On Burch’s inequality and a reduction system of a filtration

Abstract

Let F = { F n } \mathcal {F} = \{ F_n \} be a multiplicative filtration of a local ring such that the Rees algebra R ( F ) \mathrm {R}(\mathcal {F}) is Noetherian. We recall Burch’s inequality for F \mathcal {F} and give an upper bound of the a-invariant of the associated graded ring a ( G ( F ) ) \mathrm {a}(\mathrm {G}(\mathcal {F})) using a reduction system of F \mathcal {F} . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension 2 2 .