A C ] 1 J an 2 01 6 DEPTH AND REGULARITY OF POWERS OF SUMS OF

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.