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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Weighted projective space and $\mathrm{Proj}$

I'm trying to solve a problem in Jenia Tevelev's notes on GIT. (It can be found as 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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132 views

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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38 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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36 views

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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27 views

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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34 views

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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134 views

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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32 views

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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41 views

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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42 views

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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67 views

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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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 ...
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If $I$ is a homogeneous ideal of $A$ contained in $A_+$, then $\sqrt{I} = \bigcap\limits_{I\subset P\in\text{Proj }A} P$?

EDIT: This is from an exercise of Vakil's Foundations of Algebraic Geometry. 4.5.H: Suppose $I$ is any homogeneous ideal of $S$ contained in $S_+$, and if $f$ is a homogeneous element of ...
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35 views

Factoring a homogeneous element in graded ring

Let $k$ be a field, and $A = k[w,x,y,x] / (wz-xy)$, which is an integral domain. I would like to show that if $h$ is a homogeneous element in $A$, not irreducible, then it factors into a product of ...
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Question on homogeneous ideal in a graded ring

In an $\Bbb{N}$-graded ring $R=\bigoplus_nR_n$, an element is called homogenous (of degree $n$) if it is contained in $R_n$. An ideal is called homogenous if it is generated by homogenous elements. ...
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42 views

Different definitions of graded rings

In Atiyah i recently read the definition of a graded ring as a ring that can be written as $R=\displaystyle \bigoplus_{i \geq 0}^{\infty}R_i$ where each $R_i$ is an abelian subgroup of $R$ (with the ...
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64 views

Non-isomorphic algebras with equal Hilbert-Poincaré series

Let $A,B$ be two finite-dimensional graded algebras and let $P_A(x),P_B(z)$ be theirs Poincaré series. Suppose now that $P_A(x)=P_B(z)$. Question. Is it implies that $A \cong B?$
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Without loss of generality $P_1, . . . , P_s$ contain all elements of $gr_I (R)$ of positive degree, and $P_{s+1}, . . . , P_r$ do not

In the $\underline {Proposition\ 8.5.7}$ of book: Integral Closure of Ideals, Rings, and Modules, (Irena Swanson and Craig Huneke), they say: "Without loss of generality $P_1, . . . , P_s$ contain all ...
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40 views

Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra?

(I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following ...
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Equivalent definitions of homogeneous ideal

I need to show that an ideal $I$ of a $\mathbb{Z}$-graded ring R is homogeneous iff for every element $f \in I$, all homogeneous components of $f$ are in $I$. $\Leftarrow$ implication is obviously. ...
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David Eisenbud, Hilbert theorem

I just started reading D. Eisenbud Commutative algebra with a view towards algebraic geometry and I wonder about a theorem on page 42: If $M$ is a finitely generated graded module over ...
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Is the notion of strongly graded algebra a Morita invariant?

Let $G$ be a group and $A$ be a ring. $A$ is a $G$-graded ring if $A=\oplus_{g\in G} A_g$ such that $A_gA_h \subset A_{gh}$ for all $g,h\in A$. Such a ring is said to be strongly graded if ...
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67 views

What is $S_d$ in algebraic geometry?

I'm trying to read algebraic geometry on my own by doing homeworks on course hold in 2003. One of the problem is the following: Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, ...
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Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
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Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...
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In a $\mathbb{Z}$-graded ring with unity, $1$ is homogeneous of degree zero

Suppose I have a $\mathbb{Z}$-graded ring (commutative) $A$ with unity. I am sure that the unity $1 \in A$ has degree $0$. I was wondering how could one show that? (I am guessing we don't have ...
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Castelnuovo-Mumford regularity and postulation numbers

I have a problem about Castelnuovo-Mumford regularity. This is a proposition from Castelnuovo-Mumford regularity, relation types and postulation numbers by M. Brodmann and C. H. Linh. My ...
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Graded ring localization. Why is this function bijective? [duplicate]

From Hartshorne, Chapter II.2, Proposition 2.5(b). If $R$ is a graded ring and $\mathfrak a$ is a homogenous ideal, then the function defined as $$\phi(\mathfrak a) = \mathfrak aR_f\cap R_{(f)}$$ ...
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Quotient ring of a graded algebra with respect to a graded ideal

An algebra $A$ over $F$ is said to be a graded algebra if as a vector space over $F$, $A$ can be written in the form $$A=\bigoplus_{i=0}^\infty A_i$$ for subspaces $A_i$ of $A$ along with other ...
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Is a graded module over a graded ring zero when all of it's graded localizations at graded primes not containing the irrelevant ideal are zero?

Let $M$ be a graded module over an $\mathbb{N}$-graded ring $S$ and $S_+$ be the ideal of positive degree elements. Is it true that $M=0$ iff the homogeneous localization $M_{(\mathfrak p)}=0$ for ...
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A question on graded rings

For a ring $A$ and an ideal $\mathfrak{a}$ of $A$, Atiyah-Macdonald define $$A^*=\bigoplus_{n=0}^\infty \mathfrak{a}^n$$ and claim that it is a graded ring on p. 107 of their commutative algebra book. ...
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stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
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Confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky

I am confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky (Chapter 10, page 441). Which is as follows: Let $A=\oplus_{g \in G} A_g$ be a $G$-graded ...
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Extension of graded algebra by a homogeneous ideal

If an algebra is graded by the group $G$: $A=\bigoplus\limits_{d \in G} A_d$ and contains a homogeneous ideal $I \subset A$, then we have the quotient $B:=A/I$ and canonical epimorphism $\nu:A ...
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question about gradation of a ring

I was reading Mumford's 'Red book on varieties and schemes', when I came across the following paragraph: I am confused about meaning of the phrase "We let $k(X)$ be the zeroth graded piece of the ...
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Computing generators of the positive component of a graded ring

Let $R$ be a sub-algebra of $\mathbb{Q}[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$ given by finitely many generators, and let $\lambda$ be a linear form $\lambda : \mathbb{Z}^n \to \mathbb{Z}$. This defines a ...
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Extending a lemma about Castelnouvo-Mumford regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. There is a lemma: It's from "Castelnuovo-Mumford regularity postulation number and relation types" by Markus Brodmann and ...
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Castelnouvo-Mumford Regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. As u see in 2 pics below, Lemma 3.3. $(A,\mathfrak{m})$ is a Noetherian ring local, $\dim(A)=1$; $\mathfrak{q}=(x)$ is a ...
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Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
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maximal chains of graded prime ideals

In Theorem 1.5.8 [Bruns,Herzog - Cohen-Macaulay-Rings] it is proved that for a noetherian graded ring $R$, a finitely generated graded $R$-module $M$ and any chain $\mathfrak{p}_0 \subsetneq ...
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Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in ...
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Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
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Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space. Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$. However, I wish to know ...
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Natural examples of multiplicatively-graded rings

A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as $$A=\bigoplus_{n\geq 1} A_n$$ and the product satisfies $A_n \times A_m \rightarrow A_{nm}$. A natural example of ...
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Graded tensor algebra

Given a finite dimensional $\mathbb R$-vectorspace $V$ we can make $$ T(V) := \bigoplus_{n=0}^\infty V^{\otimes n}. $$ Here $V^{\otimes n} = V \otimes \cdots \otimes V$. An element of $T(V)$ looks ...