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{...
12 votes
0 answers
398 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 ...
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 ...
8 votes
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 ...
2 votes
1 answer
538 views

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 ...
2 votes
0 answers
58 views

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 ...
3 votes
1 answer
281 views

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 ...
1 vote
0 answers
91 views

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,...
1 vote
1 answer
122 views

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 ...
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 $...
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 ...
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 ...
2 votes
0 answers
69 views

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)$...
4 votes
0 answers
79 views

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 vote
1 answer
173 views

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^{...
12 votes
2 answers
716 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 \...
2 votes
0 answers
104 views

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 ...
2 votes
3 answers
413 views

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. ...
2 votes
1 answer
88 views

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 ...
2 votes
1 answer
343 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 ...
2 votes
1 answer
149 views

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 ...
3 votes
1 answer
320 views

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}...
2 votes
0 answers
97 views

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: ...
0 votes
1 answer
308 views

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). ...
1 vote
0 answers
42 views

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 ...
14 votes
2 answers
2k views

"Spec" of graded rings?

From the discussion at Hochschild cohomology and A-infinity deformations, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have ...
3 votes
0 answers
235 views

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 set ...
9 votes
2 answers
367 views

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 ...
25 votes
3 answers
1k 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 ...
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 ...
1 vote
0 answers
110 views

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 ...
2 votes
1 answer
83 views

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 ...
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 ...
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 $...
1 vote
0 answers
191 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 $B:=\{m_{1},\dots, m_{r}\}$ ...
1 vote
0 answers
184 views

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) ...
0 votes
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_{...
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: $$ A_kA_l = A_{k+l}. $$ ...
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\...
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 ...
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 ...
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 ...
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 ...
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, ...
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^{\...
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 ...
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 $\...
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 ...
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 ...
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(...