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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The degree $0$ component of a graded ring and localization

Let $S$ be a $\mathbb Z$-graded ring and $f \in S$ be a homogeneous element. Is the morphism $\mathrm{Spec}(S_{(f)}) \to \mathrm{Spec}(S_0)$ induced by the morphism of rings $S_0 \to S_{(f)}$ (which ...
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Integers modulo prime power, not a graded ring?

We can decompose Z mod p^n as a vector space over Z mod p by writing numbers base p, eg, mod 3, 52= 27 + 2*9 + 2*3 + 1, and if Rm is {0,p^m,2p^m,....(p-1)p^m}, then we have for a in Rm, b in Rk, that ...
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Interpretation of the numerator of the Hilbert series?

Let $R$ be a finitely generated graded ring over a field $k$. Let $R_\ell$ be the degree-$\ell$ homogeneous component of $R$. By the Noether normalization theorem, $R$ is finite over a graded ...
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134 views

On graded fields

Tom Marly here noted that if R is a graded field then $R$ is concentrated in degree 0, i.e., $R=R_0$ and $R_n =0$ for all $n \neq 0$. Is this proposition mentioned in any book or paper ?? ...
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Dimension of a graded module over a local$^*$ ring

Assume that $R$ is a positively graded ring which has only one maximal homogeneous ideal $\mathfrak{m}^*$. Let $M$ be a finitely generated positively graded ring over $R$ and consider $\mathcal{l}_R(M/...
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309 views

Is the formal power series ring a graded ring?

Let $k$ be a field, let $k[[t]]$ be the formal power series ring over $k$ in one variable. Does there exist a $\mathbb{Z}$-grading on $k[[t]]$? In other words, does there exist a direct sum ...
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350 views

Why are the global sections of structure sheaf of Proj$S$ just the homogenous elements of $S$?

Let $A$ be a ring and define $S = A[x_{0}, x_{1}, \ldots , x_{r}]$. Let $X = \text{Proj }S$. I would like to show that $\Gamma(X, \mathcal{O}_{X}(n)) = S_{n}$. This is Proposition II 5.13 in ...
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344 views

Proj of the Graded Ring of Global Sections

Let $R$ be a graded ring, finitely generated by $R_1$ as an $R_0$-algebra. Let $X=\mathop{\rm Proj}R$ and let $R':=\Gamma_*(\mathcal{O}_X)$ be the associated graded module of global sections of twists ...
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Which Graded and free modules over Graded PID's are Graded-free ?

Let $G$ be an abelian group and $R=\oplus_{g\in G}R_g$ be a $G$-graded, commutative ring with unity . Let us call $R$ to be a "Graded PID" if every graded ideal of $R$ is generated by a homogeneous ...
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351 views

Do How to convert degrees to decimal degrees?

Do How to convert degrees to decimal degrees? Example 1: 1. I have - 450 - degrees 2. We need get - 90 from 450 Example 2: 1. I have - 540 - degrees 2. We need get - 180 from 540
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Hilbert Coeficients; Multiplicity

Let $R$ be a graded ring and $M$ a graded $R$-module. An element $x\in R_{l}$ is said to be superficial for $M$ if $(0:_{M} x)_{n}=0$ for all but finitely many $n$. Let $M$ be a finite generated $R$-...
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Noetherianity of graded rings

I have been able to show that for a graded subring $S$ of $R[x]$ (where $R$ is a noetherian domain) that in order to show that $S$ is noetherian, it suffices to consider homogeneous ideals of $S$. ...
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What is the notation $\mathcal{A}_\infty$ for a sheaf of graded $\mathcal{O}_X$-algebras mean?

Today I encountered Hartshorne's condition $(\dagger)$ for quasi-coherent sheaves of algebras. This states that gives a graded $\mathcal{O}_X$-module $\mathcal{A}$ which has the structure of a graded ...
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572 views

graded ring and homogeneous ideal

In the Wikipedia page it says if $I$ is a homogeneous ideal in graded ring $A=\oplus_{n\ge0}A_n$, then $\frac{A}{I}$ is a graded ring decompose as: $\frac{A}{I}=\oplus_{n\ge0}\frac{A_i+I}{I}$ I ...
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47 views

Zariski cone-topology?

Let $A$ be an almost commutative algebra and write $A_0 = \text{gr} \, A/ \oplus_{i > 0} \text{gr}_i \, A$. At the bottom of p. 16 here, the author says the following: ... recall first that the ...
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Counting graded dimension of polynomial ring

The polynomial ring $k[x,y]$ has graded $k$-dimension (aka. Poincaré-series) $\frac{1}{(1-q)^2}$. This is clear since the polynomial ring in one indeterminate has $\dim_q k[x]=1+q+q^2+\cdots=\frac{1}{...
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$a^m$ homogeneous implies $a$ homogeneous?

Let $A = \bigoplus_{r \in \mathbb{Z}/n\mathbb{Z}} A_r$ be an integral $\mathbb{Z}/n\mathbb{Z}$-graded ring of characteristic $0$, and $a \in A$. Assume there exists an integer $m$ such that $a^m$ is ...
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143 views

A query about tensor product of graded modules.

Let $A=\bigoplus_{n\geq 0}A_n$, $B=\bigoplus_{n\geq 0}B_n$ be two graded Noetherian rings, where $A_n\subset B_n$ and $A_0=B_0$ is a local ring. Suppose $B$ is a finitely generated $A$ module. Let $I$ ...
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139 views

Graded vector space conditions

Wikipedia define the graded ( ring, module, vector space, ...) as here I noted that in the rings and modules it required the condition of inclusion but in the vector spaces it did not. just the ...
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161 views

(Reference) Does a ring's grading commute with the formation of fractions?

Question: Given a graded integral domain, when one forms the corresponding field of fractions, does the resulting field of fractions have a grading "compatible" with the original field of fractions?...
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Given a proper homogeneous ideal, is there always a homogeneous maximal ideal that contains it? [duplicate]

It is known that every proper ideal of a ring must be contained in a maximal ideal, by the Zorn's lemma. Is this true in general for proper homogeneous ideals in a graded ring?
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Regarding length of chains of homogeneous primes

Let $k$ be a field, and consider the polynomial ring $k[x_1,...,x_n]$. My question is this: Given a homogeneous ideal $I\subseteq (x_1,...,x_n)$, does every maximal chain of homogeneous primes $I\...
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A kind of a “ Three Space Property” for commutative strongly graded rings

Let $G$ be an abelian group ; let $R=\oplus _{g \in G} R_g$ be a commutative unital $G$-graded ring . For a subgroup $H$ of $G$ we denote by $R_H$ the $H$ graded ring as $R_H = \oplus _{h \in H} R_h$ ,...
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On commutative unital graded ring in which no non-zero homogenous element has a zero divisor in the homogenous part

A follow up of On commutative unital graded rings in which no element in any homogenous part has a zero divisor . Let $G$ be a torsion free abelian group , let $R$ be a commutative unital $G$-graded ...
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52 views

On commutative unital graded rings in which no element in any homogenous part has a zero divisor

Let $G$ be a monoid ; $R$ be a unital commutative $G$-graded ring such that for every $g \in G$ , $x_gy\ne 0 , \forall x_g \in R_g \setminus \{0\} , \forall y \in R \setminus \{0\}$ . Then is it true ...
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Exterior Algebra Graded Ring Structure?

In Hatcher's Example 3.13 he defines the Exterior Algebra $\Lambda_R[\alpha_1,...,\alpha_n]$ over a commutative ring $R$ with identity to be the free $R$-module with basis the finite products $\alpha_{...
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83 views

On giving a certain kind of grading to the polynomial ring over a commutative unital graded ring

For any commutative unital ring $R$ , $R[X]$ has a standard $\mathbb N \cup \{0\}$ grading where the $n$-th component is $R_n:= X^nR[X]$ ; however , given a commutative , unital , graded ring $R$ , I ...
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When does commutativity in each homogenous component of a unital graded ring forces commutativity of whole ring?

Let $R=\oplus_{g \in G} R_g$ be a graded unital ring , graded by a monoid $G$ . Suppose $x_gy_g=y_gx_g , \forall x_g , y_g \in R_g ; \forall g \in G$ ; then is it true that $R$ is commutative ? If ...
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Can a finitely generated graded module over commutative unital graded ring be also finitely generated by homogenous elements ?

Let $R$ be a commutative graded ring (https://en.wikipedia.org/wiki/Graded_ring) with unity , graded by a monoid (in particular $\mathbb Z$ or $\mathbb N \cup \{0\}$ if helpful for the purpose ) , ...
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170 views

Resolutions of graded modules over positively graded polynomial rings

I am looking for a proof of this fact: Let $W=k[x_1,\dots,x_n]$ be a positively graded polynomial ring. Every graded free resolution of a finitely generated $W$-module $M$ is isomorphic to the direct ...
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63 views

Maschke's theorem for $G$-graded algebras

I am reading the paper Algebras graded by a group of Knus. Immediately I run into problems, which I will now detail: Let $G$ be a group and $K$ a field. A $G$-graded algebra $A$ is a finite-...
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Quick question about Graded Rings

Recently I read about graded rings and I read old papers but I noticed something all these papers define the graded ring but there is no proves (all rings are a group graded ring and satisfy the ...
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Definition of Exterior Algebra of a Graded Module

Let $M$ be a $\mathbb{Z}$-graded module over a trivial graded ring $R=R_0$. The tensor algebra $T_R(M)$ then becomes a $\mathbb{Z}$-graded module with $i$-th graded component $$T_R(M)_i = \bigoplus_{...
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Direct Product of a Graded Ring. [duplicate]

Let $R$ be a graded ring, for example with nonnegative grading such that $$R=R_0 \oplus R_1 \oplus \dots$$ Is then $R^n= R \times \cdots \times R$ a graded ring (not with a trivial grading) ? If ...
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50 views

Graded module structure as a graded homomorphism?

Let $R$ be a ring. A $R$-module is an abelian group $M$ with an application $R\times M\longrightarrow M$, $(r, x)\longmapsto r\cdot x$ satisfying some properties. To give a $R$-module structure on ...
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Linear Resolution of a Hibi Ring

I have read somewhere that a Hibi ring has a linear resolution if and only if it's regularity is 1. Please refer me the proof of this fact. Firstly, I need to define the Hibi ring. Let $L$ be a ...
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Filtered objects and associated graded objects

Given a filtered object one can consider it's associated graded object. However, non-isomorphic objects can have the isomorphic associated graded objects. The easiest example I could think of is the ...
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Question about definition of graded ring

In the course of algebraic geometry I follow, the professor has introduced the notion of 'graded ring' to decompose polynomial ring $k[x_0, \ldots, x_n]$ as follows: $$k[x_0, \ldots, x_n] = \oplus_{d ...
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118 views

Two-sided ideal $I$ in exterior algebra $T(V)/I$.

I have a confusion regarding two definitions of the two-sided ideal in exterior algebra. Def 1) In one definition, the exterior algebra $\Lambda(V)$ is defined as $T(V)/I$, where $I$ is the two-sided ...
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Basic question on graded quotients

Let $A$ be a graded algebra, and let $I$ be a graded (two-sided) ideal. ($A=\bigoplus A_k$, $I=\bigoplus I_k$). Is it true that $A/I=\bigoplus A_k/I_k$? I can't seem to see it. Thanks for any help....
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Correspondence of grading and group actions

Let $k$ be a field of characteristic zero with $n$-th roots of unity, $R$ a $k$-algebra and $G := \mathbb{Z}/n\mathbb{Z}$. To give an action of $G$ on $R$ is equivalent to grading $R$ by $G$. (To see ...
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If $f \in \hat{m} = $ largest homogeneous ideal, then each coefficient in some generation is homogeneous.

Here is the notation & motivation.: Conjecture: If $f \in \hat{m}$, then $f = \sum g_i f_i$ where each $g_i$ is homogeneous. I've tried proof by induction on degree of $f$, number of ...
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How is this proof correct in regard to a $k$-subalgebra (Eisenbud)?

This is testing my understanding of algebras, subalgebras, & polynomial rings. Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. ...
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Basic fact used in proofs in Eisenbud's commutative algebra book. [duplicate]

Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that ...
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Question regarding Graded Rings

I am currently reading Li Huishi's book "Zariskian filtrations". I am confused by a lemma (p. 34, Lemma 2 of section 4.3 in Chapter 1) in the book. It says: Let $S = \oplus_{n \in \mathbb{Z}}S_n$ ...
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Is there a (many-sorted) first- or second-order definition for graded rings?

Usually, a graded ring $R$ over a monoid $M$ is defined as a ring which decomposes as an inner direct sum $R = \bigoplus_{m ∈ M} R_m$ of abelian subgroups such that for all $m, m' ∈ M$, $R_mR_{m'} \...
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Quotient of externally graded algebras?

Let $R$ be a commutative ring. An internally graded $R$-algebra is an $R$-algebra $A$ for which there is a family of $R$-submodules $(A_n)_{n\in\mathbb Z}$ such that (i) $\displaystyle A=\bigoplus_{...
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372 views

Is there a version of the adjunction formula for subvarieties of $\mathbb{P}^n\times\mathbb{P}^m$

I want to try and find an example of a projective variety which does not have an ample canonical or anti-canonical bundle. I think I should try and look at a subvariety of $\mathbb{P}^n\times\mathbb{P}...
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144 views

Algebraic characterization of the exterior covariant derivative

The de Rham complex on a smooth manifold $M$ is a complex of sheaves of $\mathbb{R}$-modules, which I will denote by $$\Omega^\bullet \xrightarrow{d} \Omega^{\bullet+1}.$$ A connection (not assumed ...
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180 views

Are the units of a monoid-graded ring homogeneous?

I'm just trying to find a reference for the following statement: if $R$ is an $M$-graded integral domain, where $M$ is a monoid, then every unit of $R$ is homogeneous. This source (in particular, ...

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