Shifts in the minimal free resolutions of the powers of an ideal

A celebrated result of Kodiyalam and Cutkosky, Herzog and Trung asserts that the regularity (in the sense of Castelnuovo-Mumford) of the powers of an ideal in a standard graded ring over a field is eventually a linear function of the power. In this talk, I will explain results with Amir Bagheri and Tai Ha showing that the shifts in the minimal free resolutions of powers also have such a type of linear behavior. Also that this holds in a pretty general context.