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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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 $A_{i}$ ...
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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 ...
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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 $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
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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_\...
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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 ...
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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 $R$-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $<$ ...
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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,...
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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 ...
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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.
...
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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 = ...
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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 ...
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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 ...
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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\...
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'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 ...
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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 ...
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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. ...
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'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 ...
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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 $Q_i$ ...
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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 ...
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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 ...
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$\mathbb{Z}_{(p)}$-module structure on a cyclic group
I am trying to make sense of Definition 1.9 in the following paper. See my related question from earlier today.
Let $p$ be an odd prime and $Z_{(p)}$ be the localisation of $\mathbb{Z}$ at the prime ...
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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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Possible structures of tensor product between two graded anticommutative algebras
If we have two anticommutative graded rings $M=\bigoplus_{k\geq 0} M_k$, $N=\bigoplus_{l\geq 0} N_l$ (anticommutative meaning $ab=(-1)^{deg(a)deg(b)}ba$ for $M$ and the same for $N$) such that each $...
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Cohen-Macaulay property of graded ideals
Let $R=K[x_1,\dots,x_n]$ and $I$ be a graded ideal of $R$.
My question is the following: if $R/I$ is a Cohen-Macaulay ring then $I$ is a Cohen-Macaulay ideal?
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$\mathbb Z$-graded coordinate ring $\iff$ $\mathbb C^*$ action on affine variety
Let $X$ be an affine variety with coordinate ring $\mathcal O$. I want to show that a $\mathbb C^*$ action on $X$ is equivalent to a $\mathbb Z$-grading on $\mathcal O$. I have problems with both ...
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When is a Noetherian Standard graded algebra over a field Cohen-Macaulay? Any counter-example when it is not the case?
Let $R$ be a Noetherian standard graded algebra with $R_0 = k$, a field, then it is finitely generated over $R_0$ by $R_1$ and is the homomorphic image of some $k[x_1, \ldots, x_n]$, hence isomorphic ...
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Infinite symmetric product of an $H$-space and graded product structure on homotopy groups.
Assume $X$ is an $H$-space and let's look at homotopy groups of $SP(X)$ where $SP$ here is the infinite symmetric product. By Dold-Thom we have $\pi_i(SP(X))\cong \tilde{H}_i(X)$. Since $X$ is an $H$-...
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Graded measure theory?
I've just had a thought that somebody else has definitely had before but I don't know what they might have called it.
I have a $\sigma$-algebra $(X=\mathbb{R}^n, \Sigma)$ and I want to measure it, but ...
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Hilbert polynomials of graded algebras evaluated at negative numbers
Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
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I am a bit confused on the definition of a graded ring in Eisenbud.
He defines a graded ring as $R$ together with a direct sum decomposition
$R = \bigoplus_{n \in \mathbb{N}}R_n$
where $R_n$ are abelian groups such that $R_iR_j \subset R_{i+j}$
I understand addition ...
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Classification of graded fields, semifields, and skew fields
Recently I came upon the following result:
Result 1. Let $K$ be a $\mathbb{Z}$-graded field. Then either $K$ is trivially graded (i.e. $K_k=0$ for $k\in\mathbb{Z}\setminus\{0\}$ with $K_0$ a field or ...
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The associated graded of the ungraded polynomial ring is isomorphic to the ungraded polynomial ring as modules over the ungraded polynomial ring.
Let $k$ be a commutative ring. Let $k[t]$ be the ungraded polynomial ring with $\deg t=0$. The associated graded of the ungraded ring $k[t]$ is nevertheless graded and isomorphic to the graded ...
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associated graded of ungraded polynomial ring is isomorphic to graded polynomial ring.
I would like to discuss a fine observation that I made which is not discussed in the literarure. Maybe because it is easy. Nevertheless, I think that it is quite important.
Let $k$ be a commutative ...
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Maximal homogeneous ideals in graded ring A[T]
Let $A$ be a commutative algebra with unit. Let $A[x]$ be the polynomial algebra with coefficients in $A$ with the standard gradation (by degrees).
I have the following questions.
What are maximal ...
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Quotient of Graded Algebra is Graded
Let $A$ be a graded algebra over a commutative ring, and $I$ a two sided ideal of $A$ which is not necessarily graded. We can take the quotient $A/I$ and obtain an algebra, however I am struggling to ...
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Symmetric Algebra of a Graded Module
Let $k$ be a commutative ring. Let $M$ be a $k$-module. Let $M$ be a graded module, concentrated in degree one. Then, the shifted module $M[1]$ is concentrated in degree zero. Hence, $M[1]$ is ...
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Action on associated graded algebra inducing action on filtered algebra
Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, ...
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About the number of homogeneous generators of a f.g. homogeneous ideal?
Let $G$ be an abelian group and $R$ be a $G$-graded ring. We consider the following assertions.
$(1)$ There is a map $\psi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in\mathbb{N}$, and ...
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Relation between definitions of a graded vector space and a graded module
I am new to the world of "grading", and my question is about the definition of the graded vector space. For definitions of these two, see graded vector space and graded module (they are ...
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53
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Does this map from a graded algebra have a name?
Let $k$ be a field and $A$ be a ($\mathbb{Z}$-)graded associative $k$-algebra. Suppose a $k$-linear map $\varphi : A \to k$ has the property that, whenever $a, b \in A$ are homogeneous of positive ...
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Sheaf of graded algebras
Let $U\subseteq \mathbb{R}^n$ be open and $W:=\oplus_{i\in \mathbb{Z}}W_i$ be a real graded vector space with $dim W_i<\infty$ for all $i$ and $W_0=\{0\}$. I would like to find a reasonable sheaf $\...
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Non-example of graded subring
I just want to check that I understand the definition of a graded subring, and I haven't seen anyone write down what seems like the most obvious type of non-example for a graded subring.
Inside of $k[...
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Sufficient condition for radicality of an homogeneous ideal: still true when grading over arbitrary monoid?
Let $A$ be a graded commutative ring (if necessary, with unit). Suppose we have an homogeneous ideal $I\subset A$ that satisfies the property “for all homogeneous $f\in A$ such that $f^r\in I$ for ...
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2
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Does the inverse of an invertible homogeneous element need to be homogeneous?
Let $A$ be a commutative ring, with unit, and suppose $A$ is graded over a commutative monoid $M$. (In particular $1\in A_e$, where $e\in M$ is the identity.) If $x\in A$ is invertible and homogeneous,...
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Questions about $K[x_1, \ldots , x_4]/(x_1x_2-x_3x_4)$ being a graded ring
So I was trying to understand this answer to the question of why $K[x_1, \ldots , x_4]/(x_1x_2-x_3x_4)$ ($K$ being a field) is not an UFD, and the author seems to use the fact that $K[x_1, \ldots , ...
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