All Questions
Tagged with ac.commutative-algebra graded-rings-modules
41 questions
1
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31
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Primary invariants on MAGMA for a graded ring
I have asked this question on mathstacks, but a collegue of mine recommended me to post it here.
I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
- 11
1
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0
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47
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Examples of graded subrings of $\mathbb Q(T)$
The following question came up in some discussion on some very unrelated matters.
A graded algebra $A$ is an algebra $A$ with a decomposition $A = \oplus_{i \in \mathbb Z} A_i$ such that $A_i A_j \...
- 2,029
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1
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372
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The growth of the Hilbert function of a graded ring
Let $A=\bigoplus A_i$ be a finitely generated commutative unital graded algebra over a field $k$. Let $d(i)=\dim A_i$.
In general $d(i)$ is not a polynomial in $i$ (even not eventually polynomial). ...
- 413
4
votes
1
answer
187
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Order of pole of Poincaré series
Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
- 12.9k
8
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3
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691
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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 ...
- 101
2
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2
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411
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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 ...
- 135
8
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1
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191
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Do graded-commutative rings satisfy the strong rank condition?
Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$.
It is ...
- 12.9k
1
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0
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106
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
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1
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372
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Subalgebras of quadratic algebras that are not quadratic
Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
- 16.9k
3
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4
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807
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$R$ is a UFD iff $R_{\frak{m}}$ is a UFD?
Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...
- 839
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84
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Dimension of a positively graded ring after a suitable localization
Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
1
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1
answer
182
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A question about surjective maps between quadratic algebras
Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...
- 4,600
12
votes
2
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776
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Hilbert polynomials of graded algebras evaluated at negative numbers
Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
- 50.8k
2
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1
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91
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Is the integral closure of a $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domain in a bigger $\mathbb{Z}/n\mathbb{Z}$-graded domain also graded?
Let $A\subset B$ be an inclusion of $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domains. Is the integral closure of $A$ in $B$ also $\mathbb{Z}/n\mathbb{Z}$-graded?
This is true for the $G$-graded case ...
- 16.2k
1
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0
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45
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Lifting module homomorphisms imposing conditions on characteristic polynomials
Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
- 573
9
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2
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417
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Over which (graded) rings are all modules decomposable into indecomposables?
A module is decomposable if it is the direct sum of two modules. The process of splitting summands off of a decomposable module does not need to terminate, so infinitely generated modules do not ...
- 35.2k
27
votes
3
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1k
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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 ...
- 30.3k
1
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0
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119
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Commutative monoid gradings via group scheme actions
$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and ...
- 11.8k
3
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1
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178
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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 ...
- 661
1
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0
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33
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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 $...
- 573
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0
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212
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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}\}$ ...
- 573
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0
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210
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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) ...
- 323
1
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0
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106
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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\...
- 63.9k
4
votes
1
answer
195
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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 ...
- 778
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159
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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 ...
- 14.7k
1
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1
answer
111
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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(...
- 9,061
3
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0
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62
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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: ...
- 6,700
1
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1
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646
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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 ${\...
- 661
6
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0
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407
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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 ...
- 71
6
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1
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463
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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 ...
CommunityBot
- 1
6
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1
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164
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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 ...
- 30.5k
9
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2
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513
views
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 ...
- 1,286
6
votes
1
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598
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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$ ...
- 2,928
8
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1
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399
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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 \}$? ...
- 16.1k
1
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0
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40
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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 ...
- 11
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0
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337
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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$ . ...
CommunityBot
- 1
3
votes
1
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701
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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 ...
CommunityBot
- 1
1
vote
1
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266
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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$ ...
- 6,700
1
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0
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86
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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 ...
- 13.3k
3
votes
1
answer
133
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Derivations annihilated by powers of the augmentation ideal
Consider an augmented commutative ring $R$, with augmentation ideal $\varpi$. Let $\delta$ be a derivation of $R$. The example I have in mind is $R=\mathbb F_p[x]/(x^{p^i})$ and $\delta=d/dx$, though ...
- 33.8k
5
votes
0
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917
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Height of maximal homogeneous ideals
Let $R= \oplus_{n \ge 0} R_n$ be a graded Noetherian commutative ring and suppose $R_0$ is Artinian.
Do all maximal homogeneous ideals of $R$ have the same height ?
Let $R_{>0}$ be the ideal ...
- 1,355