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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There is a known result says that for $G$ as a finite group. If $A$ be an (assocatively) $G$-graded associative algebra such that the homogeneous component $A_1$ satisfies a polynomial identity of ...
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### Do we have $\mathit\Gamma_*(\mathcal O_X)\cong S$?

In Hartshorne Proposition 5.13, the author says $r\ge 1$, but I think if $r=0$, the following proposition also holds, doesn't it? Let $A$ be a ring, let $S=A[x_0]$, and let $X=\operatorname{Proj}S$, ...
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### Do we have $\mathcal O_X(1)(X)=S_1$?

If $S$ is a graded ring which is generated by $S_1$ as an $S_0$-algebra and $X=\operatorname{Proj}S$, do we have $\mathcal O_X(1)(X)=S_1$?
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### Action of the multiplicative group induced by a grading

I found in several different fonts, (in the first section of this (https://arxiv.org/abs/alg-geom/9405004) paper, or even in this answer (https://mathoverflow.net/questions/212960/intuition-behind-...
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### $End_{R}(R)\cong R$

How can i prove that $End_{R}(R)\cong R$ as graded rings? The map of isomorphism is $$F:R→End(R),F(r)=fr,$$ where fr(a)=ar?
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### Why do we use homogeneous localisation (from a universal perspective)?

I've grown increasingly unhappy with the explanation given in introductory accounts of algebraic geometry for why we only consider the subring $S_{(f)}$ of degree $0$ elements in the localisation of a ...
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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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### 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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### 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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### 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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### 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 ...
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 ...
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 \$...