# Questions tagged [graded-rings-modules]

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56
questions

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### $\mathsf{Nil}_0(R)$ is a $p$-group when $R$ is a $\mathbb{Z}_p$-algebra

$\DeclareMathOperator\Nil{\mathsf Nil}\DeclareMathOperator\ker{ker}$I was reading through The $K$- book by Charles A. Weibel, there the category $\Nil(R)$ is defined to have objects as pairs like $(P ...

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92 views

### 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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107 views

### On the degree of the Hilbert polynomial of a graded module over the Rees algebra

If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...

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### Making a generating set of a section of a graded polynomial $R$-module coming from a quotient into a basis of a quotient by higher degree polynomial

Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $...

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84 views

### Surjection from finite rank free $R$-module to finitely generated $R$-module and basis associated to generator set

Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ ...

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### Strongly graded rings

In Theorem 3.1 of Graded rings over arithmetical orders, the authors prove that for a strongly $\mathbb{Z}$-graded ring $R$, if $R_0$ is left and right Goldie and a maximal order in its (classical) ...

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### Bigraded operadic suspension

I know from this paper by Ward that one can obtain the (signs of) the Gerstenhaber bracket using operadic suspension on any operad $\mathcal{O}$. More precisely, the insertion $\tilde{\circ}$ of the ...

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79 views

### Quotient of $\text{Proj}(A)$ by the action of a finite group

Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$.
I know I can write $X = \bigcup_{...

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311 views

### $\mathbb{Z}$-graded algebras and tensor products

Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded:
$$
A_kA_l = A_{k+l}.
$$
...

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42 views

### Homomorphisms and indecomposable decompositions of finite modules over polynomial rings [closed]

I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\...

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125 views

### Weighted blowup of a Cohen-Macaulay ring along a regular sequence is Cohen-Macaulay?

Let $(R, \mathfrak{m})$ be a Cohen-Macaulay ring, let $f_1, \dotsc, f_d \in \mathfrak{m}$ be a regular sequence, and let $n_1, \dotsc, n_d > 0$ be weights (feel free to assume that $n_1 = 1$ if it ...

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### Ring with different graded and ungraded global dimensions

Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...

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### Conceptual explanation for the sign in front of some binary operations

In several situations, I've seen that given a binary operation on a graded module $m:A\otimes A\to A$, a new operation $M(x,y)=(-1)^{|x|}m(x,y)$ is defined so that it satisfies some properties.
One ...

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417 views

### Representation theory in braided monoidal categories

The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $\text{Vect}_\mathbb{k}$, follow in more general braided monoidal categories? I ...

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141 views

### Detailed proof of $\mathfrak{s}^{-1}\mathrm{End}_V\cong \mathrm{End}_{\Sigma V}$

I asked this question on MSE but I want to ask it again here with some more context sine it received no answers. In Chapter 3 (Algebra) of the book Operads in Algebra, Topology and Physics by Markl, ...

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92 views

### Is the free algebra over an operad an algebra over that operad?

I'm asking here this question I asked on MSE that got no answers.
Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...

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168 views

### Is operadic desuspension inverse to operadic suspension?

Given a graded vector space $V$ over a field $k$, consider it's suspension $\Sigma V$ such that $(\Sigma V)^i=V^{i-1}$. For an operad of graded vector spaces over a field $\mathcal{O}$, the operadic ...

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157 views

### Polynomial identities of supercommutative-gradable algebras

All algebras below are associative, and not assumed unital, and, to fix ideas, over the complex numbers.
An algebra $A$ is supercommutative-gradable if it admits a grading $A=A_0\oplus A_1$ in $\...

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80 views

### Certain morphism between graded modules

Say we have a morphism $f=(f_{i,j}) : \oplus_{i}M_{i} \rightarrow \oplus_{j}N_{j}$ (The direct sum of graded modules is finite) and $f_{i,j}$ are morphisms of graded modules but not necessary with the ...

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74 views

### non-archimedean valuations on graded rings

Let $R$ be a commutative (non-trivially) graded ring. By a non-archimedean valuation I mean a map $v: R \to \Gamma \cup {0}$ such that for all $x,y \in R$, we have $v(x+y) \leq \max\{v(x),v(y)\}$, $v(...

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633 views

### Reason to apply the Koszul sign rule everywhere in graded contexts

The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded-commutative algebra generated by $n$ elements of ...

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326 views

### Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

This question is certainly not research level and in fact quite elementary which is why I asked it on math.stackexchange before: math.stackexchange. However it doesn't seem to get much attention there ...

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162 views

### Strongly graded algebras with no zero divisors

Let $A = \bigoplus_{i \in \mathbb{Z}} A_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that
$$
m: A_i \otimes_{A_0} A_j \to A_{i+j}
$$
is an ...

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### Asymptotic stability of associated primes of graded local cohomology modules

This question concerns the asymptotic behaviour of associated primes of graded components of local cohomology modules. A survey of this can be found in M. Brodmann, Asymptotic behaviour of cohomology: ...

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372 views

### Affine cone example

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\...

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201 views

### Homogeneous regular sequence

Consider a $\mathbb{Z}$-graded polynomial ring $R = k[x_1,\cdots,x_n]$ over a field $k$, where the elements $x_i$ are homogeneous (they may have negative degree). Let $I$ be a homogeneous ideal that ...

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124 views

### infinite fold tensor product of universal enveloping algebra

Let $\mathfrak a$ be a Lie algebra graded by the abelian semigroup $S$, then the universal enveloping algebra $U(\mathfrak a)$ of $\mathfrak a$ is $S \sqcup \{0\}$ graded. I have the following ...

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47 views

### Induced grading on free non-associative algebra

Let $A_X$ be a free non-associative algebra over a field $k$ on a set $X$ of free generators, where $X = X_0 \cup X_1$ and $X_0 \cap X_1 = \phi$. We will think of the elements of $X_0$ as even, and ...

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### Relation between the Spec and the Proj of a ring

I am reading Thaddeus' paper on GIT and flips (https://arxiv.org/pdf/alg-geom/9405004.pdf), and I am confused with a claim in the begining.
Let $R$ be a finitely generated integral algebra over an ...

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### Derivations of algebras graded by a group

Let $A$ be an algebra. A derivation of $A$ is a linear map $d:A \rightarrow A$ such that $d(ab)=d(a)b + a d(b)$ for $a, b \in A$.
If $A$ is a $\mathbb{Z}-$graded algebra, where $\mathbb{Z}$ is the ...

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### Does the associated graded functor take products of filtered k-coalgebras to graded k-coalgebras?

Let's suppose we have two noncommutative graded k-coalgebras $C_1$ and $C_2$ with respective admissible filtrations (i.e $F_{0}C_i=0$ and $\mathrm{colim}_k F_kC_i=C_i$), I would like to know if there ...

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### Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$

I read that, using the fact that $\text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $\text{Ext}^* (A,B)$ (which I understand), we can give $\bigoplus_{i} \text{Ext}^i (A,...

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### If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a ...

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### Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?

Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have:
$$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$
In a paper we ...

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695 views

### Graded analogues of theorems in commutative algebra

Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word
commutative ...

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### Global dimension of a graded algebra

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...

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### Projective dimension of graded modules

Short version:
Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
Longer version:
Let $G$ be a commutative group, let $R$ ...

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### Divided power algebra is artinian as a module over the polynomial ring

I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow.
In the paper Homological algebra on a complete intersection, with ...

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### The different gradings of a graded ring, and their schemes

Let $(A,g)$ be a graded commutative ring, where $A$ denotes the commutative ring, and $g$ its grading. What can be said about the set $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...

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### Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...

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### Hilbert functions of graded modules generated by mapped generators

I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...

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### Can the Artin-Rees lemma be derived from Krull Intersection theorem?

The Krull Intersection theorem states that : For a finitely generated module $M$ over a Noetherian ring $R$ and any ideal $I$ of $R$, we have $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...

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### Hochster-Roberts theorem

I am trying to understand this theorem, and as an example I try this case: Let $R=\mathbb{C}[z_1^{\pm},\ldots,z_n^{\pm}]^{S_n}$ be the algebra of Laurent polynomials. Now $R$ should be somehow ...

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### If $A\subset B$, what to say about their $\operatorname{Proj}$?

Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)...

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### On graded projective modules

If $R$ is a PID, then we know that any projective module over $R$ is free. Is there any graded version of this? That is, let us call a graded commutative ring $R=\oplus _{g \in G} R_g$ a graded PID if ...

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### References on infinite dimensional graded algebras

I am currently working on some generalisations of known results in 2-category theory, and I have been stymied trying to find references that discuss `graded-finite dimensional' algebras over a field - ...

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### Providing a grading for the polynomial ring over a commutative unital graded ring

Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ ...

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### Graded and projective (but not bounded below) module that is not graded-projective?

Let $A = \bigoplus_{n = 0}^\infty A_n$ be a graded algebra over a field $k$ that is locally finite: each $A_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \...

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### Can a ideal-like subset that is prime in low degrees be extended to a prime ideal ?

Let $A=\bigoplus_{n \ge 0}A_n$ be a positively graded commutative ring. Let $J_n \le A_n$ be subgroups for $j=0,\ldots,N$, satisfying
$\,\,\,A_n \cdot J_m \subseteq J_{n+m}$ for $n+m \le N$
$\,\,\,...

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### Characterization of a finitely graded (almost) domain

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the following property: if $x \in A_i$ and $y \in A_j$ and $i+j \leq N$, then
$$xy = 0 \text{ implies } x = 0 \text{ or } y = 0.$$
Hence ...