Depth and regularity of powers of sums of ideals

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k , we investigate the depth and the Castelnuovo–Mumford regularity of powers of the sum I + J in A ⊗ k B in terms of those of I and J . Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.