Questions tagged [graded-rings]

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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Property of annihilator in associated graded algebra of Noetherian rings.

Let $R$ be a Noetherian ring, $I\subset R$ is an ideal, and $S$ is a multiplicatively closed subset of $R$. Let $\mathcal{R}$ be the associated graded ring $\operatorname{gr}_I(R):=R/I\oplus I/I^2\...
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Isomorphism in Exercise 2.17(d) of Eisenbud's Commutative Algebra

Let $R$ be a $\mathbb{Z}$-graded ring, $P$ a homogeneous prime ideal of $R$, and $U$ the multiplicative subset $R \setminus P$. Let $R_{(P)}$ be the degree $0$ component of $R_P$, and let $f$ be an ...
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Dimension of filtered algebras and their associated graded algebras

Let $F$ be a finite dimensional filtered algebra and let $G$ be the associated graded algebra. Will the dimension of $G$ and $F$ coincide or differ in general? If they differ in general then what is ...
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Bounding minimal numbers of generators of a product of homogeneous ideals

Let $R$ be a standard graded ring over an infinite field $k$. Suppose $I,J\subset R$ are two homogeneous ideals with $I$ equigenerated in degree $d$ (that is, $I$ is generated by homogeneous elements ...
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Equivalent definition of the Hilbert function

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$. Let $I$ be a homogeneous ideal of $R$ with decomposition $I=\bigoplus_{n\geq0}I_n$ where each $I_n$ is a $k$-vector subspace of $I$. ...
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Equivalent condition for $k[x_1,...,x_n]/I$ to admit an $m$-linear resolution

This is part (c) of exercise 4.1.17 of Burns&Herzog's book Cohen-Macaulay Rings. Let $k$ be a field and let $R=k[x_1,...,x_n]/I$ be a homogeneous Cohen-Macaulay ring. The ring $R$ has an $m$-...
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Is module isomorphism problem easier if we have presentations?

Let $R = k[x_1,\dotsc,x_n]$ (feel free to restrict $n$ and $k$ if necessary). Assume we work in the graded setting, where everything is graded by $\mathbf{Z}^n$. Let $M, N$ be $R$-modules. In general, ...
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Quotient of graded ring is graded - confusion about the formalisms

A ring $R$ is graded if it has a direct sum decomposition $R=\bigoplus_{i\in\mathbb{Z}}R_i$ where the $R_i$ are abelian groups and $R_iR_j\subseteq R_{i+j}$. An ideal $I\subseteq R$ is graded if $I=\...
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Graded field over integers

In a $\mathbb Z$-graded field $R$, prove that $R=R_0$. Use the fact that all units in a graded domain are homogenous or otherwise. Try: Let $x_n$ be any nonzero element. Then it is homogeneous and ...
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How to regard $ \Bbb{C}[x,y,z]/(xy-z^2)$ as graded ring? [duplicate]

I heard $ \Bbb{C}[x,y,z]/((xy-z^2)$ can be regarded as grade ring, but how ? $ \Bbb{C}[x,y,z]$ can be regarded as graded ring by decomposition$ \Bbb{C}[x,y,z]= \bigoplus_{d\geq 0} A_d$, where $A_d=${...
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Necessary and Sufficient condition for Flatness of Graded Modules

I am currently self studying the textbook Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud and came across the following exercise: Exercise 6.10 (Flatness of graded modules):...
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Cohomology Ring of Formal Infinite Series (Milnor)

In chapter 4 of Milnor/Stasheff's Characteristic Classes, the authors define the ring $H^\prod(B; \mathbb{Z}/2)$ to be the ring of formal infinite series: $a = a_0 + a_1 + a_2 + \cdots$ , where $a_i \...
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Corollary 1.5 in Eisenbud's Commutative Algebra

I have two questions on Corollary 1.5 from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry. I included a screenshot of the corollary in question below the questions themselves. In ...
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Direct sum of graded modules is *isomorphic* to a graded module?

Let $R$ be a graded ring and let $M_1,\ldots,M_m$ be graded $R$-modules. We can form the direct sum $N=\bigoplus_{i=1}^m M_i$, which according to my textbook is also a graded $R$-module using the ...
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Minimal homogeneous generating sets in graded rings

Let $R$ be a Noetherian $\mathbb{N}$-graded ring where $R_0=K$ is an infinite field. I explicitly do not want to assume that $R$ is standard graded (i.e. $R\neq K[R_1]$). If $I$ is a homogeneous ideal ...
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$d: L \to L$ with graded Leibniz rule

For a graded module $L$ can one determine if there are any functions $d: L \to L$ such that they have degree 1, are linear and respect graded Leibniz rule? What does it depend on?
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Commutativity of Tor and restriction

Let $R=S[\mathbf{x},\mathbf{y}]$ be a standard bigraded polynomial ring in the sets of variables $\mathbf{x}=\{x_1,...,x_m\}$ and $\mathbf{y}=\{y_1,...,y_m\}$ (i.e., $\deg(x_i)=(1,0)$ and $\deg(y_j)=(...
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Why is $S_{0}$ a subring of the graded ring $S=\bigoplus_{d\geq 0}S_{d}$. [duplicate]

An (essentially) the same question has been asked and answered here Definition of graded rings, but the user who posted the answer has not been here for a while, and I am confused by the answer, so I ...
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Confusion on the direct sum of abelian groups and the definition of graded ring.

A graded ring is a ring $S$ together with a family $(S_{d})_{d\geq 0}$ of subgroups of the additive group of $S$, such that $S=\bigoplus_{d\geq 0}S_{d}$ and $S_{e}S_{d}\subseteq S_{e+d}$ for all $e,d\...
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Homogeneous non zero divisor in a graded module.

Let $(R_0,m_0)$ be a local ring and $R=\oplus_{n\geq 0}R_n$ a positively standard graded ring. Let $M$ be a finitely generated graded module over $R$. Let $R_+$ be the ideal $\oplus_{n>0}R_n$ and $\...
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Question about lengths in graded rings

Let $A$ be a graded, noetherian ring and $\mathfrak p$ a minimal (minimal in the set of all prime ideals) homogeneous ideal. Is it true that the rings $A_{\mathfrak p}$ and $A_{(\mathfrak p)}$ have ...
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Global dimension of a polynomial ring, knowing its graded global dimension

Let $R$ be an $\mathbb{N}$-graded noetherian ring. In what follows, "global dimension" means the supremum of the projective dimensions of $R$-modules (left or right; it doesn't matter), ...
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Let $\mathfrak p, \mathfrak p' ⊂ S$ be relevant prime ideals. If $\mathfrak p \cap S_+ = \mathfrak p' \cap S_+$, then $\mathfrak p = \mathfrak p'$.

Let $S$ be a graded ring. Let $\mathfrak p, \mathfrak p' ⊂ S$ be relevant prime ideals. If $\mathfrak p \cap S_+ = \mathfrak p' \cap S_+$, then $\mathfrak p = \mathfrak p'$. I don't see how this is ...
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Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. If $S_+$ is finitely generated, then $S$ is a finitely generated graded $A$-algebra?

Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. Suppose $S_+= (a_1, \dots, a_n)$ where $a_i$ are homogeneous of positive degree. I want to show that $S$ is a finitely generated graded $A$-...
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$\text{Homgr}_A(M,N)=\text{Hom}_A(M,N)$ if $M$ is finitely generated

Let $A$ be a ring and $M,N$ be two graded $A$-modules of type $\Delta$. For $\lambda\in\Delta$, let $X_\lambda:=\{f\in\text{Hom}_A(M,N)\ |\ \deg(f)=\lambda\}$. Write $\text{Homgr}_A(M,N):=\sum_{\...
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Defining a grading on the generalized tensor product

Let $C$ be a commutative ring, $(E_i)_{1\leq i\leq n}$ a family of graded $C$-modules such that $E_i\cong\bigoplus_{\lambda\in\Delta}M_{i\lambda}$, for each $1\leq i\leq n$. For $\gamma\in\Delta$, ...
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Image of a graded linear map is graded

Let $A$ be a graded ring, $M,N$ graded $A$-modules and $u:M\rightarrow N$ a graded linear map of degree $\delta$. I want to show that $\text{Im}(u)$ is a graded submodule of $N$. It is sufficient to ...
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How to construct a grading on a quotient module

Let $A$ be a ring, $M$ a graded $A$-module, $N$ a graded submodule of $M$ and $(M_\lambda)_{\lambda\in\Delta}$ the grading on $M$. Then $((M_\lambda+N)/N)_{\lambda\in\Delta}$ is a grading of $M/N$. ...
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Elements of degree $0$ form a subring of a graded ring

Let $A$ be a ring, $\Delta$ a commutative monoid whose elements are all cancellable and $(A_\lambda)_{\lambda\in\Delta}$ a graduation of $A$ compatible with the ring structure of $A$. I want to show ...
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Which graded commutative $R$-algebras occur as cup product algebras $H^\ast(X;R)$ of spaces $X$?

Which graded commutative $R$-algebras occur as cup product algebras $H^\ast(X;R)$ of spaces $X$? This is a question in Hatcher's Algebraic Topology (page 214 Chapter 3). Hatcher says that the problem ...
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How is the quotient of a graded ring itself graded?

I know little ring theory, but have the following general question: If $R$ is a $G$-graded ring ($R = \bigoplus_{k∈G}R_k$ where $R_iR_j ⊆ R_{i+j}$) and $I ⊆ R$ is an ideal ($I$ is closed under ...
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Do we have $A_{(\mathfrak{p})} \simeq (A_{(f)})_{\mathfrak{q}}$?

Let $A = \oplus_{d \ge 0} A_d$ be a graded ring. Let $\mathfrak{p}$ a homogeneous prime of $A$. Let $f$ be a homogeneous of $A$ of degree $\require{cancel}\cancel{d}$ $1$ such that $f \not\in \...
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Ideals in localisation of graded ring

Let $k$ be a field and $I$ a homogeneous ideal of $k[x_1,...,x_n]$. Consider the graded ring $R=k[x_1,...,x_n]/I$ with unique homogeneous maximal ideal $\mathfrak m=(x_1,...,x_n)R.$ Let $J$ be a ...
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Koszul algebra and Hilbert series

$A$ is a Koszul algebra. Its Hilbert series $h(z)=1+az+bz^2$. Prove that $h(z)$ has real roots. I know that $A=A_0\oplus A_1\oplus A_2$ and $\dim A_1=a$, $\dim A_2=b$. And it's needed to prove that $a^...
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Are there any useful generalizations of homomorphisms that can apply to a graded valuation ring?

I have two graded valuation rings, $R$ and $S$, that I want to relate by a function $f:R\rightarrow S$. The problem is that $f$ is only a group homomorphism under addition for two elements of $R$ ...
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Prove that if $\mathrm{gr}(A)$ is Noetherian without zero-divisors, then so is $A$.

Let $A$ be a filtered commutative algebra and $\mathrm{gr}(A)$ the associated graded algebra. Prove that if $\mathrm{gr}(A)$ is Noetherian without zero-divisors, then so is $A$. Associated graded ...
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Ideal generated by initial forms in the associated graded ring.

This is Exercise 14.3 of Matsumura's Commutative Ring Theory. Let $(A,m)$ be a Noetherian local ring and $G=\operatorname{gr}_{m}(A)$ be the associated graded ring, generated by direct sum of $R_0=A/m$...
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Koszul complex as a graded free resolution

I'm trying to compute the Hilbert function of a complete intersection by using the Koszul complex, but I think I'm approaching it incorrectly. If we let $R=k[x,y,z]$ and $$A= R/(f_1,f_2)$$ the Koszul ...
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Is the Weyl algebra $A_1(\Bbb{C})$ graded?

For this question, the Weyl algebra is the algebra of differential operators on $\Bbb C$: $$A_1(\Bbb{C})=\Bbb{C}\langle x,\partial\rangle/(\partial x- x\partial -1)$$ Although for a lot of purposes it ...
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An example of graded rings

Let $R$ be a commutative ring, and $I$ an ideal in $R$. Then, we have a graded ring $R_{\ast}:= \bigoplus_{n\geq 0} I^n$, where $I^0=R$. So, let $A$ be a polynomial ring with a valuable $x$ over $\...
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$\Omega^*(M)$ as a graded algebra over the ring of smooth functions. Is there a notion of basis?

In Introduction to Smooth Manifolds by John M. Lee p.360, he defines $\Omega^*(M) =\displaystyle\bigoplus_{k=0}^n\Omega^k(M)$, the differential forms on a manifold $M$, as being an associative, ...
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If $x^n \in R$ for some $n \in \mathbb{N}$, do all of the homogeneous components $x_i$ of $x$ also satisfy $x_i^n \in R$?

Let $R \subseteq S$ be an extension of $\mathbb{N}$-graded rings. Let $x = x_0 + x_1 + \dots + x_m$ ($x_i \in S_i$) satisfy $x^n \in R$ for some $n \in \mathbb{N}$. Does it follow that $x_i^n \in R$ ...
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Let $M$ be a finitely generated graded $k[x_1, \dots, x_n]$-module. Every component $M_s$ is a finite-dimensional $k$-vector space?

Let $M=\cdots \oplus M_{-1} \oplus M_0 \oplus M_1 \oplus \cdots$ be a finitely generated graded $k[x_1, \dots, x_n]$-module where $\deg x_i = 1$ for all $i$. I would like to show that each component $...
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Dimension of the associated graded module at an ideal

Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
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Is there an equivalence between a graded vector space and its degree-shifted counterpart?

I am studying graded vector spaces and I have a simple question. Let me denote by $\mathbb Z_2=\mathbb Z\text{mod}2=\{-1,0,+1\}$. Now let's perform a shift by an integer $k$ and get the set $\mathbb ...
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Prime ideal after modding out by homogenizing variable implies original ideal prime?

Let $R:=\mathbb{C}[x_1,\ldots,x_d]$ be a polynomial ring. Let $I=(f_1,\ldots,f_n)$ be an $R$-ideal. Let $f^*_i$ be the homogenization of $f_i$ in $R[t]$. Question: If $(\overline{f_1^*},\ldots,\...
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Example of a graded ring with respect to any completely additive arithmetic function - is it isomorphic to anything?

Let $\varphi : \Bbb{N} \to \Bbb{N}$ be a completely additive arithmetic function, i.e. $\varphi(nm) = \varphi(n) + \varphi(m)$ for all $n, m \in \Bbb{N}$. For example, $\Omega(n) = $ the total number ...
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Algebraically closed proof.

Let $B = \bigoplus_{i \in \mathbb Z}B_i$ be a $\mathbb Z$-graded integral domain. Given $f \in B$ define $\deg: B \rightarrow \mathbb Z \cup \{-\infty\}$ by $$\deg(f) = \mathrm{maxSupp}{(f)} \text{ if ...
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proving $\deg(fg) = \deg(f) + \deg(g)$ in a $\mathbb Z$-graded integral domain.

In a $\mathbb Z$-graded integral domain $B = \bigoplus_{i \in \mathbb Z} B_i$ my definition for the degree function is as follows: Given $f \in B,$ define $\deg : B \rightarrow \mathbb Z \cup \{-\...
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Proving that $\deg(f) = - \infty$ iff $f=0.$

In a $\mathbb Z$-graded integral domain $B = \bigoplus_{i \in \mathbb Z} B_i$ my definition for the degree function is as follows: Given $f \in B,$ define $\deg : B \rightarrow \mathbb Z \cup \{-\...
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