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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### Hartshorne I.2.6 - questions about Boercherds' solution

Hartshorne Exercise I.2.6: $Y$ is a projective variety with homogeneous coordinate ring $S(Y)$, then $\dim (Y) = \dim S(Y) + 1$. I think I'm not understanding a simple thing. It relates to the part of ...
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### Filtration on field of fractions of graded ring

Let $R = \bigoplus_{i \in \mathbb{Z}} R_i$ be a commutative graded ring, that is, $R_i \cdot R_j \subseteq R_{i+j}$ for every $i,j \in \mathbb{Z}$. Assume that $R$ is a domain, so that it has a field ...
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### If a graded ring $S$ is a UFD, then $S_0$ is a UFD

Let $S$ be a commutative graded ring ($S = \oplus_{n \in Z} S_n$). If $S$ is a UFD, can we deduce $S_0$ is a UFD? I think $S_0$ is a UFD. Because we can gather all the finite product of the ...
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### Quotient of Graded Rings $S \to S/f$ inducing Homeomorphism on Proj

Let $A$ be a Noetherian ring, and let $X$ be a closed subscheme of of $\mathbb{P}^r_A$. We define the homogeneous coordinate ring $S:=S(X)$ of $X$ for the given embedding to be $A[x_0, ..., x_r]/I$ (...
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### Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one

Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
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### Graded rings and weighted projective spaces

I am genuinely struggling with this concept, partly because of the language of graded rings. So, I will ask a simple question. If polynomial rings $\mathbb{k}[x_0,x_1]$ and $\mathbb{k}[y_0,y_1]$ with ...
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### Homogeneous elements of graded algebras as polynomials of algebra generators

Let $A = \oplus_{i > 0} A_{i}$ be a positively graded $k-$algebra and let $\mathcal{B} = \left\{b_{1}, \ldots, b_{n}\right\}$ be a generating set of $A$. Let $f$ be a homogeneous element in $A_{i}$ ...
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### How is $R_iR_j$ defined in the definition of graded rings?

In the definition of graded rings there is the condition $R_iR_j\subset R_{i+j}$, it involves the product of two subgroups $R_iR_j$. How is the set $R_iR_j$ defined? Edit: Is anyone aware of a ...
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### Prime avoidance for graded Noetherian ring with infinite residue field.

This question is related to this question. Let $R$ be a Noetherian local ring with infinite residue field. Let $\text{gr}_IR$ denote the associated graded ring $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
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### Finiteness of the homology of the Koszul complex

Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of ...
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### Curious Difficulty with the Exterior Algebra (relationship between 2 and 3 forms)

Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Then we can form the exterior algebra $\wedge V$. Let $L\subset \wedge^3 V$ be a linear subspace. I am working on the following claim. ...
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