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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Інститут прикладної математики і механіки НАН України
2007
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| Цитувати: | 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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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 |
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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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Radwan, A.E. |
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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 |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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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 |
| _version_ |
1849736014011039744 |
| fulltext |
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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.
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.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
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.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?
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.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.
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.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).
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.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. �
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.A. E. Radwan 137
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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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