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Questions tagged [graded-modules]

This tag is for questions relating to "Graded Module", extensively used in homological algebra. It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z-algebra.

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If the graded module is finitely generated then the filtration is good

Suppose $A$ is a unital Noetherian ring with an ideal $\mathfrak{q}$. Provide $A$ with its $\mathfrak{q}$-adic filtration. Let $M$ be a finitely generated $A$-module with descending filtration $(M_n)$ ...
ephe's user avatar
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If $M$ is a module and $a$ is a nonzero divisor of $M$, then $d(M)-1=d(M/aM)$

I have seen an interesting problem while reading the dimension theory of modules. Let $R$ be a local ring with maximal ideal $\mathfrak{m}$ and $M$ be finitely generated an $R$-module. Let $a\in\...
Debojyoti Pal's user avatar
5 votes
1 answer
99 views

Third term in the free Lie ring

I'm working with the Lie algebra of the free group: $$\mathscr{L}(F_n) = \oplus_{d=1}^{\infty} \mathscr{L}_d(F_n),$$ where $\mathscr{L}_d(F_n) = \gamma_d(F_n) / \gamma_{d+1}(F_n)$ and $\gamma_d(F_n)$ ...
Chase's user avatar
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2 votes
1 answer
162 views

A brief explanation on Representation Theory

I'm trying to read this beautiful paper Regularity and cohomology of determinantal thickenings by Claudiu Raicu but I'm getting in trouble with Representation Theory, since I have no knowledge of it. ...
Hola's user avatar
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Superderivative of $G^\infty$ maps $\mathbb{R}^{1,1}_\infty\to\mathbb{R}_\infty$

I am following Rogers's Supermanifolds: Theory and Applications and I might be getting something wrong, because I reach a definition that, as I understand it, doesn't imply what the author states. ...
Albert's user avatar
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29 views

Associated graded space as a (bi)algebra

A filtered bialgebra $(A,m,u,\delta,\epsilon)$ with $A_j,j\in\mathbb Z$ filtration is defined as a bialgebra so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$...
Eric Ley's user avatar
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Is each component of a graded module over a $k$-algebra a finite-dimensional vector space?

I have some problems with an argument in a proof of a lemma: Let $M = \oplus_{-\infty}^{\infty} M_n$ be a finitely generated graded $A$-module and $A=\oplus_{n\geq 0} A_n$ a graded commutative ring ...
Heraklit's user avatar
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50 views

Terminology question: "Module derivations" of the form $\mu \colon M \rightarrow M$

Let $R$ be a graded ring endowed with a graded derivation $d \colon R \rightarrow R$ of degree $k$. Let $M$ be a graded $R$-module. Is there a standard name for degree $k$ maps $\mu \colon M \...
dejavu's user avatar
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Coherent sheaves, Serre’s theorem and ext groups

Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$). Let $O_X(1)$ be a very ample invertible sheaf on $X$. Then, the ...
Walterfield's user avatar
2 votes
1 answer
70 views

Isomorphism of graded modules

Let $R$ be graded ring $M,N$ be graded $R$-modules. If $f:M \longrightarrow N$ is isomorphism of $R$-modules (NOT graded), is $f$ isomorphism of graded $R$-module? In other words, if $M,N$ are ...
AIA's user avatar
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Is Artinian assumption necessary here in Matsumura's book?

I quoted Theorem 13.2 below from Matsumura's book Commutative Ring Theory: Let $R=\bigoplus_{n\geq 0}R_n$ be a Noetherian graded ring with $R_0$ Artinian, and let M be a finitely generated graded $R$-...
William Sun's user avatar
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Graded version of Lazard's criterion

Lazard's criterion says that a module over a commutative ring is flat if and only if it is a filtered colimit of free modules. Does the graded version hold, i.e.: A graded module over a graded ...
Bubaya's user avatar
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3 votes
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Is $k[x, x^{-1}]$ a (graded) injective $k[x]$-module

Consider $k[x]$ with the usual grading, and the graded $k[x]$-module $k[x, x^{-1}]$. Is it injective? I suppose yes, because it is torsion free and graded divisible (i.e., divisible by homogeneous ...
Bubaya's user avatar
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Characterisation of graded homorphisms from a graded module to $\mathbb{R}[t]$

Let $\mathbb{R}[x, y]_d$ denote the vector space of homomogeneous polynomials in the indeterminates $x, y$ of degree $d$ with real coefficients. As a graded ring $A :=\mathbb{R}[x, y] = \bigoplus_{d = ...
Colin Tan's user avatar
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What does 'desuspension' mean in the context of homological algebra/graded modules?

I am familiar with suspensions/reduced supensions/desuspensions etc. in topology, but not in the context of homological algebra and graded modules. What does suspension and desuspension mean in the ...
user829347's user avatar
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1 answer
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Treating graded rings and modules as sequences of abelian groups connected by maps

I've been reading about $\mathbb{Z}$-graded rings and modules recently and have attempted to understand these as sequences of abelian groups with connecting maps $A_i\times M_j\to M_{i+j}$ etc. This ...
user829347's user avatar
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0 votes
1 answer
129 views

Kernel of morphism of graded modules is graded submodule

This is related to my earlier question regarding the category of graded modules. Let $A$ be a commutative unital $\mathbb{Z}$-graded ring, and $f:M\to N$ be an $A$-linear map where $f=\sum_if_i$ and ...
user829347's user avatar
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1 vote
1 answer
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Hom functor on the category of graded modules

I am trying to make sense of the introduction to Section 2 of the following paper. Let $R$ be a $\mathbb{Z}$-graded ring and $\text{Mod}_R$ be the 'category of graded $R$-modules'. Let $D$ be the ...
user829347's user avatar
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1 vote
1 answer
360 views

Prove that an epimorphism in $\text{Mod}_R$ is surjective

Let $R$ be a unital ring and $f:M\to N$ be an epimorphism of modules. I know how to prove that a morphism of modules is a monomorphism iff it is injective, and that a surjective morphism is an ...
user829347's user avatar
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1 vote
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associated graded of ungraded polynomial ring is isomorphic to graded polynomial ring.

I would like to discuss a fine observation that I made which is not discussed in the literarure. Maybe because it is easy. Nevertheless, I think that it is quite important. Let $k$ be a commutative ...
Flavius Aetius's user avatar
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Symmetric Algebra of a Graded Module

Let $k$ be a commutative ring. Let $M$ be a $k$-module. Let $M$ be a graded module, concentrated in degree one. Then, the shifted module $M[1]$ is concentrated in degree zero. Hence, $M[1]$ is ...
Flavius Aetius's user avatar
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0 answers
103 views

Relation between definitions of a graded vector space and a graded module

I am new to the world of "grading", and my question is about the definition of the graded vector space. For definitions of these two, see graded vector space and graded module (they are ...
Zed Fang's user avatar
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1 answer
88 views

Symmetrization map over the polynomial ring of a vector space.

Let $V$ be a finite-dimensional complex vector space. Is the morphism \begin{gather*} \mathrm{Sym}^{\bullet}(V \oplus V^*) \to D(V) \cong \frac{\mathrm{T}^{\bullet}(V\oplus V^*)}{I} \,, \\[0.5em] (v_1,...
Flavius Aetius's user avatar
1 vote
0 answers
79 views

Hilbert's Syzygy Theorem for bigraded modules

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
  • 719
3 votes
0 answers
43 views

Cohomology groups inherit Grading from Complex

Let $R$ be a $\mathbb{Z}^n$-graded ring with unique maximal ideal $R_{+}$ generated by all elements positive degree in $(\mathbb{N}_{>0})^n$ and we consider a bounded complex of $\mathbb{Z}^n$-...
user267839's user avatar
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2 votes
1 answer
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The category of graded $\mathbb S$-modules form a monoidal category

I am reading paragraph 6.2 in Algebraic Operads by Jean-Louis Loday and Bruno Vallette. Proposition 6.2.2 states: The category of graded $\mathbb S$-modules, with the (composite) product $\circ$ and ...
Maxim Nikitin's user avatar
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0 answers
129 views

Isomorphism of associated graded of modules with increasing exhaustive filtrations induces an isomorphism of the modules

Let $R$ be an ungraded commutative ring and let $M$ and $N$ be $R$- modules equipped with increasing and exhaustive filtrations, i.e $\{0\}\subset F_0P\subset\dots \subset F_nP\subset\dots\subset P$ ...
Flavius Aetius's user avatar
0 votes
1 answer
228 views

Isomorphism of associated graded vector spaces implies an isomorphism of vector spaces

Let $R$ be a commutative ring and $M$ be a module with decreasing filtration $M=F_0M\supset F_1M\supset\dots$. Assume that $N$ is a $\mathbb Z$-graded $R$-module such that there is an isomorphism of ...
Flavius Aetius's user avatar
12 votes
0 answers
277 views

when are graded injective modules graded and injective?

Define a graded injective module over a graded ring $R$ to be an injective object in $GrMod-R$ (the category of right graded $R$-modules). From the little research I have done, a graded injective ...
RumDiary's user avatar
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1 vote
1 answer
91 views

Condition for isomorphism of $R$-module homomorphism.

In the proof of theorem 9.4.7 of Charles Weibel's "An Introduction to Homological Algebra" it is claimed to aid the proof that an $R$-module homomorphism $\psi$ is an "isomorphism if ...
andrew's user avatar
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0 answers
181 views

Necessary and Sufficient condition for Flatness of Graded Modules

I am currently self studying the textbook Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud and came across the following exercise: Exercise 6.10 (Flatness of graded modules):...
Oakley_Edens's user avatar
1 vote
1 answer
309 views

About twisted graded modules

While reading the notes from Tom Marley about graded rings and modules I came across the following statement: "Define M(n) (read "M twisted by n") to be equal to M as an R-module, but ...
Fynn B.'s user avatar
  • 21
0 votes
2 answers
531 views

Direct sum of graded modules is *isomorphic* to a graded module?

Let $R$ be a graded ring and let $M_1,\ldots,M_m$ be graded $R$-modules. We can form the direct sum $N=\bigoplus_{i=1}^m M_i$, which according to my textbook is also a graded $R$-module using the ...
SlugMan523's user avatar
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0 answers
89 views

Example of a module over $k$-algebra whose length is infinite but dimension 0.

This is came from the definition of system of parameters in Stanley's 1996 book "Combinatorics and Commutative Algebra." In this book at page 33, he defines the (homogeneous) system of ...
user124697's user avatar
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2 votes
0 answers
140 views

Computing Persistent Barcodes.

I'm currently reading the following paper on persistent homology: https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf. Given a filtration of a simplicial complex, $K$, $$\{0\}=K^0\subseteq K^...
Jhon Doe's user avatar
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1 vote
1 answer
96 views

Betti number of graded ideal and graded quotients

For a monomial ideal $I$ of a polynomial ring $S$ with degree $b$, Theorem 1.34 of the ``Combinatorial Commutative Algebra" says that $$\beta_{i,b}(I) = \beta_{i+1,b}(S/I)$$ with a proof stating ...
user124697's user avatar
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1 vote
1 answer
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Reference request modules over a small category

I am currently writing up one of my results where I am using modules over small $\mathbb{k}$-linear categories. I so far sadly did not find a proper introduction of them to refer the reader to and was ...
Felix's user avatar
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2 votes
1 answer
346 views

Questions about Hartshorne Proposition I.7.4 and Theorem I.7.5

I have a few questions regarding the proofs of proposition 7.4 and theorem 7.5 in Hartshorne's Algebraic Geometry 1.7 Intersections in Projective Space. Here is the beginning of proposition 7.4 and ...
Pakchoiandbacon's user avatar
-2 votes
2 answers
136 views

Homogeneous non zero divisor in a graded module.

Let $(R_0,m_0)$ be a local ring and $R=\oplus_{n\geq 0}R_n$ a positively standard graded ring. Let $M$ be a finitely generated graded module over $R$. Let $R_+$ be the ideal $\oplus_{n>0}R_n$ and $\...
yo yo's user avatar
  • 536
2 votes
0 answers
68 views

How are $R$-modules with negativity condition $R_iM_j\subset M_{j-i}$ related to graded $R$-modules?

Let's say we have a graded ring $R=\bigoplus_{i=0}^n R_i$ ($R_m=0$ for $m > n$), in particular we have $R_i R_j \subset R_{i+j}$. We now have a $R$-module $M$ which is also graded $M=\bigoplus_{i=0}...
LegNaiB's user avatar
  • 2,767
1 vote
1 answer
100 views

Defining a grading on tensor product of algebras

Let $A$ be a commutative ring and $(\Delta_i)_{i\in I}$ a finite family of commutative monoids. Write $\Delta:=\prod_{i\in I}\Delta_i$. For each $i\in I$, let $E_i$ be a graded $A$-algebra of type $\...
user829019's user avatar
1 vote
0 answers
80 views

Defining a grading on the algebra of a magma

Let $\Delta$ be a commutative monoid and $S$ a magma. Assume $\varphi:S\rightarrow\Delta$ is a homomorphism. Define $S_\lambda:=\varphi^{-1}(\{\lambda\})$ for all $\lambda\in\Delta$. Clearly $S_{\...
user829019's user avatar
1 vote
0 answers
36 views

$\text{Homgr}_A(M,N)=\text{Hom}_A(M,N)$ if $M$ is finitely generated

Let $A$ be a ring and $M,N$ be two graded $A$-modules of type $\Delta$. For $\lambda\in\Delta$, let $X_\lambda:=\{f\in\text{Hom}_A(M,N)\ |\ \deg(f)=\lambda\}$. Write $\text{Homgr}_A(M,N):=\sum_{\...
user829019's user avatar
0 votes
0 answers
32 views

Defining a grading on the generalized tensor product

Let $C$ be a commutative ring, $(E_i)_{1\leq i\leq n}$ a family of graded $C$-modules such that $E_i\cong\bigoplus_{\lambda\in\Delta}M_{i\lambda}$, for each $1\leq i\leq n$. For $\gamma\in\Delta$, ...
user829019's user avatar
0 votes
1 answer
126 views

Image of a graded linear map is graded

Let $A$ be a graded ring, $M,N$ graded $A$-modules and $u:M\rightarrow N$ a graded linear map of degree $\delta$. I want to show that $\text{Im}(u)$ is a graded submodule of $N$. It is sufficient to ...
user829019's user avatar
0 votes
1 answer
73 views

How to construct a grading on a quotient module

Let $A$ be a ring, $M$ a graded $A$-module, $N$ a graded submodule of $M$ and $(M_\lambda)_{\lambda\in\Delta}$ the grading on $M$. Then $((M_\lambda+N)/N)_{\lambda\in\Delta}$ is a grading of $M/N$. ...
user829019's user avatar
2 votes
1 answer
344 views

Hilbert series of a tensor product

Let $k$ be a field. Let $A,U,V$ be $\mathbb{N}$-graded-commutative $k$-algebras. Suppose there are homogeneous algebra maps $A\rightarrow U$ and $A\rightarrow V$ giving $U$ and $V$ the structure of ...
user15160811's user avatar
1 vote
0 answers
28 views

Are there any useful generalizations of homomorphisms that can apply to a graded valuation ring?

I have two graded valuation rings, $R$ and $S$, that I want to relate by a function $f:R\rightarrow S$. The problem is that $f$ is only a group homomorphism under addition for two elements of $R$ ...
matty_k_walrus's user avatar
5 votes
0 answers
123 views

Why does this diagram commute?

I am trying to understand the commutativity of diagram (2) of Proposition 3.11 from this paper. It should be a simple computation but I don't understand how they do it and when I try to do it myself I ...
Javi's user avatar
  • 6,303
1 vote
1 answer
463 views

Koszul complex as a graded free resolution

I'm trying to compute the Hilbert function of a complete intersection by using the Koszul complex, but I think I'm approaching it incorrectly. If we let $R=k[x,y,z]$ and $$A= R/(f_1,f_2)$$ the Koszul ...
new2java's user avatar

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