Depth and regularity of powers of sums of ideals (Q269883)

Let \(k\) be a field, and \(A, B\) be two polynomial rings over \(k\). Let \(I\) and \(J\) be two homogeneous ideals in \(A\) and \(B\), respectively. The authors study the depth and the Castelnuovo-Mumford regularity of the \(n\)-th power of the sum \(I+J\) in \(A\otimes _k B\) and express the results in terms of the corresponding invariants of \(I\) and \(J\). They determine upper or lower bounds for depth and regularity. For a sufficiently large \(n\), for these invariants some formulae are determined. They give an estimation for the least number \(m\) such that \(\mathrm{reg}(I+J)^n\) is a linear function, for \(n\geq m\). The authors also compute the depth and the Castelnuovo-Mumford regularity of \((I+J)^n/(I+J^{n+1})\).