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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)

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $R_i$ such that $R_i R_j \subset R_{i+j}$. Reference: Wikipedia

The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid or group. The direct sum decomposition is usually referred to as gradation or grading.

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The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known 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 of $S_{(f)}$, the subring of $S_f$…
Zhen Lin
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Is the radical of a homogeneous ideal of a graded ring homogeneous?

Let $S = \sum_{n \in \mathbb{Z}} S_n$ be a commutative graded ring. Let $I$ be a homogeneous ideal of $S$. Let $J$ be the radical of $I$, i.e. $J = \{x \in S| x^n \in I$ for some $n > 0\}$. Is $J$ a homogeneous ideal?
Makoto Kato
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An elegant description for graded-module morphisms with non-zero zero component

In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the following class of morphisms: $\Sigma=\left\lbrace f\in…
BBischof
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Showing that a homogenous ideal is prime.

I'm trying to read a proof of the following proposition: Let $S$ be a graded ring, $T \subseteq S$ a multiplicatively closed set. Then a homogeneous ideal maximal among the homogeneous ideals not meeting $T$ is prime. In this proof, it says "it…
DA55
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"Graded free" is stronger than "graded and free"?

This topic suggested me the following question: If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a basis consisting of homogeneous elements)? In general…
user89712
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Quotient ring of a graded algebra with respect to a graded ideal

An algebra $A$ over $F$ is said to be a graded algebra if as a vector space over $F$, $A$ can be written in the form $$A=\bigoplus_{i=0}^\infty A_i$$ for subspaces $A_i$ of $A$ along with other properties. And a graded ideal $I$ in a graded algebra…
user782220
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Minimal systems of generators for finitely generated algebras over commutative (graded) rings

Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that any two minimal systems of generators of $R$ over…
Will
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Weighted projective space and $\mathrm{Proj}$

I'm trying to solve a problem from Jenia Tevelev's notes on GIT. (Problem 5 at the end of this pdf.) Compute $$\operatorname{Proj}\frac{\mathbb{C}[x,y,z]}{(x^5+y^3+z^2)}$$ where $\operatorname{wt.}x,y,z=12,20,30$. Here is what I have so far. Let…
Tim
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Why is cohomology the direct product of the $H^n$?

During a talk I mentioned in passing Borel's result that for $G$ a connected Lie group, $H^*(BG;\mathbb Q)$ is a polynomial ring. An audience member corrected me in very short order that no, it's in fact a power series ring. I responded that while…
jdc
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Global sections of Proj

In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } S_{(f)}$, the latter denoting the degree zero…
Future
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Why is the topology on $\operatorname{Proj} B$ induced from that on $\operatorname{Spec}(B)?$

In the proof of Lemma $3.36$ in Algebraic Geometry and Arithmetic Curves, it is stated that, if $B=\oplus_{d\ge0}B_d$ is a graded algebra over a ring $A,$ and if $I$ is an ideal of $B,$ then $$V(I)\cap\operatorname{Proj}(B)=V_+(I^h),$$ where…
awllower
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Are minimal prime ideals in a graded ring graded?

Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minimal prime in $A$. Is $\mathfrak p$ a graded ideal? Intuitively, this means the irreducible components of a projective variety are also projective varieties. When $A$ is Noetherian, I…
YZhou
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If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

Quick question (hopefully): What is the correct definition of a tensor product of two graded $R_*$-modules and/or graded $R_*$-algebras $M_*$ and $N_*$ over the graded ring $R_*$? $M_* \otimes_{R_*} N_* = ?$ If R is not graded I know how to do…
Elle Najt
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Two definitions of graded rings

So, there are two types of definitions of graded rings (I will consider only commutative rings) that I have seen: 1) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = \bigoplus_{n \in \mathbb{Z}} R_n$, where for all $m,n…
Rankeya
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Exercise 4.5.E a) in Ravi Vakil's Foundations of Algebraic Geometry.

Hi! I am following the hint given in Exercise 4.5.E in Vakil's Foundations of Algebraic Geometry, but I am stuck trying to prove that if $a_1,a_2 \in Q_i$, then $a_1^2 + 2a_1 a_2 + a_2^2 \in Q_{2i}$. What I have tried so far: We have $$a_1^2 + 2a_1…
Oliver E. Anderson
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