimes_k B$. Under reasonable conditions on $k, I$ and $J$, we provide exact formulas and describe the asymptotic behavior of the depth and the regularity of the powers of $P$ in terms of the data of $I$ and $J$. Thereby, we strengthen previous work of H.T. H\`a, N.V. Trung and T.N. Trung. Our main technical result says that, under the aforementioned conditions, for all $s\ge 0$ and all $n\ge 1$, the simple decomposition $I^sP^n=I^{s+1}P^{n-1}+I^sJ^n$ yields a Betti splitting for $I^sP^n$. A decomposition of an ideal $L$ as a sum of two subideals is called a Betti splitting if the minimal free resolution of $L$ is completely determined by those of the summands and their intersection.