# 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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### 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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### 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 ...
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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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### 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 ...