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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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 ...
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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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62 views

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 ...
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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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32 views

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 ...
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30 views

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 } ...
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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 ...
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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 ...
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Question about the classification theorem for finitely generated graded $F[t]$-modules

I am in the beginnings on learning persistence homology, and as a start I'm studying Gunnar Carlsson's survey "Topology and Data". Theorem 2.10 states the following: "Suppose $M_{\star}$ is a ...
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69 views

For graded algebras over a field, does finite Krull dimension imply finite generation?

On p. 37 of his book Algorithms in Invariant Theory, Bernd Sturmfels writes, Let $R$ ... be a graded $\mathbb{C}$-algebra of dimension $n$. This means... that $n$ is the maximal number of ...
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Why is a graded $\mathbb{C}$-algebra free as a module over the subring generated by a regular sequence?

I am reading Bernd Sturmfels' book Algorithms in Invariant Theory. On p. 38 he makes the following assertion: "If $\theta_1,\dots,\theta_n$ are algebraically independent over $\mathbb{C}$, then the ...
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39 views

Description of Blow up via Ress Algebra

Let $X=\mathrm{Spec}(R)$ be an affine scheme and $Z=\mathrm{Spec}(R/I)$ be a closed subscheme of $X$ defined by an ideal $I$ of $R$. The blow up of $X$ along $Z$ is ...
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Is there an ideal, which is homogenous over $\overline F$, generated by elements of $F[X]$, but not by homogenous elements of $F[X]$?

Suppose that $F$ is a field, and $\overline F$ is it's algebraic closure. Suppose further, that $I \subseteq \overline F[x_0, \ldots, x_d]$ is a homogenous ideal, that is, it is generated by ...
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44 views

Associated graded ring isomorphic to polynomial ring implies regularity

Let $(A, \mathfrak{m}, k)$ be a Noetherian local ring of dimension $d$. I would like to prove, or rather understand, why the following holds: If the associated graded ring ...
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Coalgebra differential on reduced symmetric algebra

Definitions/setup (I am using "Formality and star products" by Cattaneo) Let $g=\bigoplus_{i\in \mathbb{Z}} g^i$ be a graded vector space over some field. We can then look at the reduced symmetric ...
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Cohomology groups of the bar construction of free graded algebras

Let $A= \bigwedge \langle x_n \rangle $ be the free graded commutative algebra with differential zero ($x_n$'s are elements with positive grading )over a field of characteristic zero. I want a ...
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Iff criterion for augmentation ideal to be finitely generated.

Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace$$A = \bigoplus_{i \ge 0} A_i.$$One can write$$A = A_0 \oplus A_{> 0},$$where we have$$A_0 = k^0[x_1, \dots, x_n] ...
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Isomorphism (?) of polynomial rings with different gradings and their Proj

It's a (relatively) well-known fact that, if $a_0,\ldots,a_n\in\mathbb{N}$ share a common factor $d\in\mathbb{N}$ then $$\operatorname{Proj}k_a[x_0,\ldots,x_n] \cong ...
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276 views

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 ...
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dual algebra of a coalgebra

Given a coalgebra $A$ over a field $F$ (for example, $H_*(X:F)$, i.e. the homology of a space $X$ equipped with $\Delta_*$, where $\Delta: X\to X\times X$, $x\to (x,x)$) how to obtain its dual $A^*$ ...
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Homogeneus primes in a graded ring

Let $B=\oplus_{n\in\mathbb Z} B_n$ be a graded ring (commutative with 1). We know that $B_0$ is a subring of $B$, so we have the inclusion $B_0\hookrightarrow B$. My question is: Is every prime ...
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44 views

Decomposition of a homogeneous polynomial

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!
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Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
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A regular sequence in a determinantal ring

Let $S=K[X_{ij}\colon 1\le i\le m, 1\le j \le n, m\le n]$ be a ring of polynomials with coefficients in a field, $X=(X_{ij})$ a matrix of indeterminates, $I$ the ideal of maximal minors and $R=S/I$. ...
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Hartshorne Exercise 2.6: what gradation does $S(Y)_{x_i}$ inherit from $S$?

Let $S=k[x_0,\dots,x_n]$ be the "homogeneous polynomial ring" of $\mathbb P^n$ and let $S(Y)_{x_i}$ denote the localization at the image of $x_i\in S(Y)$ of $S(Y)=S/I(Y)$. In Hartshorne, he asks us to ...
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What is the meaning of 'homogeneous' here? And what does it mean by 'degree'?

This is a part extracted from a textbook that has many definitions that I was confused and failed to find. Let $\displaystyle A=\oplus_{n=0}^\infty A_n$ be a Noetherian graded ring. Then $A_0$ is ...
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Tensor algebra becomes a graded $R$-algebra short proof verification

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
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Quotient of graded rings by graded ideals are again graded rings

I want to prove the following statement. Let $A = \bigoplus_{n=0}^{\infty}A_n$ to be a graded $R$-algebra and $I$ a graded ideal of $A$. Let $(A/I)_n = (A_n + I)/I$ be the image of $A_n$ in $A/I$. ...
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Why in a graded ring $A$ finitely generated that's an algebra over a field $K$ every maximal ideal is a $K$-subspace?

Probably this question has already been asked, but I'm very bad in find old question and I searched for half an hour, so I'm asking it again. I suppose that's true beacuse my professor used this ...
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1answer
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Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...
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About the meaning of associated graded ideal

Let $G$ be any multiplicative group (abelian or not). Suppose that $R$ is a $G$-graded ring, i.e., there exists a family of additive subgroup $\{R_g\}_{g\in G}$ such that $R=\bigoplus_{g\in G}R_g$ ...
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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 ...
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What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...
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Do non-graded rings exist?

I've been reading about graded rings recently, and I was wondering if there exist commutative or non-commutative rings for which no non-trivial gradings exist? That is, a ring for which if ...
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First order differential calculus over algebra $A$

Let $A$ be a unital algebra and $d_u$ is universal differential given by $d_u(x)=1 \otimes x-x \otimes 1$. It is viewed as the map $A \to \Omega_u^1(A)$ where $\Omega_u^1(A)=\ker m$ is the kernel of ...
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1answer
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When is a D-module holonomic?

Suppose that I have a small family of partial differential equations such that $y(x)$ must be a solution to $$ P_i(x,D) y(x)=0,\ \ \ \ \ \ i=1,...,m $$ for all $i=1,...,m$, where I am using ...