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 can give a proof, as follows. There is some filtration of $A$, as an […]

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: $\Sigma=\left\lbrace f\in \hom_{\text{gr}R\text{-mod}}\left(A,B\right) \ | \ \ker\left(f\right)_0\neq 0, \ \mathrm{coker}\left(f\right)_0\neq 0\right\rbrace$ (by quotient I mean simply that […]

Let $R$ be a non-negatively graded Noetherian ring such that $R_{0}$ is Artinian and $R_{+}$ is a nilpotent ideal. Prove that $R$ is Artinian. Give an example to show that this is false if the Noetherian property is removed. This is an exercise from a note that I saw on the Internet. I can not […]

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 suffices to show that if $a,b \in S$ are homogeneous and $ab […]

Let $S = \sum_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $> 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra […]

Let $G$ be a monoid ; $R$ be a unital commutative $G$-graded ring such that for every $g \in G$ , $x_gy\ne 0 , \forall x_g \in R_g \setminus \{0\} , \forall y \in R \setminus \{0\}$ . Then is it true that $R$ is an integral domain ? I am only able to show […]

Let $S$ be a $\mathbb{Z}$-graded ring and $F$ a $\mathbb{Z}$-graded module that is free of finite rank $n$. Then we can write $F = \oplus_{i=1}^n S(\nu_i)$, where $S(\nu_i)$ is a graded ring isomorphic to $S$ given by $S(\nu_i)_k = S_{k+\nu_i}$. Question 1: the ring isomorphism $S(\nu_i) \rightarrow S$ given by $x \mapsto x$ is not […]

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!

Here is one exercise from some notes on graded rings. I tried but I got no idea to solve it. Please help me. Thanks. Let $R$ be a graded ring. Prove that $R$ is Noetherian (Artinian) if and only if $R$ satisfies the ascending (descending) condition on homogeneous ideals.

Suppose you have a filtration $R=R^0\supset R^1\supset R^2\supset\cdots$ on a commutative ring $R$. This gives the associated graded ring $$ \text{gr}(R)=\bigoplus_{n=0}^\infty R^n/R^{n+1}. $$ From my reading, I know multiplication is defined on homogeneous elements in the following way. If $a\in R^m$ and $b\in R^n$, then $a+R^{m+1}\in R^m/R^{m+1}$ and $b+R^{n+1}\in R^n/R^{n+1}$, then $$ (a+R^{m+1})(b+R^{n+1})=(ab+R^{m+n+1}). $$ Something […]

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