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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Connection between twisted graded modules and twisted sheaves
I came across the definitions of graded twisted modules while reading about the syzygy theorem. In the meantime I also attended an algebraic geometry course where twisted sheaves occured. For both ...
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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 $(M_n)$ ...
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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\...
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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 $\gamma_d(F_n)$ ...
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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.
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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.
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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 $m(A_k\otimes A_{n-k})\subset A_n$...
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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 ...
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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 \...
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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 ...
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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 ...
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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 $R$-...
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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 ...
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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 ...
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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 = ...
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