Filtered and graded Procesi extensions of rings

Filtered and graded Procesi extensions of rings

In this paper, we introduce filtered and graded Procesi extensions of filtered and graded rings as a natural modification of Procesi extensions of rings. We show that these extensions behave well from the geometric point of view.

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Datum:2007
1. Verfasser: Radwan, A.E.
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Veröffentlicht: Інститут прикладної математики і механіки НАН України 2007
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Zitieren:Filtered and graded Procesi extensions of rings / A.E. Radwan // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 131–137. — Бібліогр.: 15 назв. — англ.

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spelling nasplib_isofts_kiev_ua-123456789-1523742025-02-23T20:23:44Z Filtered and graded Procesi extensions of rings Radwan, A.E. In this paper, we introduce filtered and graded Procesi extensions of filtered and graded rings as a natural modification of Procesi extensions of rings. We show that these extensions behave well from the geometric point of view. 2007 Article Filtered and graded Procesi extensions of rings / A.E. Radwan // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 131–137. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:13A30, 13B35, 13G10. https://nasplib.isofts.kiev.ua/handle/123456789/152374 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description In this paper, we introduce filtered and graded Procesi extensions of filtered and graded rings as a natural modification of Procesi extensions of rings. We show that these extensions behave well from the geometric point of view.
format Article
author Radwan, A.E.
spellingShingle Radwan, A.E.
Filtered and graded Procesi extensions of rings
Algebra and Discrete Mathematics
author_facet Radwan, A.E.
author_sort Radwan, A.E.
title Filtered and graded Procesi extensions of rings
title_short Filtered and graded Procesi extensions of rings
title_full Filtered and graded Procesi extensions of rings
title_fullStr Filtered and graded Procesi extensions of rings
title_full_unstemmed Filtered and graded Procesi extensions of rings
title_sort filtered and graded procesi extensions of rings
publisher Інститут прикладної математики і механіки НАН України
publishDate 2007
url https://nasplib.isofts.kiev.ua/handle/123456789/152374
citation_txt Filtered and graded Procesi extensions of rings / A.E. Radwan // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 131–137. — Бібліогр.: 15 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT radwanae filteredandgradedprocesiextensionsofrings
first_indexed 2025-11-25T04:42:06Z
last_indexed 2025-11-25T04:42:06Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2007). pp. 131 – 137 c© Journal “Algebra and Discrete Mathematics” Filtered and graded Procesi extensions of rings A. E. Radwan Communicated by D. Simson Abstract. In this paper, we introduce filtered and graded Procesi extensions of filtered and graded rings as a natural modifi- cation of Procesi extensions of rings. We show that these extensions behave well from the geometric point of view. 1. Introduction For basic notions, conventions and generalities, which we need here in this paper, we refer to [7] and [8], see also [1-6, 13, 14, 15]. There are many different ways of describing the ring extensions and their applications; several of these were thought to be different in some papers, see [2,4,5,6,9,10,11,12,14]. If R and S are not necessarily commutative rings, then the prime ideal structures of R and of S are, in general, rather poorly related. It appears that if one restricts to so-called filtered and graded Procesi extensions of rings R and S, as we see in this paper, then things improve considerably over the filtered and graded levels. In fact, the usefulness of the topological spec-map appears in the study of ring extensions, the theory of schemes which centers around sheaves and in geometrical applications. We study the Procesi extensions of filtered and graded rings and show that these extensions behave well in constructing geometric spaces. First, we introduce the effect of existence of the extensions ϕ : R → S; filtered ring homomorphism, with S = ϕ(R).SR on a filtered ring S. This will allow to study the transfer of properties from the filtration of FS to the 2000 Mathematics Subject Classification: 13A30, 13B35, 13G10. Key words and phrases: filtered and graded extensions, affine schemes. Jo u rn al A lg eb ra D is cr et e M at h .132 Filtered and graded Procesi extensions of rings filtration of F ′S. Next, we prove that if S is a filtered Procesi extension of R then, by using the Rees-level, G(S) is again a graded Procesi extension of G(R). Finally, we show that our extension, over the associated graded level can be applied to the affine schemes. 2. Filtered Procesi Extensions Throughout this paper R and S will denote filtered rings with unit and Z− filtrations FR = {FnR}n∈Z, FS = {FnS}n∈Z respectively. R-filt will denote the category of left filtered unitary R-modules. Let ϕ : R → S be a filtered ring homomorphism in R-filt; then ϕ is said to be filtered Procesi extension if S = ϕ(R).SR where SR = {s ∈ S : sϕ(r) = ϕ(r)s ∀ r ∈ R}. Hence S may be viewed as a filtered ring through ϕ as: F ′nS = Fn(ϕ(R)).SR = (ϕ(R) ∩ FnS).SR; n ∈ Z. It is equally straightforward to prove the following: 2.1. Lemma. Under the assumption and notation mentioned above we have: (a) S, with the filtration F ′S, is a filtered ring, (b) S, with filtration F ′S, is left a filtered R-module, and (c) HOMFR,FS(R, S) ⊆ HOMFR,F ′S(R, S). � A morphism f : M → N between filtered R-modules is said to be strict if f(FnM) = f(M)∩FnN , for all n ∈ Z. For a complete information on filtered and graded ring theory, the reader is referred to [1, 3, 7, 8]. It is easy to prove the following characterizations: 2.2. Proposition. Let ϕ : R → S be a filtered Procesi extension as above. (a) If ϕ is a monomorphism and R a strongly filtered ring, in the sense that (FnR)(FmR) = Fn+mR, forall n, m ∈ Z, then F ′nS ∼= FnR.SR, for all n ∈ Z, and F ′S is a strong filtration on S. (b) If ϕ is strict and R a strongly filtered ring then S is strongly filtered with respect to F ′S. (c) If ϕ is a monomorphism and I any two sided ideal in R then IS = SI. (d) ϕ(Z(R)) ⊆ Z(S) ⊆ SR, where Z(R) is the commutative subring in R of all the central elements, with the induced filtration. (e) If ϕ is an epimorphism and S strongly filtered ring with respect to FS then S = S.Z(S) is strongly filtered with respect to F ′S; F ′nS = FnS.Z(S), for all n ∈ Z. Proof. Straightforward. A filtration FM on M ∈ R-filt is said to be discrete if there is an Jo u rn al A lg eb ra D is cr et e M at h .A. E. Radwan 133 integer α such that FnM = 0 for all n < α, separated if ⋃ n∈Z FnM = 0 and exhaustive if M = ⋃ n∈Z FnM . Finally M is said to be filtered complete if M ∼= M∧F = lim ←−n M/FnM . In other words, M is complete if FM separated and all Cauchy sequences in the FM -topology of M converge, [1, 3]. We now come to the main results of this section: 2.3. Proposition. With notations as above. (a) If FS on S is discrete then F ′S on S discrete. (b) If FS is separated then F ′S separated. (c) If FS is exhaustive then F ′S exhaustive. (d) If FS is exhaustive, separated and complete then S is filtered and complete at F ′S. Proof. (a) We may take αF = αF ′ . Then we have F ′nS = (FnS ∩ ϕ(R)).SR = (0 ∩ ϕ(R)).SR = 0, for all n < α. (b) Let t ∈ ⋂ n∈Z F ′nS = ⋂ n∈Z ((ϕ(R) ∩ FnS).SR). Then t = yn1.x1; yn1 ∈ FnS ∩ ϕ(R), x1 ∈ SR, for all n ∈ Z. This implies that t = 0. (c) Since ⋃ n∈Z FnS = S, then (( ⋃ n∈Z FnS) ∩ ϕ(R)).SR = ϕ(R).SR = S. On the other hand, ⋃ n∈Z F ′nS = ⋃ n∈Z [(FnS ∩ ϕ(R)).SR] = = [( ⋃ n∈Z FnS) ∩ ϕ(R)].SR = ϕ(R).SR = S. This yields the assertion. (d) By using (b) and (c) we conclude that all Cauchy sequences in the F ′S-topology of S converge. From this, the result easily follows. 2.4. Open question. Two filtrations FM and F ′M on an R- module M , are said to be equivalent if there exists some α ∈ N such that Fn−αM ⊆ F ′nM ⊆ Fn+αM , for all n ∈ Z, see [3]. With these notations, is it true that FS is equivalent to F ′S? Jo u rn al A lg eb ra D is cr et e M at h .134 Filtered and graded Procesi extensions of rings 3. Graded Procesi Extensions If ϕ : R → S is a filtered Procesi extension with S = ϕ(R).SR as above then we get the associated graded extension morphisms ϕ̃ : R̃ → S̃ (respectively, G(ϕ) : G(R) → G(S)) in R̃−gr (respectively, in G(R)−gr) in a natural way. Now, obviously, if S = ϕ(R).SR, then S̃ = ⊕ n∈Z (ϕ(R) ∩ FnS).SR ∼= ∑ n∈Z (ϕ(R) ∩ FnS).SRXn e ; Xe = Xe.1 ∈ (S̃)1, where Xe is the central element of degree one in S̃. Therefore S̃ = (ϕ(R).SR)∼ = ϕ̃(R̃).S̃R̃; S̃R̃ = {s̃ ∈ S̃ : ϕ̃(r̃) = s̃ϕ(r̃) for all r̃ ∈ R̃} = {s ∈ S : ϕ(r)s = sϕ(r) for all r ∈ R}∼ = (SR)∼. We now come to the main result of this paper. 3.1. Proposition. With notations and conventions introduced above, let S̃ = ϕ̃(R̃).S̃R̃, i.e. is a ϕ̃ graded Procesi extension. Then ¯̃ϕ : R̃/XR̃ → S̃/XS̃ is a graded Procesi extension in R̃/XR̃ − gr. In other words, if S̃ = ϕ̃(S̃).S̃R̃ then ¯̃S = ¯̃ϕ( ¯̃R). ¯̃S ¯̃ R, where ¯̃S = S̃/XS̃ and ¯̃R = R̃/XR̃. Proof. Consider the following commutative diagram R̃ −→ ¯̃ϕ S̃ ηR̃ ↓ ↓ ηS̃ G(R) ∼= R̃/XR̃ −→ ¯̃ϕ S̃/XS̃ ∼= G(R); with ¯̃ϕ(¯̃r) = ϕ̃(r̃) + XS̃ , for all ¯̃r ∈ R̃/XR̃. Now, let ¯̃s = s̃ + XS̃, s̃ = smXm = ϕ̃(r̃).z̃ = z̃ϕ̃(r̃); sm ∈ FmS and z̃ ∈ S̃R̃. Hence s̃ϕ̃(r̃)+XS̃ = ϕ̃(r̃)s̃+XS̃ and s̃ϕ̃(r̃)− ϕ̃(r̃)s̃ ∈ XS̃. This implies that ¯̃s ∈ ¯̃ϕ( ¯̃R). ¯̃S ¯̃ R, where ¯̃S ¯̃ R = {¯̃s ∈ ¯̃S : ¯̃ϕ(¯̃r).¯̃s = ¯̃s. ¯̃ϕ(¯̃r) , for all ¯̃r ∈ ¯̃R}, and ¯̃s = ϕ̃(r̃)z̃ + XS̃. Conversely, let ¯̃t ∈ ¯̃ϕ( ¯̃R). ¯̃S ¯̃ R; ¯̃t = (ϕ̃((r̃) + XS̃).¯̃z = (ϕ̃(r̃) + XS̃).(s̃ + XS̃) = ϕ̃(r̃).s̃ + XS̃ = s̃ϕ̃(r̃) + XS̃. Therefore ¯̃t ∈ ¯̃S. Hence, we conclude that ¯̃S = ¯̃ϕ( ¯̃R). ¯̃S ¯̃ R and ¯̃ϕ : R̃/XR̃ → S̃/XS̃ ia a graded Procesi extension in R̃/XR̃ − gr. Jo u rn al A lg eb ra D is cr et e M at h .A. E. Radwan 135 3.2. Remark. With notations and conventions as above, let S = ϕ(R).SR, i.e. ϕ ∈ R-filt is a filtered Procesi extension and consider the commutative diagram: R −→ ϕ S σR ↓ ↓ σR G(R) −→ G(ϕ) G(S) Then one may derive G(S) = G(ϕ)(G(R)).G(S)G(R) = G(ϕ(R).SR), giving the same result as in 3.1. 4. Geometric Implications In the sequel of this section, R, S will be filtered rings with unit such that G(R), G(S) are Noetherian domains. Now, let ϕ : R → S be filtered Procesi extension as above such that S = ϕ(R).SR. Hence we get a graded Procesi extension T = G(ϕ) : G(R) → G(S) such that G(S) = T (G(R)).G(S)G(R). It is straight forward to show that: (1) T (Z(G(R))) ⊆ Z(G(S)) ⊆ G(S)G(R). (2) The inverse image T−1(p) of a (graded) prime ideal of G(S) is a (graded) prime ideal of G(R). Let us endow the graded prime spectrum X = Specg(G(R)) (similar to Y = Specg(G(S))) with the so-called Zariski topology, by letting the open sets (then basic affine Noetherian open sets, see [9,10]) for this topology to be the sets X(f) = {p ∈ X : f 6∈ p}, where f runs through the homogeneous elements of G(R). In general, Specg(G(R)) is not a scheme. However in case G(R) is positively graded, then we write Proj(G(R)) for the Zariski open sub- set of X consisting of the graded prime ideals not containing G(R)+ =⊕ n>0 G(R)n, and in this case the closed set V (G(R)+) in X is nothing but Spec(G(R)0). Therefore P (X) = Proj(G(R)) = {p ∈ X : G(R)n 6⊂ p for some n > 0}. It is clear that X = p(X) if and only if G(R)n.G(R)−n = G(R) for all n > 0. We then have the following result: 4.1. Proposition. Any filtered Procesi extension ϕ : R → S induces a continuous morphism aT : Specg(G(S)) = Y → X = Specg(G(R)), p 7→ T−1(p) = G(ϕ)−1(p). Jo u rn al A lg eb ra D is cr et e M at h .136 Filtered and graded Procesi extensions of rings Proof. Let p ∈ Y = Specg(G(S)), and assume r1.G(R).r2 ⊂ T−1(p) = aT (p), for some r1, r2 ∈ h(G(R)) such that r2 6∈ T−1(p). Now, T (r1)G(S)T (r2) = T (r1)T (G(R))G(S)G(R)T (r2) = T (r1)T (G(R))T (r2)G(S)G(R) ⊆ p G(S)G(R) = p, where T (r1), T (r2) ∈ h(G(S)) and T (r2) 6∈ p ∈ Y . Then T (r1) ∈ p and so r1 ∈ T−1(p). Therefore T−1(p) ∈ X. On the other hand, we leave it as straightfor- ward verification, that for any f ∈ h(G(R)) we have aT−1(X(f)) = Y (T (f));T (f) ∈ h(G(S)), which shows that aT is continuous. This proves our assertion. Let us consider the behaviour of the Procesi extensions with respect to affine schemes: If f ∈ h(G(R)) ∩ Z(G(R)) then, by the exactness of the localization functors, G(ϕ) = T : G(R) → G(S) induces a graded Procesi extension, over the localization level, G(R)f −→ Tf G(S)T (f) such that G(S)T (f) = Tf (G(R)f ).(G(S)T (f)) G(R)f , and G(R)f = Qg f (G(R)), G(S)T (f) = Qg T (f)(G(S)); the graded localization at f and T (f), respectively. Thus T = G(ϕ) induces a graded ring extension Og Y (Y (f)) −→ Og X(X(T (f))) which already is compatible with the restriction graded homomorphism of the graded structure sheaves Og Y , Og X . Hence we have a graded sheaf extension T sheaf : Og Y −→ Og X . Again, by the exactness of the localiza- tion functors associated to p and aT (p) = q, T induces a graded local ring extension of the stalks; T sheaf p : Og Y,p −→ Og X,aT (p). Therefore we have proved the following useful result. 4.2. Proposition. The filtered Procesi extension ϕ : R → S such that S = ϕ(R).SR induces a graded Procesi extension (in the above sense) (Y = Specg(G(S)), Og Y ) → (X = Specg(G(R)), Og X) of graded affine schemes. � Jo u rn al A lg eb ra D is cr et e M at h .A. E. Radwan 137 References [1] M. J. Asonsio, M. Van den Bergh and F. Van Oystaeyen, A new algebraic approch to microlocalization of filtered rings, Trans. Amer. Math. Soc. 316 (1989), 15-25. [2] M. H. Fahmy and S. Mohammady, Normalizing extensions of regular and π- regular P.I.-rings, Proc. Math. Phys. Soc., Egypt, 67(1992), 39-46. [3] Li-Huishi, Zariskian filtrations, Ph.D. Thesis, University of Antwerp, UIA, Bel- guim, 1991. [4] A. N. Mohammed and F. Van Oystaeyen, Algebraically birational extensions, UIA Preprined, Belguim, 1992. [5] L. M. Merino and A. Verschoren, Strongly normalizing extensions, J. Pure Ap- plied Algebra, 92(1994), 161-172. [6] L. M. Merino, A. E. Radwan and A. Verschoren, Strongly normalizing modules and sheaves, Bull. Soc. Math. Belg., 44(1992), 273-291. [7] C. Nǎstǎsescu and F. Van Oystaeyen, Graded and filtered rings and modules, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1979. [8] C. Nǎstǎsescu and F. Van Oystaeyen, Graded ring theory, Math. Library 28, North Holland, Amsterdem, 1981. [9] A. E. Radwan, Filtered and graded micro-affine schemes, J. Inst. Math. & Comp. Sci., 5(1994), 73-81. [10] A. E. Radwan and F. Van Oystaeyen, Microstructure sheaves, formal schememes and quantum sections over projective schemes, Anneaux et modules collection travaux en cours, Hermann, 1996. [11] A. E. Radwan, Base change and the microstructure sheaves, to appear. [12] A. E. Radwan and N.M. Noureldeen, Base change and the formal structure sheaves, accepted in REV. Roumaine of Math. Pure Appl. Math., 2007. [13] F. Van Oystaeyen, Prime spectra in non-commutative algebra, Lecture Notes in Mathematics, Vol.444, Springer, Berlin, 1975. [14] F. Van Oystaeyen, Birational extensions of rings, Proceedings of the 1978 Antwerp Conference, Lecture Notes in Pure and Applied Math. Vol.51, Marcel Dekker, New York, 1979, 287-328. [15] F. Van Oystaeyen and A. Verschoren, Non-commutative algebraic geometry, Lec- ture Notes in Mathematics, Vol.887, Springer, Berlin, 1981. Contact information A. E. Radwan Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt. E-Mail: zezo41058@yahoo.com Received by the editors: 11.05.2007 and in final form 08.04.2008.

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