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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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Graded endomorphisms factoring through nilpotent ones

Let $M$ be a finitely generated 2-graded $k[x,y]$-module, concentrated in nonnegative degrees, and $f$ be a degree $(n,m)$-endomorphism of $M$. We can consider $f$ as a morphism $M\langle n,m\rangle\...
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$X \in \sqrt{{\rm Ann\,( gr\,} M)} \iff \exists m \in \mathbb{N} \text{ such that } \forall j \in \mathbb{Z} ,\; x^mM_j \subset M_{j + mn - 1}$

Let $R$ be a (not necessarily commutative) ring with unity and $M$ a left $R-$module.$\DeclareMathOperator{\Ann}{Ann}\DeclareMathOperator{\gr}{gr}$ Let $(R_i)$ be a $\mathbb{Z}$ filtration on $R$ and ...
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Associated graded ring of an ideal is reduced, then the ideal is normal.

Let $I$ be an ideal in a Noetherian ring $R$ such that $\mathrm{gr}_I(R)$ is a reduced ring. Prove that $I$ is normal. What is the error in this proof? We proceed by induction over $n$. If $n=1$ we ...
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Quotient of Graded ring is Graded ring

I'm checking the detail of the proof of the following property in Dummit and Foote Page 444 : Proposition 33. Let $ S= \oplus_{k=0}^{\infty} S_k $ be a graded ring, let $ I $ be a graded ideal in $ ...
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How to understand $\mathbb{Z}^n$-graded ring?

I am reading Ringel's note and I encountered a question I have never met. The question is what's the meaning of a $\mathbb{Z}^n$-graded ring? This is from the following: "Note that the rings $U_q (n_ ...
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naturally graded algebras

I found a good explanation on graded algebras Understanding of graded algebra But I am confused about difference between "naturally graded" and "graded". May you please clarify the notion of ...
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Can this graded ring with this Poincare series be finitely generated?

I'm having a bit of a hard time figuring this out, but I have a family of graded rings with the following Poincare series: $$\frac{1+t}{1-(n-1)t}$$ where $n$ is a positive integer. Assume the ...
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Understanding the Yoneda product defined in terms of morphisms of projective resolutions.

On the wikipedia page for the Ext functor, they say that one can equip the graded abelian group $\operatorname{Ext}^*:=\bigoplus_{i=0}^{\infty}\operatorname{Ext}^i(A,A)$ with the structure of a ring (...
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Embed a weighted projective space into an unweighted projective space.

To show is the following. Let $X = P(a_0,\dotsc,a_n)$, $a_i \geq 1$ be a weighted projective space (that is $X = \operatorname{Proj} k[x_0,\dotsc,x_n]$, where $\operatorname{deg} x_i = a_i$). ...
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associated graded ring is PI implies original ring is PI

This seems like it should be a known result but I didn't find it in a couple of standard references on noncommutative Noetherian rings. Recall that a unital associative ring $R$ is a polynomial ...
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Lifting graded ideal to ideal in filtered ring

Let $R$ be a Noetherian filtered ring. Let $P\subset \operatorname{gr} R$ be a homogeneous ideal. Does there exists an ideal $\tilde{P}\subset R$ such that $\operatorname{gr}\tilde{P}$ (with the ...
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Prime ideal of ring without gradation generating the prime homogeneous ideal contained in the same ring with gradation?

Let $R=\oplus_{i\geq 0}R_i$ be a graded ring. Denote $S$ as the ring $R$ without gradation structure. Suppose $p\in Spec(S)$. I want to consider $Q=\oplus_ip\cap R_i$. Suppose $ab\in Q\subset p$ with $...
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difference between graded ring and its twisted global sections

Let $S_{\bullet}$ be a graded ring, generated in degree $1$ with $S_0 = k$ (a field). One can associate to $S_{\bullet}$ the twisted graded ring $$ \Gamma_{\bullet} = \left( \ \Gamma(\mathrm{Proj} S_{\...
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$(S_f)_0$ is a finitely generated algebra if $S$ is. [duplicate]

Let $A, S$ be commutative rings with identity, and assume $S$ is a finitely generated $\mathbb{Z}^{\geq 0}$-graded $A$-algebra. If $f\in S$ is a homogeneous element of positive degree, $S_f$ is a $\...
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Homogeneous ideal of height $2$ in $\mathbb C[X,Y]$ [closed]

If $J$ is a homogeneous ideal of height $2$ in $\mathbb C[X,Y]$ such that $J\subseteq (X,Y)$, then does there necessarily exist an integer $n\ge 1$ such that $X^n,Y^n \in J$ ?
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Reference request for a “freeness” property of graded monoids

Let $I$ be a monoid and $G$ be an $I$-graded monoid, with multiplication $$ ( - \cdot - ) : G_i \times G_j \to G_{i+j}. $$ I'm interested in the following property of $G$: P: for any two indices $i,...
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1answer
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Intermediate step solving Hartshorne Ex II-3.12 a)

In this exercise we have a surjective graded ring homomorphism $\varphi:S\to T$. This induces a morphism $f:$Proj$(T)\to $Proj$(S)$ by contraction of ideals. I'm asked to show that $f$ is a closed ...
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1answer
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Passing from a set of generators to a set of homogeneous generators

Consider a graded ring $R=R_0\oplus R_1\oplus\dots$. Suppose the irrelevant ideal $R_1\oplus R_2\oplus \dots$ is finitely generated. How come we can assume that it is generated by a finite number of ...
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Irreducible highest weight representations as a graded algebra

Let $L$ be a semisimple Lie algebra and let $V(\lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $\lambda$. How can we view the sum \begin{align*} \oplus_{n\in\mathbb{N}...
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Graded global sections of Proj(S) for S a polynomial ring and more general

Throughout, suppose $S$ is a graded ring which is finitely generated by $S_{1}$ and an $S_{0}$-algebra. Let $X = \text{Proj} S$. There is the usual associated graded module given by $$ \Gamma_{\bullet}...
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Support of a graded module of a ring concentrated in non-negative dimensions

I wanted to prove the following equivalence. Consider $R$ a graded commutative Noetherian ring such that $R^{<0}=0$ and $M$ a graded, finitely generated $R$-module. Then $M^i=0$ for $i \gg 0$ if ...
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Isomorphism between localizations of graded ring $S_{(P)} \cong [S_{(f)}]_{PS_f \cap S_{(f)}}$

I know that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the prime ideals ...
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Ascending / descending chain condition on graded modules.

Let $R = \bigoplus_{n \in \mathbb{N}} R_n$ be a graded commutative ring. Then $R$ is noetherian / artinian if and only if it has the ascending / descending chain condition for homogeneous ideals, see ...
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Tensor product of graded algebras 3

Let $A$ and $B$ be $\mathbb{Z}_2$-graded algebras, i.e. $A=A_0 \oplus A_1$, $B= B_0 \oplus B_1$. I am trying to show that the graded tensor product $A \otimes B = (A \otimes B)_0 \oplus (A \otimes B)...
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When a homogeneous ideal is written as a product of two ideals, then each of two ideals is homogeneous?

I know that in a graded domain, if a homogeneous element is written as a product of two elements, then each of two elements is also homogeneous. That is, the set of all the homogeneous elements of ...
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Indecomposable graded $A$-modules with $\operatorname{End}_A(M)/\mathfrak m\neq k$

Let $A$ be a $k$-algebra, for instance $A=k[x]$. Then we know that a module $M$ is indecomposable iff $\operatorname{End}_A(M)$ is a local ring (whose maximal ideal is denoted by $\mathfrak m$). This ...
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Constructing a graded ring from abelian groups - defining the unit.

On page 631 of Lang's Algebra, he gives a construciton Let $G$ a commutative monoid. Suppose for each $r,s \in G$, we have abelian groups $A_r$, and maps $A_{r} \times A_s \rightarrow A_{r+s}$, $...
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How does coassociativity of a coalgebra $C$ imply that the derivation on $\Omega C$ is a differential?

I am trying to show that $d²=0$ where $d$ is the derivation on $T(s^{-1}\bar{C})$ induced by the map $s^{-1}\bar{C}\to T(s^{-1}\bar{C})$ defined by $$s^{-1}x\mapsto -\sum (-1)^{|x_{(1)}|}s^{-1}x_{(1)}\...
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what's the meaning of $B_m/f^n$?

In Page 82 of Qing Liu's book "Algebraic Geometry and Arithmetic Curves", $B$ is a graded ring and $f\in B_+$ is a homogeneous element, it says $B_{(f)}$ is a direct factor of $B_{f}=B_{(f)}\oplus(\...
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Ideal is graded iff it is generated by homogeneous elements

Consider the polynomial ring $\mathbb{F}[x_{1},...,x_{n}]$ in $n$ variables and let $I \subset \mathbb{F}[x_{1},...,x_{n}]$ be an ideal. We call $I$ graded if we can decompose it into it's homogeneous ...
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Definition of “degree zero” in the localization of a graded ring

Let $S$ be a graded ring and $f$ is an element of degree $d>0$. Then $S_{(f)}$ is defined as the subring of elements of degree $0$ in the localized ring $S_{f}$ (Hartshorne, pp. 77). But the ...
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Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a graded (non-commutative) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) Noetherian as a ring. In the ...
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On the structure of $R_0$ in graded PID $R=\bigoplus_{n \ge 0} R_n$

Let $R=\bigoplus_{n \ge 0} R_n$ be a graded integral domain. If $R$ is a PID, then is $R_0$ a field ? Since $R$ is Noetherian, I know that $R_0$ is Noetherian and $R$ is a finitely generated $R_0$-...
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The Endomorphism algebra of graded vector space

Let $G$ be a group. A linear map $f:V\rightarrow W$ of $G$-graded vector spaces is said to be homogeneous of degree $g$ if $f(V_{h}) \subseteq W_{g\cdot h}$ for all $h\in G$. We denote the space all ...
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Are the associate primes of a graded module homogeneous?

Let $R$ be a $\mathbf N^r$-graded ring, for instance a polynomial ring in $r$-variables. A prime ideal $\mathfrak p\subseteq R$ is associated to a graded $R$-module $M$ if there is a (not necessarily ...
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Doubt on twisting sheaves definition

I am reading Hartshorne, and I don't understand something about twisting sheaves. He defines, for $S$ a graded ring, the $n$-th twisting sheaf $O_X(n)$ as "$S(n)^{\tilde}$. I tried to interpret this ...
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Binomial coefficients undefined in in the Hilbert polynomial for projective space

Let $k$ be a field and let $X= \mathbb{P}_{k}^{r}$ be the projective space (as a scheme) of dimension $r$ over $k$. Let $\mathcal{O}(d)$ denote the degree $d$ twisted structure sheaf. Then we define ...
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In a $\mathbb Z$-graded ring we have $IR \cap R_0 = I$

I have a problem with an exercise from Tom Marley which is: Let $R$ be a $\mathbb Z$-graded ring and $I$ an ideal of $R_0$. Prove that $IR \cap R_0 = I$. For $I \subset IR \cap R_0 $, we can ...
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Units are homogeneous in $\mathbb Z$-graded domains

I am confused by an exercise from Tom Marley which is: Let $R$ be an arbitrary $\mathbb Z$-graded domain: $1)$ Prove that all units in $R$ are homogeneous. $2)$ By using $1$, if $R$ is a field, ...
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On “homogeneous” height and “homogeneous” Krull-dimension?

Let $R=\oplus_{n \ge 0} R_n$ be a commutative graded ring. For a prime ideal $P$ of $R$, let $ht P$ denote the usual height of a prime ideal $P$ of $R$. Now let $P$ be a prime ideal of $R$ which is ...
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Adams operations and an artificial grading on K-theory

In this article by Snaith (p. 575) appears the following comment: ... these transgressive elements [...] can be located by means of the Adams operations [...]. These operate (unstably) in both the ...
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Localization of a graded ring at degree zero

Let $S$ be a $Z^{\ge 0}$-graded ring and $f,g$ be two homogeneous elements of positive degree in $S$. I wonder if $$(S_{fg})_0 \cong [(S_f)_0]_{g^{\text{deg} f}/f^{\text{deg} g}}$$ is true (and how ...
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Properties of the Zariski topology on Proj

Let $S_\bullet$ be a $(\mathbb{Z}_{\geq 0})$-graded ring, $f \in S_+$ be a homogenous element, $I \subseteq S_+$ any homogenous ideal, $V_+(I) := \{p \in ProjS_\bullet | I \subseteq p \}$. I'm ...
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The super group $GL(1|1)$

It is difficult to find information on super groups and I have built my knowledge from various sources. I have the following questions. $GL(1|1)$ is defined as the group of invertible linear ...
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Bihomogeneous Nullstellensatz

I'm reading 'Arithmetically Cohen-Macaulay Sets of Points in $\mathbb P^1\times\mathbb P^1$' by Elena Guardo and Adam Van Tuyl. (One can read it partially on Google books.) I doubt whether the '...
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1answer
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High-degree pieces of graded ideal with coprime generators

(I have a couple of questions about graded ideals and I would appreciate any help/ideas anyone may have on the following. I had posted this one earlier, but nothing came of it, so deleted it, trimmed ...
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Direct sum and the inclusion property

Let $R$ be a ring and if $R= \bigoplus R_i$ as additive subgroups for each $i \in I$ where $I $ is a finite group Is that implies $R_i R_j $ must be contained in $R_p$ for any $p \in I$ ?? In ...
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Extending a linear action to monomials of higher degree

Let $R$ be a commutative ring and let $M$ be an $m\times n$ matrix with entries in $R$. $M$ defines a map $\widehat{M}:R[x_1,\ldots,x_n]\to R[y_1,\ldots,y_m]$ by extending the linear action on degree $...
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93 views

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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1answer
82 views

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 ...