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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A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element.

I expect that the following result is true, but i can't prove it. A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. I need some help to prove this.
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Structure constants in a finitely generated $\mathbb{k}$-algebra

Let $\mathbb{k}$ be a field of characteristic $0$. Suppose we have a finitely generated graded $\mathbb{k}$-algebra $A= \bigoplus_{i=0}^{\infty}A_i$ which is free of finite rank as a module over a ...
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On Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
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the standard even grading on $M_2(A)$ and $A\otimes \mathbb{K}$

I have a question about a passage in Blackadar's book about K-Theory. Let $A$ be a (ungraded) $C^*$-algebra. There is a grading on $M_2(A)$ with $M_2(A)^{(0)}$ the diagonal matrices and $M_2(A)^{(1)}$...
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Infinite direct sum of abelian groups

While reading about graded rings, I read that a graded ring $R$ is an infinite direct sum of abelian groups $\displaystyle R=\bigoplus_{i \in \mathbb Z} A_i$ together with a bilinear map $A_i\oplus ...
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Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
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Characterization of prime homogeneous ideals

Let $R$ be a graded ring and $I$ ideal in $R$ and homogeneous. $I$ is prime if and only if for all $a, b\in R$ homogeneous such that $ab\in I$ then $a\in I$ or $b\in I$. Let $ab\in I$ and $a = ...
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Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
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Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$ [closed]

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.
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Sum of Hilbert functions of a finite exact sequence of finitely generated graded modules

Let $A = \bigoplus_{n\geq 0} A_n$ be a graded ring that is generated as an $A_0$-algebra by a finite collection of elements of $A_1$, where $A_0$ is artinian. I wish to show that if $$ 0 \to M(1) \...
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$G_{\mathfrak a}(A)$ integral domain and $\bigcap \mathfrak a^n = 0$ implies $A$ is integral domain

This is Lemma 11.23 in Atiyah: For an ideal $\mathfrak a \subseteq A$, define $G_{\mathfrak a} (A) = \bigoplus _{n=0} ^\infty \mathfrak a^n / \mathfrak a^{n+1}$. The statement of the Lemma: ...
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Reflexive Graded Module

Let $R=k[x_1,\dots,x_d]$ be a polynomial ring and $M=M_0\oplus M_1\oplus M_2\oplus\cdots$ be a graded $R$-module. Is it true that $M$ is reflexive as an $R$-module if and only if $M_i$ is reflexive as ...
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is there a ring homomorphism $f:H^*(T^n;R)\to H^*(T^n;R)$ of graded rings which is not induced by a continuous map $T^n\to T^n$?

Let $R$ be a commutative ring with unit $1_R$ and let $\xi\in H^1(S^1;R)$ be a generator ($H^1(S^1;R)$ is the first singular cohomology group of $S^1$). Let $p_i:(S^1)^n\to S^1$ be the projection onto ...
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Algorithmic computing kernel of a graded homomorpism

For computing kernel of a module homomorphism we can use module-Grobner basis such as described in notes talking about computing SyZyGies. How can we compute kernel of a homomorphism between a graded ...
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Noetherian assumptions in basic properties of coherent sheaves of modules

Using Hartshorne's definition of 'coherent sheaf': Proposition 5.11c Let $S$ be a graded ring, $M$ a graded $S$-module, $X=\operatorname{Proj} S$. Then $\tilde M$ is a quasi-coherent $\mathscr O_X$...
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Cones of max-Spec

Let $k$ be an algebraically closed field and $R=k\oplus R_1\oplus R_2\oplus \ldots$ be a graded commutative ring that's finitely generated by elements of positive degree. If $M$ is a finitely ...
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A proposed criterion for finding when an homogenous ideal is radical

Let $X$ be a projective variety over an algebraically closed field, and $I$ be the homogenous ideal of $X$ and $J$ be an ideal with the same zero set. Suppose that I know $I=\langle f_1,...f_n \rangle$...
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Example of a graded module that is not a ring.

I'm looking for an example of a graded module, that is not a ring. All the examples of graded modules that I have come across, like $k[x_1,x_2,\dots,x_n]$ are all graded rings. Thanks in advance
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When a graded ring is Cohen-Macaulay?

I am trying to solve exercise 19.10 from Eisenbud's Commutative Algebra. I want to show that if $R=k[x_0,...,x_n]/I$ is a graded ring, then $R$ is Cohen-Macaulay iff $R_{\mathfrak p}$ is Cohen-...
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Support of a tail of a graded module.

Suppose that $R$ is a non-negatively, graded commutative ring. I have been trying to decide if the following is true for a graded $R$-module $M$ (not necessarily finite over $R$): $$\text{Supp}_R M=\...
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Zero divisors in a finite dimensional Poincaré duality algebra

Let $A= \bigoplus_{i=0}^{n}A_i$ be a finite dimensional algebra over a field $\mathbb{k}$ such that $A_0 \cong \mathbb{k} \cong A_n$. Consider the bilinear form $$\varphi: A_i \times A_{n-i} \to A_n$$...
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Finitely graded ring and zero divisors

Let $R= \bigoplus_{i=0}^{d} R_i$ be a finitely graded ring such that $R_0$ is a field and $R_d \cong R_0$. I'm trying to understand how zero divisors work in such a ring; when is it true that for $r \...
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Finite generation and associated graded modules

My question is as follows: Let $R$ be a ring and let $M$ a right $R$-module. Suppose that $(F_i)_{i \geqslant 0}$ is a filtration of $R$ and $(M_i)_{i \geqslant 0}$ is a compatible filtration of $...
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The intersection and sum of irrelevant ideals are also irrelevant

Definition: A homogenous ideal $I \subset K[x_0,\dots,x_n]$ is irrelevant if $\left <x_0^r,\dots,x_n^r \right> \subset I$ for some $r > 0$. For $I \cap J$, this is probably circular logic, ...
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Is $\mathbb{Z}[x,y,z]$ an $\mathbb{N}$-graded ring?

I know that my question seems to be obvious, but I really need the answer Let $\mathbb{Z}[x, y, z]$ be a polynomial ring. I know that it is a $\mathbb{Z}$-graded ring, but is it not in fact $\mathbb{...
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Homogeneous localization and regularity

Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I ...
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Algorithm for computing the inverse limit of a finite inverse system

Let $k$ be a field (finite if you'd like), let $(I,\le)$ be a finite directed poset with $|I|=n$, and let $(A_i,f_{ij})_{i\le j\in I}$ be an inverse system of finitely generated, graded, commutative $...
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Algorithm for computing an inverse image

Let $k$ be a field (finite if you'd like), and let $f:A\to B$ be a map of graded, commutative $k$-algebras. Suppose further that $A$ is finitely generated and choose a presentation $$A:=k[x_1,\ldots,...
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Poincaré series and Hilbert polynomial of some graded modules [closed]

Let $k$ be a field, and let $k[X, Y ]$ be the polynomial ring in two variables equipped with the usual grading such that $\deg(X) = \deg(Y ) = 1$. Consider the ideals $I = (X, Y^2)$ and $J = (X^2, Y^2)...
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problems to understand a special definition of “free graded commutative algebra” from lecture

I have problems to understand a definition from lecture: Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., ...
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where do elements go under multiplication in a graded module?

Assume that $M = \bigoplus_{n = 0}^\infty M_n$ is a graded $A$-module, where $A = \bigoplus_{n = 0}^\infty A_n$ is a graded ring. We have by definition $A_m M_n \subset M_{m + n}$. Does this mean that ...
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Explanation of a proof about graded module structure

Let $\Bbb F$ be a field and $M$ a finitely generated $\Bbb F[x]$-module. The structure theorem for modules over a PID says that $$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb F[x]/(f_j(x))\...
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Adjoint to $\mathsf{Proj}$? - A quest to understand categories of graded objects.

I've been having a hard time with graded objects in algebraic geometry for some time. Lately I realized a lot of my difficulties come from not having any idea at all of where graded objects live. ...
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How to find a counter-example that the centralizer of differential graded algebras does not preserve quasi-isomorphism?

Let $A^{\bullet}$, $B^{\bullet}$ be two differential graded algebras (dga) and $f: A^{\bullet}\to B^{\bullet}$ be a differential graded homomorphism between them. Now let $R$ be another algebra ($R$ ...
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If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$?

If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$? This gives a graded ring, but it is not quite the tensor product since we ...
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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_*} ...
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$0 \in S_k$ for which k?

If $S$ is a graded ring, for which $k \in \mathbb{Z}$ do we have $0 \in S_k$? I think there shouldn't exist such a k. So as 0 is the empty sum we don't need this?
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Is it true that taking injective hull commutes with the tensor product?

Let $M$ and $N$ be two modules (can assume them to be finitely generated if need be) over the ring $A=k[x_0,...,x_n]$. Denote by $E(M)$ the injective hull of $M$. We work in the category of positively ...
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Radical of an ideal in $R [x]$

Let $\frak {I}$ be an ideal of $R[x]$, the polynomial ring over a commutative ring with identity $R$. Is it true that the radical of $\frak{I}$, the intersection of all prime ideals containing $\frak{...
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Shift of a simple graded module

I am trying to understand the simple graded modules over a graded ring $R$ (all the gradings over $\mathbb{Z}$). I know that there exists a bijection between the simple graded $R$-modules and simple ...
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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 ...
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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_{(...
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Graded modules/algebras on commutative monoids?

In the book Algebra of the Bourbaki group they deal with graded modules/algebras which are graded on a commutative monoid. What is the need to require the commutativity condition? Thanks.
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Constructing a ring consisting of formal infinite series from a given ring

Let $A$ be an $\mathbb{N}$-graded $\Bbbk$-algebra, where $\Bbbk$ is a field, and where $\dim_\Bbbk A_n < \infty$ for all $n \in \mathbb{N}$. I can't see anything preventing me from constructing a ...
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Bijective correspondence of rational points in projective space

Let $k$ be a field and consider an arbitrary point $\alpha = (\alpha_0, \dots, \alpha_n) \in \mathbf{P}(k^{n+1})$. Then there is a bijection $\rho: \mathbf{P}(k^{n+1}) \to \mathbf{P}_k^n (k)$, the ...
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Principal open sets in graded rings

I am interested in Prop II.2.5b in Hartshorne stating that if $D_+ (f) = \{p \in \textrm{Proj } B \mid f \notin p \}$ then there is a canonical homeomorphism $D_+ (f) \cong \textrm{Spec } B_{(f)}$ the ...
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$\textrm{Dim } S(Y) = 1 + \textrm{Dim } Y$

Let $k$ be algebraically closed, and $S = k[Y_0, ... , Y_n]$, and let $\mathscr Y$ be a variety in projective $n$ space, corresponding to the homogeneous prime ideal $\mathscr P$ of $S$. Let $U_0 = \{...
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Can a graded $k$-algebra have torsion over $k[\theta]$ for $\theta$ a non-zerodivisor?

Let $k$ be a field and let $R$ be an $\mathbb{N}$-graded $k$-algebra such that $R_0=k$. Let $\theta$ be a not necessarily homogeneous element of $R$ that is transcendental over $k$ and is a non-...
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$A$-module is free if and only if equation involving Hilbert-Poincaré series holds.

Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector spaces $A_i$ and $M_j$ are ...