Rees' theorem for filtrations, multiplicity function and reduction criteria

Abstract

Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty.$ Rees proved that for all $n\gg0$, $\lambda(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$ if and only if deg $P(I/J)(X)\leq$ dim $R-1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg $P(I/J)$ achieves its maximal degree. On the other hand, for ideals $J\subset I$ in a formally equidimensional local ring, we consider the multiplicity function $e(I^n/J^n)$ which is a polynomial in $n$ for all large $n.$ We explicitly determine the deg $e(I^n/J^n)$ in some special cases. For an ideal $J$ of analytic deviation one, we give characterization of reductions in terms of deg $e(I^n/J^n)$ under some additional conditions.