Saturation degree of powers of graded ideals

Abstract

Let $I$ be a non-nilpotent graded ideal of a standard graded algebra over a Noetherian commutative ring. We show that the saturation degree sdeg $I^n$ is asymptotically a constant function or a linear function whose slope is one of the generating degrees of $I$. Furthermore, we give sufficient (and necessary) conditions for a numerical functions to be the function sdeg $I^n$ of an ideal $I$ generated by forms of the same degree. Our results may provide a better understanding of the initial behavior of the regularity function reg $I^n$.