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Questions tagged [graded-modules]

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1answer
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Hilbert Syzygy Theorem for non-graded modules

The statement of Hilbert Syzygy Theorem is as follows: Let $R = k[x_1 , \ldots , x_n]$ be a polynomial ring over a field $k$ and $M$ be a finitely generated graded $R$-module. Then $\text{pd }M \leq n$...
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Prove that a finitely generated $\mathbb F_p[t]$-module is a free $\mathbb F_p[t]$-module.

Specifically, I am asked to show that if $(G,\omega)$ is a $p$-valued group of finite rank, (meaning that the associated graded group $grG$ is finitely generated as an $\mathbb F_p[t]$-module), then $...
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Projective Resolution of exterior algebra as a module over divided polynomial algebra

Let $\Lambda_\mathbb{Z}[x]$ be an exterior algebra on one generator with $|x|=n$, let $\Gamma_\mathbb{Z}[x]$ be a divided polynomial algebra with $|x_k|= kn$, and suppose that $\Lambda_\mathbb{Z}[x]$ ...
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For which algebra $A$: $\mathrm{vect} \cong A-$mod

In general there is an equivalence between a module category (see Module Category nLab) over a monoidal category $\mathcal{C}$ and $A-$mod, where $A$ is an algebra over $\mathcal{C}$: $$\mathcal{M}_{\...
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Module category

I am currently reading about module categories and have been not very successful. In this case a module category is a category with an action of a monoidal category. (more information on nLab) In ...
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Support of a graded module of a ring concentrated in non-negative dimensions

I wanted to prove the following equivalence. Consider $R$ a graded commutative Noetherian ring such that $R^{<0}=0$ and $M$ a graded, finitely generated $R$-module. Then $M^i=0$ for $i \gg 0$ if ...
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38 views

Dual of a differential graded module

Let $(M, d_M)$ be a differential $\mathbb{Z}$-graded module over a differential graded algebra (over a field) $(A, d_A)$. I am wondering if there is a canonical way of looking at the dual $\hom_A(M,A)...
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Divided power algebra is artinian as a module over the polynomial ring

In the paper Homological algebra on a complete intersection, with an application to group representations I found the following argument that I do not understand: Suppose $B$ is a local artinian ring ...
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Ascending / descending chain condition on graded modules.

Let $R = \bigoplus_{n \in \mathbb{N}} R_n$ be a graded commutative ring. Then $R$ is noetherian / artinian if and only if it has the ascending / descending chain condition for homogeneous ideals, see ...
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Quotient of graded modules

I am trying to understand why the quotient of two graded modules is graded. I am struggling with the part to show that $A^p\cdot M^q/N^q\subseteq M^{p+q}/N^{p+q}$ with $N^p=N\cap M^p$. Any tips how? ...
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The Drinfeld centre of G-graded vector spaces

I'm currently working on understanding the Drinfeld centre construction. My professor gave me the exercises to understand an example: Take $G$ a group and $Vec_G$ the category of $G$-graded vector ...
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Tensor product of graded algebras 3

Let $A$ and $B$ be $\mathbb{Z}_2$-graded algebras, i.e. $A=A_0 \oplus A_1$, $B= B_0 \oplus B_1$. I am trying to show that the graded tensor product $A \otimes B = (A \otimes B)_0 \oplus (A \otimes B)...
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Singular chain complex as a graded algebra

Let $X$ be a topological space and denote $S_*(X)$ the singular chain complex of $X$. There is a chain map (The Eilenberg-Zilber map) $$E: S_*(X)\otimes S_*(X) \rightarrow S_*(X \times X)$$ which ...
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Proving that $\operatorname{Tor}_n^R$ is a bifunctor

$\newcommand{\Tor}{\operatorname{Tor}}$ Ex10.2 pg 615: For a ring $R$ and fixed $k \ge 0$, prove that $\Tor_n^R(-,-)$ is a bifunctor. I am aware of this post. I am also not satisfied with the ...
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Modules can be viewed as monoid objects in some appropriate monoidal category?

I need to find a solution to have an inverse structure form: not classic modules over a monoid but monoids over modules. I had received this answer from here monoid objects are the minimal ...
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The Endomorphism algebra of graded vector space

Let $G$ be a group. A linear map $f:V\rightarrow W$ of $G$-graded vector spaces is said to be homogeneous of degree $g$ if $f(V_{h}) \subseteq W_{g\cdot h}$ for all $h\in G$. We denote the space all ...
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Meaning of “graded $R$-modules” in May's “A Concise Course in Algebraic Topology” on Page 89

The confusing passage is also here on page 2. May's usage of graded $R$-modules appears to differ from the standard definition, as far as I can tell. He is taking graded $R$-modules to mean the ...
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Are the associate primes of a graded module homogeneous?

Let $R$ be a $\mathbf N^r$-graded ring, for instance a polynomial ring in $r$-variables. A prime ideal $\mathfrak p\subseteq R$ is associated to a graded $R$-module $M$ if there is a (not necessarily ...
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In a $\mathbb Z$-graded ring we have $IR \cap R_0 = I$

I have a problem with an exercise from Tom Marley which is: Let $R$ be a $\mathbb Z$-graded ring and $I$ an ideal of $R_0$. Prove that $IR \cap R_0 = I$. For $I \subset IR \cap R_0 $, we can ...
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Units are homogeneous in $\mathbb Z$-graded domains

I am confused by an exercise from Tom Marley which is: Let $R$ be an arbitrary $\mathbb Z$-graded domain: $1)$ Prove that all units in $R$ are homogeneous. $2)$ By using $1$, if $R$ is a field, ...
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On “homogeneous” height and “homogeneous” Krull-dimension?

Let $R=\oplus_{n \ge 0} R_n$ be a commutative graded ring. For a prime ideal $P$ of $R$, let $ht P$ denote the usual height of a prime ideal $P$ of $R$. Now let $P$ be a prime ideal of $R$ which is ...
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Nakayama Lemma, graded version

I try to understand the proof of Nakayama Lemma, graded version: $S$ is the polynomial ring and $m$ is its maximal homogeneous ideal. Let $M$ be a finitely generated graded $S$-module and $U$ a ...
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Dimension of a graded module over a local$^*$ ring

Assume that $R$ is a positively graded ring which has only one maximal homogeneous ideal $\mathfrak{m}^*$. Let $M$ be a finitely generated positively graded ring over $R$ and consider $\mathcal{l}_R(M/...
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Which Graded and free modules over Graded PID's are Graded-free ?

Let $G$ be an abelian group and $R=\oplus_{g\in G}R_g$ be a $G$-graded, commutative ring with unity . Let us call $R$ to be a "Graded PID" if every graded ideal of $R$ is generated by a homogeneous ...
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Hilbert Coeficients; Multiplicity

Let $R$ be a graded ring and $M$ a graded $R$-module. An element $x\in R_{l}$ is said to be superficial for $M$ if $(0:_{M} x)_{n}=0$ for all but finitely many $n$. Let $M$ be a finite generated $R$-...
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Minimal graded projective resolution

Let $M$ be a finitely generated graded $k[x]$-module for some field $k$. Let $$\cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$$ be a minimal graded projective ...
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1answer
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A query about tensor product of graded modules.

Let $A=\bigoplus_{n\geq 0}A_n$, $B=\bigoplus_{n\geq 0}B_n$ be two graded Noetherian rings, where $A_n\subset B_n$ and $A_0=B_0$ is a local ring. Suppose $B$ is a finitely generated $A$ module. Let $I$ ...
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Graded torsion module is zero in high enough degree

If $R = k[x,y]$ with the usual grading for some field $k$, we say that a graded $R$-module $M$ is graded torsion if $M$ is annihilated by some power of $\mathfrak m = (x,y)$. Is it true that there ...
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Graded vector space conditions

Wikipedia define the graded ( ring, module, vector space, ...) as here I noted that in the rings and modules it required the condition of inclusion but in the vector spaces it did not. just the ...
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1answer
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Maximal degree of a minimal homogeneous generator smaller than regularity

Let $S=k[x_1,\dots,x_n]$ be a standard graded polynomial ring and $I\subset S$ a homogeneous ideal. Denote $\mathrm{maxdeg}(I)$ the maximal degree of an element in a minimal system of homogeneous ...
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1answer
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Can a finitely generated graded module over commutative unital graded ring be also finitely generated by homogenous elements ?

Let $R$ be a commutative graded ring (https://en.wikipedia.org/wiki/Graded_ring) with unity , graded by a monoid (in particular $\mathbb Z$ or $\mathbb N \cup \{0\}$ if helpful for the purpose ) , ...
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1answer
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Well-definedness of the morphism for the chain complex $\text{Hom}(V,W)$

Let $R$ be a unital commutative ring. Suppose that for the $\mathbb{Z}$-graded $R$-modules $V$ and $W$, $(V,d)$ and $(W,d)$ are chain complexes. Let $\text{Hom}(V,W)$ be the module of homogeneous ...
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Resolutions of graded modules over positively graded polynomial rings

I am looking for a proof of this fact: Let $W=k[x_1,\dots,x_n]$ be a positively graded polynomial ring. Every graded free resolution of a finitely generated $W$-module $M$ is isomorphic to the direct ...
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Definition of Exterior Algebra of a Graded Module

Let $M$ be a $\mathbb{Z}$-graded module over a trivial graded ring $R=R_0$. The tensor algebra $T_R(M)$ then becomes a $\mathbb{Z}$-graded module with $i$-th graded component $$T_R(M)_i = \bigoplus_{...
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Graded module structure as a graded homomorphism?

Let $R$ be a ring. A $R$-module is an abelian group $M$ with an application $R\times M\longrightarrow M$, $(r, x)\longmapsto r\cdot x$ satisfying some properties. To give a $R$-module structure on ...
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Finding kernel of a map in a minimal free resolution of a graded ideal

Let $I=(x_{1}^{2}-x_{2}x_{3},x_{3}^{2}x_{4},x_{1}x_{2}x_{3},x_{4}^{3})\subset S=K[x_{1},x_{2},x_{3},x_{4}]$ be an ideal. I am trying to compute its minimal graded free resolution. Since $I$ has four ...
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1answer
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Basic question on graded quotients

Let $A$ be a graded algebra, and let $I$ be a graded (two-sided) ideal. ($A=\bigoplus A_k$, $I=\bigoplus I_k$). Is it true that $A/I=\bigoplus A_k/I_k$? I can't seem to see it. Thanks for any help....
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gradings of quotients of graded modules

If $M$ is a graded $R$-module ($R$ a graded ring) $\bigoplus$$M$$_i$ and $N$ is a graded $R$-submodule of $M$, $\bigoplus$$N$$_i$, then how do we write the grading on the quotient module $M/N$? I have ...
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Question regarding Graded Rings

I am currently reading Li Huishi's book "Zariskian filtrations". I am confused by a lemma (p. 34, Lemma 2 of section 4.3 in Chapter 1) in the book. It says: Let $S = \oplus_{n \in \mathbb{Z}}S_n$ ...
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1answer
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Is there a version of the adjunction formula for subvarieties of $\mathbb{P}^n\times\mathbb{P}^m$

I want to try and find an example of a projective variety which does not have an ample canonical or anti-canonical bundle. I think I should try and look at a subvariety of $\mathbb{P}^n\times\mathbb{P}...
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1answer
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Multigraded free module over multigraded ring has multihomogeneous basis?

Let $R$ be an $\mathbb{N}^m$-graded ring and let $M$ be an $\mathbb{N}^m$-graded module over $R$. Supposing that $M$ is free as an $R$-module, does there necessarily exist an $R$-basis homogeneous ...
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What is the internal grading of Ext

I have just learned the definition for $Ext_{A}^i(M,N)$ where $M,N$ are $A$-modules. However, in the book Quadratic Algebras by Polishchuk and Positselski they sometimes write $Ext_A^{ij}(M,N)$ (for ...
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1answer
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Computation of basic Hilbert functions

I want to compute the Hilbert function for the ring $$M:=\frac{k[x,y,z,w]}{(x,y)\cap(z,w)}$$ and compare it to the Hilbert function for the ring $$N:=k[x,y].$$ I tried computing the bases for each $...
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1answer
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Why is a left basis also a right basis?

Let $k$ be a field and $A$ a graded $k$-algebra with $A_0=k$, $A^i=0$ for $i<0$ and $\dim_kA^i<\infty$. Let $B$ be a subalgebra of $A$ which is normal, i.e. $AB^{\geq1}=B^{\geq1}A$. Assume that ...
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Let R be a Z-graded ring. Prove that R is Noetherian if and only if R satisfies the ascending condition on homogeneous ideals.

The problem is "Let $R$ be a $\mathbb{Z}$-graded ring. Prove that $R$ is Noetherian if and only if $R$ satisfies the ascending condition on homogeneous ideals." I know it sounds a lot like Ascending ...
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Hilbert Polynomial of zero dimensional module

Let $M$ be a finitely generated graded $A$-module with $\dim M=0.$ Let $M$ be $\mathbb Z$-graded and $A$ be $\mathbb N$-graded Noetherian ring. Then how can we say that the Hilbert function of $M$ is ...
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Relationship between associated graded module and a quotient

Let $R$ be a ring, $J \subset I$ ideals. Let $\mathfrak{I} = R\supset I \supset I^2 \supset \cdots$ be the $I$-adic filtration of $R$. I want to compare the associated graded module $\operatorname{gr}...
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79 views

Coproduct of graded modules

Let $M=\oplus_{i\ge 0}M_i$ and $N=\oplus_{i\ge 0}N_i$ two graded modules over a commutative ring $R$. What is their coproduct? Is it their tensor product $M\otimes_k N$? Thank you!
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1answer
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Counit for the filtration adjunction

This is a follow-up to my question here, which was answered by Eric Wofsey. Let $\mathsf{A}$ be an abelian category, such as $R \mathsf{Mod}$ for concreteness. We can think of the category of $\...
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1answer
167 views

Does the usual filtration on graded objects satisfy a universal property?

Let $\mathsf{A}$ be an abelian category, such as $R \mathsf{Mod}$ for concreteness. We can think of the category of $\mathbb{Z}$-graded objects in $\mathsf{A}$ as the functor category $\mathsf{A}^\...