Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [graded-algebras]

A graded module that is also a graded ring is called a graded algebra.

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $R_{i}$ such that $ R_{i}R_{j}\subseteq R_{i+j}$. An algebra $A$ over a ring $R$ is a graded algebra if it is graded as a ring.

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Existence of $\mathbb{N}$-grading compatible with LNDs.

Let $B$ be a finitely generated integral $\mathbb{C}$-domain. Let $\partial:B\to B$ be a LND, locally nilpotent derivation, i.e. a $\mathbb{C}$-linear map satisfying Leibniz rule: $\partial(fg)=f\partial(g)+\partial(f)g$ for $f,g\in B$; locally…
Yikun Qiao
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Associated Graded Algebra

I'm trying to work through Exercise III.27 of Lang's Algebra: Let $A$ be a filtered algebra, $A=\bigcup_{j\geq 0}A_{j}$. For $j\geq 0$, define $R_{j}=A_{j}/A_{j-1}$, with $A_{-1}=\{0\}$. Let $R=\bigoplus_{j\geq 0}R_{j}$. Define a natural product on…
user608571
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When is $\text{gr}(V)\cong V$?

Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded space $\text{gr}V=\bigoplus_j V_j/{V_{j-1}}$,…
Eric
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Alternative definition of Koszul algebra by injective resolutions?

Let $A$ be a positively graded algebra. This means that $A$ is a $k$-algebra graded non-negatively and $A_0 \cong k \times \dots \times k$ such that each degree is finite-dimensional. From here, we say $A$ is Koszul if for all simple $A$-modules…
Molang
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Exterior algebra as a quotient is the same as exterior algebra as a vector subspace of the tensor algebra.

I am writing an expository paper, and in it I defined $\Lambda^k(V)$ as the subspace of alternating tensors of order $k$, i.e. as a vector subspace of $V^{\otimes^k}$, where $V$ is a $\mathbb{K}=\mathbb{C},\mathbb{R}$-linear vector space. I then…
Chris
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Reason to apply the Koszul sign rule everywhere in graded contexts

I'm copy-pasting this question I asked in MO that received no answer. 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…
Javi
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Tensor product being a coproduct in the category of anticommutative graded algebras

This is from Rotman's Advanced Modern Algebra, Part I. A positively graded algebra $A = \bigoplus_{p \geq 0} A_p$ over a commutative ring $R$ is anticommutative if $ab = (-1)^{pq}ba$ for $a \in A_p, b \in A_q$, alternating if $a^2 = 0$ for all $a…
Jxt921
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Explicit formula for the equalizer of coalgebras

The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is explicit and claims that, given $f,g:C\rightarrow D$…
Victor TC
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How can I show that the kernel of the cup product is the same as the image of this morphism?

Let $X$ be a topological space and $F$ a field. Suppose the singular cohomology of $X$, $H^n(X;F)$, is finitely generated in every dimension. In this case, by Künneth's formula: $$H^*(X\times X;F)\cong H^*(X;F)\otimes H^*(X;F),$$ we know that the…
YM78
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On the inductive (?) definition of the interior product derivation

Given a manifold $ M $ and a $ \mathrm TM $-valued $ (k + 1) $-form $ K $ on $ M $ one can define a $ k $-derivation $ i_K $ of $ \Omega(M) $ by some messy formula. The $ i_K $ should reduce to the interior product when $ K\in \Omega^0(M;\mathrm TM)…
GeometriaDifferenziale
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Canonical form of presentation matrix (over non-PID)

$\newcommand\im{\operatorname{im}}$Let $R = k[x_1,\ldots,x_n]$ as $\mathbf{Z}^n$-graded $k$-algebra, and $M$ be a finitely generated graded $R$-module. Then it is standard that there is a morphism $\phi\colon F \to G$ of finite rank free modules…
Bubaya
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Wedderburn theorem version for superalgebras

I am looking for an example of usage of wedderburn theorem version for superalgebras (which is a $\mathbb{Z}_2-$graded algebra). The theorem states that if $ A$ is finite dimensional $\mathbb{Z}_2-$graded simple associative algebra (it can be…
matan
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Is the Weyl algebra $A_1(\Bbb{C})$ graded?

For this question, the Weyl algebra is the algebra of differential operators on $\Bbb C$: $$A_1(\Bbb{C})=\Bbb{C}\langle x,\partial\rangle/(\partial x- x\partial -1)$$ Although for a lot of purposes it seems that people like to use the Bernstein…
Eric Nathan Stucky
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Computation of Associated Graded Module

I am trying to compute $\mathrm{gr}_m(P)$ where $m=\langle X,Y\rangle $ and $P=\langle X^2-Y^3\rangle$ in the power series ring $R=\mathbb C[[X,Y]]$ with the $m$-adic filtration and show that it is not a prime ideal. I know…
user6
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Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A differential graded algebra is graded algebra with chain…
Andrews
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