Zip Property of Graded and Filtered Affine Schemes

Abstract

In this paper we study the transfer of zip property between filtered (graded) rings and affine graded (filtered) structure schemes. Under some conditions, the zip property of filtered (graded) rings is preserved under their graded and filtered affine schemes. One may apply these results up to the formal level as in [8].
Introduction:  Consider a zariskian filtered ring S such that the associated graded ring G(S)=⊕(F_n S)/(F_(n-1) S)≅S ̃/(XS ̃ )  is commutative Noetherian domain; [10]. This includes many more geometric applications, i.e. this situation is general in the sense that it allows application of the results to most of the important examples. The topological base space T will be  Spec^g of G(S). The canonical element of degree one in S ̃=⊕F_n S≅∑_(n∈Ζ)▒F_n  SX^n≤S[X,X^(-1)] is the  1∈F_1 S in S, we write it as  X.
For moment let S be a graded ring. For a homogenous element a∈S, the annihilator ideal ann^g (a)={s∈S:sa=0} is a homogenous ideal, as is the ideal annihilator  ann^g (A)={s∈S: sA=0}; A⊆S a set of homogenous elements and as is the ideal annihilator ann^g (I)={s∈S:sI=0}; I▁(⊲) S an ideal of homogenous elements.
A graded ring S is said to be zip if ∀ A⊆S:  ann^g (A)=0⇒∃ A_0⊆A, finite subset of homogenous elements: ann^g (A_0)=0. In this definition, we can equivalently need to use that A is a graded ideal of S. We need only zip expression of commutative case.
For elementary notions, conventions and generalities, which we need here in this paper we refer to the list of references.
Objectives: In this paper, we study the transfer of zip property from filtered (graded) rings to the graded and filtered structure affine schemes.
Results: According to the work of Leroy and Matczuk ([4] , Theorem 3.2(1) ), who investigated the behavior of the zip property for a localization of a ring , we extend this result for graded and filtered affine schemes.
Conclusion: In this research, we investigate the zip property of filtered (graded) rings is preserved under their graded and filtered affine schemes. In the forthcoming work, we hope to come back to introduce the same results on the formal level, one may make this by [8].