Research on the associated graded ring of a general filtration

In this research we aimed to generalize the经典理论on the Rees algebras of ideals so that它可以应用于Rees algebras of filtrations，where the word "filtration" means a family F = {Fn}_<n∈n > of ideals such that a = f_0⊃f_1⊃f_2⊃…andFmFn⊂Fm+n for any m,n∈n . we first introduced a new notion that is called the reduction system of F.Under some good conditions on the reduction system of Fwe could study the Cohen-Macaulay or the Gorenstein property of the associated graded ring of F. thegood conditions stated above are the conditions concerning(i) the localization at prime idealscontaining f_1 and(ii) the cohomological property on A/ Fn for finite number of n.l let l be thenumber of elements forming the reduction system of F and s be the height of f_1 . If -s【less than or】equal】1,then our result is very practical. In fact, In that case，我们发现了许多应用程序的结果，我们可以建立一个satisfactory theorywithout assuming any conditions on -s. Thus we may say that our purpose of this研究has beenalmost achieved。

In this research we aimed to generalize the classical theory on the Rees algebras of ideals so that it can be applied to the Rees algebras of filtrations, where the word "filtration" means a family F = {Fn}_<n∈N> of ideals such that A=F_0⊃F_1⊃F_2⊃・・・and FmFn⊂Fm+n for any m,n∈N.We first introduced a new notion that is called the reduction system of F. Under some good conditions on the reduction system of F, we could study the Cohen-Macaulay or the Gorenstein property of the associated graded ring of F. The good conditions stated above are the conditions concerning(i) the localization at prime ideals containing F_1 and(ii) the cohomological property on A/ Fn for finite number of n.Let l be the number of elements forming the reduction system of F and s be the height of F_1. If l-s【less than or equal】1, then our result is very practical. In fact, in that case, we found a lot of applications of the result. Furthermore we could establish satisfactory theory without assuming any conditions on l-s. Thus we may say that our purpose of this research has been almost achieved.