Questions tagged [graded-rings-modules]

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On a lemma of projective dimension

Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...
Noto_Ootori's user avatar
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1 answer
171 views

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 ...
Tim Campion's user avatar
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graded reps of Lie algebras literature

I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2 grading), such as ...
iron's user avatar
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1 answer
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Is every graded hereditary ring hereditary?

Let R be a graded (associative, unital) ring. If R is left graded hereditary (i.e. its left graded global dimension is 0 or 1), does it follow that R is left hereditary (i.e. its left global dimension ...
Ralle's user avatar
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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,...
uno's user avatar
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A non-example of a graded Frobenius algebra

Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not ...
Lorenzo Del Vecchiopontopolos's user avatar
3 votes
1 answer
362 views

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 $...
Lorenzo Del Vecchiopontopolos's user avatar
1 vote
0 answers
58 views

Twisting a graded algebra by an automorphism (Transitivity)

Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra ...
Lumi's user avatar
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Literature on Clifford modules

I encountered Clifford modules in the book Heat Kernels and Dirac Operators. I am particularly interested in the definition of the isomorphism $$\mathrm{End}(E)\cong C(V)\otimes \mathrm{End}_{C(V)}(E)$...
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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 ...
Sourjya Banerjee's user avatar
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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^{...
Pierre Dubois's user avatar
12 votes
2 answers
715 views

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 \...
walkar's user avatar
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Filtrations and Koszul algebras

I was looking at this question and asked my self the following: Let $A$ be graded algebra, which is also an $\mathbb{N}_0$-filtered algebra. If its associated graded algebra $\mathrm{gr}(A)$ is ...
Didier de Montblazon's user avatar
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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 ...
Heer's user avatar
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Relation(s) between units and nilpotent elements in graded noncommutative rings

In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of ...
M.G.'s user avatar
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3 votes
1 answer
325 views

Hilbert's Syzygy Theorem in the bigraded case

I've been recently wondering how to prove the existence of a Hilbert polynomial for finitely generated bigraded modules $M$ over a polynomial ring $R=k[X_0,...,X_n,Y_0,...,Y_m]$ with the usual ...
Carnby 's user avatar
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0 answers
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Is the associated algebra of a quadratic filtered algebra again quadratic

Let $A$ be a filtered quadratic algebra. Let $G(A)$ be the associated graded algebra. Will $G(A)$ again be a quadratic algebra or can higher relations appear when passing to the graded seeting? EDIT: ...
Didier de Montblazon's user avatar
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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 ...
Hvjurthuk's user avatar
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Graded global dimension of a graded algebra

Let $k$ be an algebraically closed field of characteristic $0$. Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...
Walterfield's user avatar
3 votes
4 answers
723 views

$R$ 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 ...
It'sMe's user avatar
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9 votes
2 answers
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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 ...
Tilman's user avatar
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2 votes
3 answers
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Proving the graded structure of the tensor algebra from only the universal property

When the tensor algebra is presented, it is usually constructed as the direct sum of all tensor powers of the space. By this construction, the graded structure of the tensor algebra is easy to prove. ...
sudgy's user avatar
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1 answer
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Is there an operad homotopifying the Koszul rule?

In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One ...
Emily's user avatar
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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 ...
Emily's user avatar
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6 votes
1 answer
270 views

Connection between the classifications of group extensions and group-graded algebras in terms of non-abelian cohomology

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension ...
Johannes Hahn's user avatar
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397 views

Rational points of weighted projective spaces

[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.] Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
Laurent Moret-Bailly's user avatar
2 votes
1 answer
342 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 ...
Louis 's user avatar
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3 votes
1 answer
165 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 ...
Louis 's user avatar
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1 vote
0 answers
30 views

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 $...
Hvjurthuk's user avatar
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0 answers
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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 $B:=\{m_{1},\dots, m_{r}\}$ ...
Hvjurthuk's user avatar
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0 answers
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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) ...
a196884's user avatar
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2 votes
1 answer
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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 ...
Javi's user avatar
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0 answers
97 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_{...
waphety's user avatar
7 votes
1 answer
570 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: $$ A_kA_l = A_{k+l}. $$ ...
Piet Bongers's user avatar
1 vote
0 answers
85 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\...
SCM's user avatar
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4 votes
1 answer
188 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 ...
One More Question's user avatar
6 votes
0 answers
148 views

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 ...
Andrea Ferretti's user avatar
8 votes
2 answers
401 views

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 ...
Javi's user avatar
  • 489
5 votes
2 answers
534 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 ...
Ted Jh's user avatar
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2 votes
1 answer
191 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, ...
Javi's user avatar
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0 votes
0 answers
138 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^{\...
Javi's user avatar
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5 votes
1 answer
290 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 ...
Javi's user avatar
  • 489
3 votes
2 answers
203 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 $\...
YCor's user avatar
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1 vote
0 answers
116 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 ...
Paulo Rossi's user avatar
1 vote
1 answer
106 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(...
Leray Jenkins's user avatar
7 votes
2 answers
2k 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 ...
Javi's user avatar
  • 489
6 votes
2 answers
531 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 ...
Jakob Werner's user avatar
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1 vote
1 answer
191 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 ...
Fofi Konstantopoulou's user avatar
3 votes
0 answers
58 views

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: ...
Fred Rohrer's user avatar
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1 vote
1 answer
572 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 ${\...
user267839's user avatar
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