We introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a∗-invariant of powers of homogeneous ideals. Specifically, for an equigenerated homogeneous ideal I in a standard graded algebra over a Noetherian ring, we give bounds for the smallest values of power q starting from which a∗(Iq) and reg(Iq) become linear functions.