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Questions tagged [graded-rings]

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $R_i$ such that $R_i R_j \subset R_{i+j}$. (Def: http://en.m.wikipedia.org/wiki/Graded_ring)

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Hartshorne I.2.6 - questions about Boercherds' solution

Hartshorne Exercise I.2.6: $Y$ is a projective variety with homogeneous coordinate ring $S(Y)$, then $\dim (Y) = \dim S(Y) + 1$. I think I'm not understanding a simple thing. It relates to the part of ...
eggselent's user avatar
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Filtration on field of fractions of graded ring

Let $R = \bigoplus_{i \in \mathbb{Z}} R_i$ be a commutative graded ring, that is, $R_i \cdot R_j \subseteq R_{i+j}$ for every $i,j \in \mathbb{Z}$. Assume that $R$ is a domain, so that it has a field ...
Henrique Augusto Souza's user avatar
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If a graded ring $S$ is a UFD, then $S_0$ is a UFD

Let $S$ be a commutative graded ring ($S = \oplus_{n \in Z} S_n$). If $S$ is a UFD, can we deduce $S_0$ is a UFD? I think $S_0$ is a UFD. Because we can gather all the finite product of the ...
Functor's user avatar
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Homogenization of a prime ideal is prime

I'm studying from Qing Liu's book and I'm a bit stuck in his construction of $\operatorname{Proj}(B)$. In particular Lemma 2.3.35 says that if $I$ is a prime ideal then $I^h= \bigoplus _{d\geq 0} I\...
Iñaki Mendieta's user avatar
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There can't only be non-homogenous prime ideals between two homogenous primes. (Vakil 12.2.G, part c)

I'm stuck and looking for advice on part c of Question 12.2.G of Vakil's FOAG. The question states: (a) Suppose $X \subset \mathbb{P}^n$ is an irreducible projective $k$-variety. Show that the affine ...
Ice2water's user avatar
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1 answer
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Blowup of a simplicial affine toric variety at the fixed point of the torus action

In this question, all cones are strongly convex, rational, polyhedral cones. We shall adopt the convention that, if a lowercase Greek letter $\sigma$ denotes a simplicial cone, then the uppercase ...
isekaijin's user avatar
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What is $\operatorname{Proj}A[x]$ when $A[x]$ has the trivial grading?

$\newcommand{\Proj}{\operatorname{Proj}} \newcommand{\p}{\mathfrak{p}} \newcommand{\Spec}{\operatorname{Spec}}$ Let $A$ be a commutative ring, and $A[x]$ has the trivial grading where every element ...
Chris's user avatar
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Proj construction and Nilpotent Homogenous Elements in Graded Ring [duplicate]

Let $A= \oplus_{n \ge 0} A_n$ a Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined map $q^*: \...
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Quotient of Graded Rings $S \to S/f$ inducing Homeomorphism on Proj

Let $A$ be a Noetherian ring, and let $X$ be a closed subscheme of of $\mathbb{P}^r_A$. We define the homogeneous coordinate ring $S:=S(X)$ of $X$ for the given embedding to be $A[x_0, ..., x_r]/I$ (...
user267839's user avatar
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1 answer
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Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one

Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
Lorenzo Andreaus's user avatar
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24 views

Correspondence between $G$-graded $(A,A)$-bimodules and $G$-graded (left) $A^e$-modules

Let $G$ be a group (note $G$ is not necessarily abelian), and let $A$ be a $G$-graded algebra over a field $k$. We know that a $k$-central $(A,A)$-bimodule corresponds to a (left) $A^e$-module via $a \...
YSB's user avatar
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On the homeomorphism $U_f\subset \operatorname{Proj}A\leftrightarrow \operatorname{Spec}(A_f)_0$

Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this ...
Chris's user avatar
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Name for algebra which is commutative up a group action

I am wondering if there is a name for an algebra which is commutative up to some group action. To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, ...
Minkowski's user avatar
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Projective and free modules

In the graded context, if $R = K[x_1,...,x_n]$ where $K$ is a field, is a projective R-module a free R-module?
Cib's user avatar
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1 answer
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On the bijection between homogenous prime ideals of $A_f$ and prime ideals of $(A_f)_0$

Let $A$ be a $\mathbb{Z}^{\geq0}$ graded ring, and $f$ an element of positive degree. It is well known that the homogenous prime ideals of $A_f$ are in bijection with prime ideals of $(A_f)_0$, that ...
Chris's user avatar
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Is the intersection of a homogenous ideal with a non homogenous ideal homogenous?

Let $A$ be a commutative graded ring, $I\subset A$ a homogenous ideal, and $J\subset A$ a non homogenous ideal. Is $I\cap J$ homogenous? I feel like it should be. Indeed consider the following proof: ...
Chris's user avatar
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Graded pieces are finitely generated as $R_0$-modules

Let $R=R_0\oplus R_1 \oplus \ldots$ be a Noetherian graded ring and $M$ a graded $R$-module such that $M$ is finitely generated as an $R$-module. I want to show that for each $n$, $M_n$ is finitely ...
kubo's user avatar
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Example of non-graded ideal

Let $R=k[x,y]$ and $M=(x^2+y^3)$, and take the usual grading on $R$ (i.e. $R_n$ are the homogeneous polynomials of degree $n$). We have that $M$ is not a graded submodule. This is because if we define ...
kubo's user avatar
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Is each component of a graded module over a $k$-algebra a finite-dimensional vector space?

I have some problems with an argument in a proof of a lemma: Let $M = \oplus_{-\infty}^{\infty} M_n$ be a finitely generated graded $A$-module and $A=\oplus_{n\geq 0} A_n$ a graded commutative ring ...
Heraklit's user avatar
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Degree of generators of intersection of ideals

I found myself unable to answer the following question: Let $I,J$ be two ideals in $K[x_1, \dots, x_n]$ such that $I$ has a set of generators of degree $\leq d_I$ and $J$ has a set of generators of ...
LurchiDerLurch's user avatar
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88 views

Computing the associated graded ring and its Poincaré/Hilbert series

Suppose $R=k[x,y]/(f)$, where $k$ a field, $f=y^d+g\in k[x,y]$ and $g\in(x,y)^{d+1}$ with $d$ a fixed positive integer. The ideal $I=(x,y)$ is maximal in $R$. The associated graded ring $A=G_I(R)$ ...
Algebear's user avatar
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Is the category of graded modules over a graded-commutative ring an AB5 category?

Is the category of $\mathbb{Z}$-graded modules over a graded-commutative ring an AB5 category? It is abelian, the subobjects of each object form a set, and it admits arbitrary coproducts. But I don't ...
user829347's user avatar
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Weighted projective spaces Iano-Fletcher

I believe a statement in this Ian fletcher notes is Iano fletcher paper on Weighted projective spaces is wrong. For lemma 5.5 he states that the Proj$S(a_0,...,a_n)$ is isomorphic to Proj$S(qa_0,...,...
ben huni's user avatar
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Graded rings and weighted projective spaces

I am genuinely struggling with this concept, partly because of the language of graded rings. So, I will ask a simple question. If polynomial rings $\mathbb{k}[x_0,x_1]$ and $\mathbb{k}[y_0,y_1]$ with ...
ben huni's user avatar
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1 answer
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Homogeneous elements of graded algebras as polynomials of algebra generators

Let $A = \oplus_{i > 0} A_{i}$ be a positively graded $k-$algebra and let $\mathcal{B} = \left\{b_{1}, \ldots, b_{n}\right\}$ be a generating set of $A$. Let $f$ be a homogeneous element in $A_{i}$ ...
Anfänger's user avatar
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2 answers
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How is $R_iR_j$ defined in the definition of graded rings?

In the definition of graded rings there is the condition $R_iR_j\subset R_{i+j}$, it involves the product of two subgroups $R_iR_j$. How is the set $R_iR_j$ defined? Edit: Is anyone aware of a ...
Random Seed's user avatar
3 votes
1 answer
104 views

Prime avoidance for graded Noetherian ring with infinite residue field.

This question is related to this question. Let $R$ be a Noetherian local ring with infinite residue field. Let $\text{gr}_IR$ denote the associated graded ring $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
Display name's user avatar
2 votes
0 answers
85 views

Problem showing the saturation map of a graded ring f.g. in degree 1 induces an isomorphism of projective schemes and O(1) (The Rising Sea 15.6.G)

I've had problem working on this exercise from the July 31, 2023 version of Vakil's The Rising Sea: 15.6.G. Exercise. Show that the map of graded rings $S_\bullet \to \Gamma_\bullet\widetilde{S_\...
AprilGrimoire's user avatar
2 votes
1 answer
70 views

Isomorphism of graded modules

Let $R$ be graded ring $M,N$ be graded $R$-modules. If $f:M \longrightarrow N$ is isomorphism of $R$-modules (NOT graded), is $f$ isomorphism of graded $R$-module? In other words, if $M,N$ are ...
AIA's user avatar
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0 answers
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Is Artinian assumption necessary here in Matsumura's book?

I quoted Theorem 13.2 below from Matsumura's book Commutative Ring Theory: Let $R=\bigoplus_{n\geq 0}R_n$ be a Noetherian graded ring with $R_0$ Artinian, and let M be a finitely generated graded $R$-...
William Sun's user avatar
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1 vote
0 answers
23 views

Graded version of Lazard's criterion

Lazard's criterion says that a module over a commutative ring is flat if and only if it is a filtered colimit of free modules. Does the graded version hold, i.e.: A graded module over a graded ...
Bubaya's user avatar
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3 votes
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58 views

Is $k[x, x^{-1}]$ a (graded) injective $k[x]$-module

Consider $k[x]$ with the usual grading, and the graded $k[x]$-module $k[x, x^{-1}]$. Is it injective? I suppose yes, because it is torsion free and graded divisible (i.e., divisible by homogeneous ...
Bubaya's user avatar
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0 votes
0 answers
75 views

Projective-injective modules over $k[x]$

I am looking at graded $k[x]$-modules. I am also interested in the subcategory of modules $M$ all whose graded components are finite dimensional, but that's an optional restriction. In either case, I ...
Bubaya's user avatar
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0 votes
0 answers
92 views

units in a $\mathbb{Z}$ graded ring

Let $R$ be a graded ring $R=\bigoplus^{\infty}_{i=-\infty} R_{i}$. Let $f=\sum f_{n}$ be a unit element in $R$. Suppose $f_{0}$ is not contained in any mininal (homogeneous) prime ideal of $R$, then ...
MATHQI's user avatar
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Non-toric ring which is Cohen-Macaulay

I know a lot of examples of classes of binomial ideals $I$ in $S=K[x_1,\dots,x_n]$ whose $S/I$ is a Cohen-Macaulay domain. Basically, if $I$ is a toric ideal and there exists a monomial order $<$ ...
Hola's user avatar
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1 answer
45 views

How to understand Free Module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$

How can i understand of the free module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$ where $e_i=(0,\ldots ,1, \ldots 0) \in K[x]^r $ denotes the i–th canonical basis vector of $K[x]^r$. We call $x^\alpha e_i=(0,...
Kevin Duran's user avatar
1 vote
0 answers
70 views

Finiteness of the homology of the Koszul complex

Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of ...
Plafonddeplatre's user avatar
1 vote
0 answers
63 views

Curious Difficulty with the Exterior Algebra (relationship between 2 and 3 forms)

Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Then we can form the exterior algebra $\wedge V$. Let $L\subset \wedge^3 V$ be a linear subspace. I am working on the following claim. ...
nonial's user avatar
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0 answers
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Characterisation of graded homorphisms from a graded module to $\mathbb{R}[t]$

Let $\mathbb{R}[x, y]_d$ denote the vector space of homomogeneous polynomials in the indeterminates $x, y$ of degree $d$ with real coefficients. As a graded ring $A :=\mathbb{R}[x, y] = \bigoplus_{d = ...
Colin Tan's user avatar
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1 answer
46 views

Treating graded rings and modules as sequences of abelian groups connected by maps

I've been reading about $\mathbb{Z}$-graded rings and modules recently and have attempted to understand these as sequences of abelian groups with connecting maps $A_i\times M_j\to M_{i+j}$ etc. This ...
user829347's user avatar
  • 3,440
0 votes
1 answer
129 views

Kernel of morphism of graded modules is graded submodule

This is related to my earlier question regarding the category of graded modules. Let $A$ be a commutative unital $\mathbb{Z}$-graded ring, and $f:M\to N$ be an $A$-linear map where $f=\sum_if_i$ and ...
user829347's user avatar
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2 votes
1 answer
43 views

Direct limits $A_0\to A_1\to... A$ with split monomorphisms.. should the maps $A_i\to A$ be split monos also?

Let $\mathcal{C}$ be the category of $\mathbb{Z}$-graded rings. I have a sequence $A_i$ of $\mathbb{Z}$-graded rings, and split monomorphisms $\varphi_i:A_i\to A_{i+1}$ i.e. $$A_0\rightarrowtail A_1\...
user829347's user avatar
  • 3,440
1 vote
1 answer
46 views

'zero elements in different degrees are considered distinct' in graded rings/modules.. how does this work?

I am trying to make sense of section 3 of the following paper. I quote Remark 3.5 below, which is causing me some confusion: We implicitly formulate the theory of graded groups in such a way that the ...
user829347's user avatar
  • 3,440
1 vote
1 answer
364 views

Prove that an epimorphism in $\text{Mod}_R$ is surjective

Let $R$ be a unital ring and $f:M\to N$ be an epimorphism of modules. I know how to prove that a morphism of modules is a monomorphism iff it is injective, and that a surjective morphism is an ...
user829347's user avatar
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1 vote
0 answers
56 views

Graded version of Baer's criterion for injectivity

I am trying to make sense of Definition 3.1 of the following paper. See my related question from earlier today. Here is the definition below: Let $R$ be a graded ring and $I$ be a graded $R$-module. ...
user829347's user avatar
  • 3,440
1 vote
0 answers
39 views

'Shift' of a graded ideal

I am trying to understand Definition 3.1 of the following paper. We have a unital ring $R$ with a $\mathbb{Z}$-grading $(R_i)$ that is commutative i.e. $xy=(-1)^{|x||y|}yx$, and an $R$-module $I$ with ...
user829347's user avatar
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1 vote
1 answer
57 views

Does an algebra over a $\mathbb{F}_2$ of countable dimension have a grading where each component is finite?

I have a commutative unital $\mathbb{F}_2$-algebra $Q$ of countably infinite dimension, and I want to see if it has a $\mathbb{Z}$-grading where $Q_i=0$ for $i<0$ and $Q_0=\mathbb{F}_2$ and $Q_i$ ...
user829347's user avatar
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6 votes
1 answer
207 views

Is it true that $S(n)\otimes_S S(m)\cong S(n+m)$?

Convention: rings are unital and commutative. Let $S=\bigoplus_{d\geq 0}S_d$ be a graded ring. For $n\in\mathbb Z$ let $S(n)$ be the graded $S$-module defined by $S(n)_d=S_{n+d}$. It is claimed in ...
Sha Vuklia's user avatar
  • 4,062
2 votes
0 answers
63 views

Dimension of a positively graded ring after a suitable localization

Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
Sourjya Banerjee's user avatar
2 votes
0 answers
50 views

Graded ring defined via a sequence of abelian groups

I am trying to make sense of Definition 1.9 of the following paper. A graded ring is defined via a sequence of abelian groups; I outline the details below. Let $p$ be an odd prime. Define $J_0:=Z_{(p)}...
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