Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 404 https://internationalpubls.com Zip Property of Graded and Filtered Affine Schemes Nawal M. NourEldeen1,3,*, Helmy A. A.2, A. E. Radwan 2 1Department of Mathematics, College of Science, Taibah University, Madinah, Kingdom of Saudi Arabia. neldeen@taibahu.edu.sa 2Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt. helmy5962@hotmail.com, zezorawan@yahoo.com 3Department of Mathematics, Women’s College of Arts, Science and Education, Ain Shams University, Egypt. *Corresponding Author. Article History: Received: 18-09-2024 Revised: 26-10-2024 Accepted: 06-11-2024 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) =βŠ• 𝐹𝑛𝑆 πΉπ‘›βˆ’1𝑆 β‰… οΏ½ΜƒοΏ½ 𝑋�̃� 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 𝑆𝑝𝑒𝑐𝑔 of 𝐺(𝑆). The canonical element of degree one in οΏ½ΜƒοΏ½ =βŠ• 𝐹𝑛𝑆 β‰… βˆ‘ πΉπ‘›π‘›βˆˆπ›§ 𝑆𝑋𝑛 ≀ 𝑆[𝑋, π‘‹βˆ’1] is the 1 ∈ 𝐹1𝑆 in 𝑆, we write it as 𝑋. For moment let 𝑆 be a graded ring. For a homogenous element π‘Ž ∈ 𝑆, the annihilator ideal π‘Žπ‘›π‘›π‘”(π‘Ž) = {𝑠 ∈ 𝑆: π‘ π‘Ž = 0} is a homogenous ideal, as is the ideal annihilator π‘Žπ‘›π‘›π‘”(𝐴) = {𝑠 ∈ 𝑆: 𝑠𝐴 = 0}; 𝐴 βŠ† 𝑆 a set of homogenous elements and as is the ideal annihilator π‘Žπ‘›π‘›π‘”(𝐼) = {𝑠 ∈ 𝑆: 𝑠𝐼 = 0}; 𝐼 ⊲  𝑆 an ideal of homogenous elements. A graded ring 𝑆 is said to be zip if βˆ€β€„π΄ βŠ† 𝑆: β€„β€„π‘Žπ‘›π‘›π‘”(𝐴) = 0 β‡’ βˆƒβ€„π΄0 βŠ† 𝐴, finite subset of homogenous elements: π‘Žπ‘›π‘›π‘”(𝐴0) = 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]. Keywords: Graded annihilator, Affine schemes, Zip property. mailto:helmy5962@hotmail.com mailto:zezorawan@yahoo.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 405 https://internationalpubls.com 1. Introduction Throughout the paper 𝑆 will denote a zariskian filtered ring such that the associated graded ring 𝐺(𝑆) = βŠ• 𝐹𝑛𝑆 πΉπ‘›βˆ’1𝑆 β‰… οΏ½ΜƒοΏ½ 𝑋�̃� 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 𝑆𝑝𝑒𝑐𝑔 of 𝐺(𝑆). The canonical element of degree one in οΏ½ΜƒοΏ½ =βŠ• 𝐹𝑛𝑆 β‰… βˆ‘ πΉπ‘›π‘›βˆˆπ›§ 𝑆𝑋𝑛 ≀ 𝑆[𝑋, π‘‹βˆ’1] is the 1 ∈ 𝐹1𝑆 in 𝑆, we write it as 𝑋. For moment let 𝑆 be a graded ring. For a homogenous element π‘Ž ∈ 𝑆, the annihilator ideal π‘Žπ‘›π‘›π‘”(π‘Ž) = {𝑠 ∈ 𝑆: π‘ π‘Ž = 0} is a homogenous ideal, as is the ideal annihilator π‘Žπ‘›π‘›π‘”(𝐴) = {𝑠 ∈ 𝑆: 𝑠𝐴 = 0}; 𝐴 βŠ† 𝑆 a set of homogenous elements and as is the ideal annihilator π‘Žπ‘›π‘›π‘”(𝐼) = {𝑠 ∈ 𝑆: 𝑠𝐼 = 0}; 𝐼 ⊲  𝑆 an ideal of homogenous elements. A graded ring 𝑆 is said to be zip if βˆ€β€„π΄ βŠ† 𝑆: β€„β€„π‘Žπ‘›π‘›π‘”(𝐴) = 0 β‡’ βˆƒβ€„π΄0 βŠ† 𝐴, finite subset of homogenous elements: π‘Žπ‘›π‘›π‘”(𝐴0) = 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. 2. Zip graded affine schemes A graded sheaf 𝑂𝑇 𝑔 of graded rings, over a topological space 𝑇, is zip graded sheaf over 𝑇 if locally is zip sheaf i.e. βˆ€β€„π‘ƒ ∈ 𝑇 β‡’ 𝑂𝑇,𝑃 𝑔 is zip graded ring. As in section one, we consider 𝑇 = 𝑆𝑝𝑒𝑐𝑔(𝐺(𝑆)), the graded prime spectrum of 𝐺(𝑆). Write 𝛽 for the basis of the Zariski topology on 𝑇 consisting of the basic open sets 𝑇(𝑓) = {𝑃 ∈ 𝑇; 𝑓 βˆ‰ 𝑃}; 𝑓 homogenous element. We may define graded structure sheaf on 𝑇: we may associate to 𝑇(𝑓) the graded ring 𝑄𝑓 𝑔(𝐺(𝑆)) = 𝑆𝑓 βˆ’1 β‹… 𝐺(𝑆); just, by inverting the homogenous set 𝑆𝑓 = {1, 𝑓, 𝑓2, β‹― } of 𝐺(𝑆) in the classical way and we obtain the graded structure sheaf 𝑄𝑇 𝑔 on 𝑇 having as the stalk at 𝑃 ∈ 𝑇 the graded local ring 𝑄𝑃 𝑔(𝐺(𝑆)) = 𝑆𝑃 βˆ’1 β‹… 𝐺(𝑆); 𝑆𝑃 = β„Ž(𝐺(𝑆) βˆ’ 𝑃). Proposition 1. With the same consideration If 𝑆 is zip commutative filtered ring, then οΏ½ΜƒοΏ½ is zip commutative graded ring. Proof: Follows from [4], [5] just at the filtered (graded) level. Proposition 2. Under the assumption and notation mentioned above we have: i. 𝐺(𝑆) and οΏ½ΜƒοΏ½ (1 βˆ’ 𝑋)οΏ½ΜƒοΏ½ ⁄ are zip graded rings. ii. 𝑄𝑓 𝑔(𝐺(𝑆)) is zip graded ring; 𝑓 ∈ 𝐺(𝑆). iii. 𝑄𝑃 𝑔(𝐺(𝑆)) is zip graded ring; 𝑃 ∈ 𝑇. Proof: i. By assumption, since 𝐺(𝑆) and οΏ½ΜƒοΏ½ (1 βˆ’ 𝑋)οΏ½ΜƒοΏ½ ⁄ are commutative Noetherian domains. ii. or iii. This is an adaptation of Theorem 3.2. (1) of [4]; indeed, this theorem is phrased for ring theory but the fact that we work with graded objects. The proof of Theorem 3.2. (1) of [4] then carries over after the common modifications of graded nature. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 406 https://internationalpubls.com Now, we come the first main result of this paper which comes directly from definition and the above proposition. Proposition 3. With conventions notation as before: (𝑇, 𝑂𝑇 𝑔 ) is zip graded affine scheme. Also, we may define the graded structure sheaf 𝑂𝑛,𝑇 𝑔 on 𝑇: we may associate to 𝑇(𝑓); 𝑓 ∈ 𝐺(𝑆) homogenous element, the graded ring 𝑆�̃�(𝑛) βˆ’1 ( οΏ½ΜƒοΏ½ 𝑋𝑛�̃� ), where 𝑆�̃�(𝑛) is the homogenous image of 𝑆�̃� in οΏ½ΜƒοΏ½ and we obtain 𝑂𝑛,𝑇 𝑔 on 𝑇 having as stalk at 𝑃 ∈ 𝑇, the graded local ring 𝑄𝑝(𝑛) 𝑔 (οΏ½ΜƒοΏ½ 𝑋𝑛�̃� ⁄ ) = 𝑆�̃�(𝑛) βˆ’1 (οΏ½ΜƒοΏ½ 𝑋𝑛�̃� ⁄ ). Proposition 4. Under the same assumptions: If οΏ½ΜƒοΏ½ is zip domain, then βˆ€β€„π‘› ∈ 𝛧+, οΏ½ΜƒοΏ½ 𝑋𝑛�̃� ⁄ = οΏ½Μ„ΜƒοΏ½(𝑛) zip graded ring. Proof. It is easily checked that, if 𝐼(𝑛) β€„βŠ²   𝐼 �̄̃�(𝑛): β€„π‘Žπ‘›π‘›π‘” (𝐼(𝑛)) = 0; 𝐼(𝑛) = 𝐼/𝑋𝑛�̃�; 𝐼 β€„βŠ²   𝐼 �̃� .Then π‘Žπ‘›π‘›π‘”(𝐼) = 0;. in 𝑆 . Then there exists 𝐼0 βŠ²β€„πΌ(𝑓𝑖𝑛𝑖𝑑𝑒) ∢ π‘Žπ‘›π‘›π‘”(𝐼0) = 0. Then there exists 𝐼0/𝑋𝑛�̃� =  𝐼0 β€„βŠ²   𝐼: π‘Žπ‘›π‘›π‘”(𝐼0) = 0 and we have οΏ½Μ„ΜƒοΏ½(𝑛), βˆ€π‘›, is zip. Proposition 5. (Graded version of Theorem 3.2.(1) in [4]) Under the same consideration: 𝑄 οΏ½Μ„ΜƒοΏ½ 𝑔 (οΏ½Μ„ΜƒοΏ½) is zip graded ring. Again, we can mention the second main important result of this paper which comes directly from definition and the above proposition. Proposition 6. With conventions and notations as before: If οΏ½ΜƒοΏ½ is zip domain then the graded structure affine scheme (𝑇, 𝑂𝑛,𝑇 𝑔 ), βˆ€β€„π‘› ∈ 𝛧+, is zip graded affine scheme. A similar result holds in case of the Rees graded micro-localization rings οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½); 𝑆𝑓 = {𝑓, 𝑓2, β‹― } in 𝐺(𝑆). That is οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) = lim 𝑔 οΏ½βƒ–οΏ½ 𝑄 οΏ½Μ…ΜƒοΏ½(𝑛) 𝑔 (οΏ½Μ„ΜƒοΏ½(𝑛)) = lim 𝑔 οΏ½βƒ–οΏ½ (𝑓(̅𝑛) )βˆ’1(οΏ½Μ„ΜƒοΏ½(𝑛)) ; οΏ½Μ„ΜƒοΏ½(𝑛) = οΏ½ΜƒοΏ½ 𝑋𝑛�̃� ⁄ , see [1] and [9]. Now, if 𝐼 οΏ½ΜƒοΏ½ πœ‡ βŠ²β€„ οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) such that π‘Žπ‘›π‘›π‘” (𝐼 οΏ½ΜƒοΏ½ πœ‡ ) = 0, 𝐼 οΏ½ΜƒοΏ½ πœ‡ = 𝑄 οΏ½ΜƒοΏ½(𝑛) 𝑔 (𝐼 οΏ½ΜƒοΏ½ πœ‡ ) and 𝐼 οΏ½ΜƒοΏ½ πœ‡ βŠ²β€„ οΏ½ΜƒοΏ½(𝑛) =  𝑄 οΏ½ΜƒοΏ½(𝑛) 𝑔 (οΏ½ΜƒοΏ½), representing 𝐼 οΏ½ΜƒοΏ½ πœ‡ at a level 𝑛 in the inverse limit. Hence π‘Žπ‘›π‘›π‘” (𝑄 οΏ½ΜƒοΏ½(𝑛) 𝑔 (𝐼 οΏ½ΜƒοΏ½ πœ‡ )) = 0 and there exists 𝐼0 πœ‡ (𝑛) β€„βŠ²   οΏ½ΜƒοΏ½(𝑛) (as above) finite such that π‘Žπ‘›π‘›π‘” (𝑄 οΏ½ΜƒοΏ½(𝑛) 𝑔 (𝐼0(𝑛))) = 0. Then there exists 𝐼 0οΏ½ΜƒοΏ½ πœ‡ β€„βŠ²   𝐼 οΏ½ΜƒοΏ½ πœ‡ (finite) such that π‘Žπ‘›π‘›π‘” (𝐼 0οΏ½ΜƒοΏ½ πœ‡ )  = 0 and οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) is zip ring. It is noted that one may start the argument at any π‘š larger than 𝑛. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2s (2025) 407 https://internationalpubls.com Now, associating to an open set 𝑋(𝑓) the micro-localizations οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½), res. 𝑄𝑓 πœ‡ (𝑆) we obtain sheaf �̃�𝑋 πœ‡ , res. 𝑂𝑋 πœ‡ having as the completed stalks at 𝑃 ∈ 𝑆𝑝𝑒𝑐𝑔(𝐺(𝑆)) (or 𝑇 = π‘ƒπ‘Ÿπ‘œπ‘—π‘”β€„πΊ(𝑆) the ring �̃�𝑝 πœ‡ (οΏ½ΜƒοΏ½),res. 𝑄𝑃 πœ‡ (𝑆), see [9]. Note that 𝑂𝑋 πœ‡ is a sheaf of zariski rings and the 𝑋 βˆ’adic completion �̃�𝑇,𝑝 πœ‡ βˆ§π‘‹ = lim 𝑔 οΏ½βƒ–οΏ½ lim 𝑔 𝑓 οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) 𝑋𝑛 lim𝑔 𝑓 οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) = lim 𝑔 οΏ½βƒ–οΏ½ lim𝑔 𝑓 οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) 𝑋𝑛 οΏ½ΜƒοΏ½ οΏ½ΜƒοΏ½ πœ‡ (οΏ½ΜƒοΏ½) = lim 𝑔 οΏ½βƒ–οΏ½ lim𝑔 𝑓 οΏ½Μ…ΜƒοΏ½ οΏ½ΜƒοΏ½(𝑛) πœ‡ ( οΏ½ΜƒοΏ½Μ…) = �̃�𝑝 πœ‡ (οΏ½ΜƒοΏ½) , the micro-localization at β„Ž(𝐺(𝑆) βˆ’ 𝑃). As above, at a level 𝑛, obtaining 𝑄𝑃 πœ‡ (𝑆) from �̃�𝑝 πœ‡ (οΏ½ΜƒοΏ½) and one may easily prove that �̃�𝑝 πœ‡ (οΏ½ΜƒοΏ½), res. 𝑄𝑃 πœ‡ (𝑆), are zip rings. Hence the following result holds: Proposition 7. i.(𝑇, �̃�𝑇 πœ‡ ) is zip graded affine scheme. ii.(𝑇, 𝑂𝑇 πœ‡ ) is zip filtered affine scheme. References [1] Asensio M. J., Van den Bergh M. and Van Oystaeyen F., A new algebraic approach to micro-localization of filtered rings, Trans. Amer. Math. Soc. 316 (1989), 15-25. [2] Hortshorne R., Algebraic geometry, G.T.M. 52, Springer Verlag, New York, 1977. [3] Huishi L. and Van Oystaeyen F., Zariskian filterations, Comm. In algebra, 17(12) (1989). [4] Leroy A. and Matczuk J., Zip property of certain extensions, Journal of Pure and Applied Algebra, Vol. 220, No. 1(2016), P. 335-345. [5] Lunqun O., Jinwang L. and Yuemin X., Extension of zip modules, J. of Advances in Mathematics (China), 43(5) (2014), 683-694. [6] Nastasescu C. and Van Oystaen F., Graded and filtered rings and modulus, L.N. in Mathematics, Springer-verlag, Berlin, Heidelberg, New York, 1977. [7] Nastasescu C. and Van Oystaen F., Graded ring theory, M. Library 28, North Holland, Amesterdam, 1981. [8] Nawal M. NourEldeen, Radwan A. E., and Ahmed Aboubakr, On micro-localization of graded and filtered formal modules, accepted and to appear in Applied Mathematics and Information Sciences (2024). [9] Radwan A. E. and Van Oystaeyen F., Micro-structure sheaves, formal schemes and quantum sections over projective schemes, In P of contact, France-Belgium, 1992. [10] Radwan A. E., Filtered and graded micro-affine schemes, J. Inst. Math. And Comp. Sci., 5(1994), 73-81. [11] Sharp R. Y., Graded annihilators and uniformly F-compatible ideals, Acta Math. (Vietnamica), 40 (2015), 179-195.