Articles of graded rings

Basic question regarding a finitely generated graded $A$-algebra

Let $S = \oplus_{n \geq 0} S_n$ be a graded ring. Let $S$ be a finitely generated $A$-algebra, where $A = S_0$, a commutative ring with unity. Then there exists $t_1, .., t_M$ homogeneous elements of positive degree that generate $S$ over $A$. It then follows that $S$ is isomorphic to $k[x_1, …, x_M]/I$ as […]

Is the length of the composition series of a free module identical to the number of its bases?

Let $A_0$ be an Artinian ring, $M$ a free $A_0$-module. Then, is the length of the composition series of $M$ identical to the number of its bases? It seems to me that it is not. If $\mathfrak a$ is an ideal of $A_0$ and $e$ is a basis of $M$ then ${\mathfrak a}e$ is a […]

Hartshorne's proof of Proposition 2.5, Chapter II of his book Algebraic Geometry

This question already has an answer here: The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$ 2 answers

Relation between stalks of twisted sheaf and structure sheaf

Let $A$ be a ring, $B = A[T_0,\dots, T_d]$, and $X = \textrm{Proj } B$. Then at every point $x \in X$, $$\mathcal{O}_X (n)_x \cong \mathcal{O}_{X,x}$$ Let $x$ correspond to a homogeneous prime ideal $p$. So by definition, $\mathcal{O}_X(n) = B(n)^\tilde{}$ thus taking stalks, we get $(\tilde{B(n)})_p = B(n)_{(p)}$. The latter is represented by $a/s$ […]

Notation for free graded resolutions of graded modules?

I am now reading a paper about Castelnuovo-Mumford regularity and in this paper, there is a notation as following: Let $S=k[x_1,…,x_{n}]$, by Hilbert’s syzygy theorem, if $N$ is a graded module over $S$ then $N$ has a free graded resolution over $S$ of the form : $$0\longrightarrow F_{k}\longrightarrow …\longrightarrow F_{1}\longrightarrow F_0\longrightarrow N\longrightarrow 0$$ where $F_{i}=\bigoplus_{j=1}^{t_{i}}S(-a_{ij})$. […]

An ideal is homogenous iff it is invariant under a certain automorphism.

I’m working on the following. Let $R=R_0+R_1+ \cdots $ be a graded ring and $u$ a unit of $R_0$. Then the map $T_u$ defined by $T_u(x_0+x_1+ \cdots +x_n) = x_0+x_1u+ \cdots + x_n u^n$ is an automorphism of $R$ (this is clear). If $R_0$ contains an infinite field $k$, then an ideal $I$ of $R$ […]

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

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

Question on Noetherian/Artinian properties of a graded ring

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 […]

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