MathOverflow

Questions tagged [graded-rings-modules]

Ask Question

The tag has no usage guidance.

39 questions
Newest
Active
Bountied
Unanswered
Filter by
Sorted by
Tagged with
0
votes
0answers
69 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 ...
1
vote
1answer
64 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(...
1
vote
2answers
247 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 ...
3
votes
2answers
278 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 ...
0
votes
0answers
29 views

Degeneracy for pairings for $\mathbb{N}_0$-graded algebras

Let $A = \bigoplus_{i \in \mathbb{N}_0} A_i$ be an $\mathbb{N_0}$-graded algebra such that $A_n \neq 0$, and $A_k = 0$, for all $k > n$, and which is generated in degrees $0$ and $1$. If we assume ...
1
vote
1answer
128 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 ...
3
votes
0answers
36 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: ...
1
vote
1answer
264 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 ${\...
5
votes
0answers
162 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 ...
2
votes
1answer
105 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 ...
0
votes
0answers
46 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 ...
1
vote
1answer
377 views

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 ...
2
votes
0answers
72 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 ...
5
votes
0answers
214 views

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 ...
0
votes
0answers
179 views

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,...
4
votes
1answer
179 views

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 finite-...
6
votes
1answer
148 views

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 ...
20
votes
3answers
603 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 ...
4
votes
1answer
195 views

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(...
5
votes
1answer
167 views

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 $R$ ...
9
votes
2answers
395 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 ...
8
votes
1answer
215 views

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 $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...
6
votes
1answer
227 views

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) ...
1
vote
0answers
31 views

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 ...
5
votes
0answers
246 views

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 $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
0
votes
0answers
88 views

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 ...
1
vote
1answer
175 views

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)...
2
votes
1answer
353 views

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 ...
1
vote
0answers
55 views

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 - ...
1
vote
1answer
153 views

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 $S$ , $a \in S$ ...
8
votes
1answer
187 views

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 = \...
0
votes
0answers
26 views

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 $n+m \le N$ $\,\,\,...
1
vote
0answers
81 views

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 ...
0
votes
0answers
170 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). ...
3
votes
1answer
82 views

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 ...
5
votes
0answers
424 views

Inverse limit of graded rings

Let $(I,\le)$ be a directed set and let $(\rho^{\beta\alpha}: R^\beta \to R^\alpha)_{\alpha \le \beta}$ be an $I$-directed system of $\mathbb{Z}$-graded rings whose multiplication is denoted by $$\...
5
votes
0answers
653 views

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 ...
12
votes
2answers
1k views

“Spec” of graded rings?

From the discussion here, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have nonzero degree. So I have some naive and maybe stupid ...
4
votes
0answers
421 views

Tor over graded rings

Let $R$ be a graded ring (concentrated in nonnegative dimensions and maybe bounded from above). For every positive natural number $n$, denote by $R\to\tau_{\leq n}R$ the $n$-truncation and by $\tau_{\...