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Questions tagged [graded-algebras]

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

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Finding a good homogeneous coordinate ring for a smooth projective variety

TL;DR Given a smooth projective variety $X\subseteq \mathbb{P}^m$ (I guess this means that the homogeneous coordinate ring $S(X)$ is regular whenever localizing at a non-maximal graded prime ideal). ...
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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 ...
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Non trivial massey triple product [closed]

Suppose that $(A, d)$ is graded commutative algebra with three generators $a, b, c$, each of degree $1$ and $da = db =0$, $dc = ab$. Then, how can I construct a non-trivial Massey product? What ...
trying's user avatar
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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\...
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Homogeneous elements of graded algebras as polynomials of algebra generators

Let $A = \oplus_{i > 0} A_{i}$ be a positively graded $k-$algebra and let $\mathcal{B} = \left\{b_{1}, \ldots, b_{n}\right\}$ be a generating set of $A$. Let $f$ be a homogeneous element in $A_{i}$ ...
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homogeneous subalgebra of a Lie superalgebra

In the case of Lie superalgebras which are defined as $L=L_{\bar{0}} \oplus L_{\bar{1}}$, I am a bit confused about the term "homogeneous subalgebra". Does it mean that the subalgebra which ...
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How does the following representation of $\mathbb{C}l(3)$ decompose into irreducibles?

Consider the three dimensional complex Clifford algebra $\mathbb{C}l(3)$ and the following representation $$S=\text{Span}\{|0\rangle,c^\dagger|0\rangle,\gamma^3|0\rangle,\gamma^3c^\dagger|0\rangle\},$$...
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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 ...
Molang's user avatar
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Universal Property of tensor product of $\mathbb Z_2$-graded algebras.

If $A$ and $B$ are two $k$ algebra's with a $\mathbb{Z}_2$-grading then I know that a $\mathbb{Z}_2$-graded structure can be defined on their tensor product. One does this by altering the ...
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Literature for $G$-graded rings [closed]

I am searching for a book which contains very basic statements (with proofs) of $G-$ (respectively $\mathbb{Z}-$) graded rings and ideals. (For example that $R/I$ is a $G$-graded ring if $I$ is a ...
Laak's user avatar
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Twisting a graded algebra by an automorphism (Transitivity)

Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra ...
Lumiel's user avatar
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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 $\mathbb{Z}_2-$...
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Possible structures of tensor product between two graded anticommutative algebras

If we have two anticommutative graded rings $M=\bigoplus_{k\geq 0} M_k$, $N=\bigoplus_{l\geq 0} N_l$ (anticommutative meaning $ab=(-1)^{deg(a)deg(b)}ba$ for $M$ and the same for $N$) such that each $...
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Quotient of an exterior algebra over $\mathbb{F}_2$

Let $E$ be the exterior algebra over $\mathbb{F}_2$ on countably infinitely many generators $x_0,x_1,x_2\ldots$ as discussed in Proposition 1.5 of the following paper. $E$ is a commutative unital $\...
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Exterior Algebra as a quotient is or is not compatible with Exterior Algebra as a vector subspace space with $\det$ convention of wedge product?

Ok, so I asked a similar question here, but after the first response it quickly devolved into an unfocused mess. I now believe I can adequately formulate my confusion. Let $V$ be $\mathbb{K}=\mathbb{C}...
Chris's user avatar
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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}=\...
Chris's user avatar
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Quotient of Graded Algebra is Graded

Let $A$ be a graded algebra over a commutative ring, and $I$ a two sided ideal of $A$ which is not necessarily graded. We can take the quotient $A/I$ and obtain an algebra, however I am struggling to ...
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Symmetric Algebra and Graded Symmetric Algebra of a Graded Module

Let $k$ be a commutative ring. Let $M=\oplus_iM_i$ be a $\mathbb Z$-graded $k$-module. There is the symmetry algebra of $M$, defined as $$\mathrm{Sym}^{\bullet}(M):=T^{\bullet}(M)/\langle x\otimes y-y\...
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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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Graded algebras in Tu's book "An introduction to manifolds"

I am reading Tu's book "An introduction to Manifolds". Specifically, I am reading about the the exterior algebra of multicovectors (subchapter 1.3) and I happened to come across a definition ...
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The dimension of the graded algebra of a commutative ring. [duplicate]

Here is the question I am trying to understand it is solution: (Poincare series of a graded algebra) Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all ...
Intuition's user avatar
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Poincare Series of a graded algebra (revisited)

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
Intuition's user avatar
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Poincare series of a graded algebra

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
Intuition's user avatar
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163 views

Sheaf of graded algebras

Let $U\subseteq \mathbb{R}^n$ be open and $W:=\oplus_{i\in \mathbb{Z}}W_i$ be a real graded vector space with $dim W_i<\infty$ for all $i$ and $W_0=\{0\}$. I would like to find a reasonable sheaf $\...
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A basic question about the big bracket and Lie algebra

Definition: The big bracket is the graded Lie algebra structure on algebraic functions on $V\oplus V^*$ defined as follows: $\bullet$ For $u,v\in B^{-1}\oplus B^{-1}=k\oplus V\oplus V^*$: $$[u,v]=\...
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A $k$-linear graded automorphism of $A$ is not necessarily an automorphism of algebra.

Hi I'm studying the paper "Twisted Graded Algebras and Equivalences of Graded Categories" by J.J Zhang. At one point the statement is made "Note that a $k$-linear graded automorphism of ...
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Rees algebra is finitely generated if associated graded algebra is

$ \newcommand{\cn}{\colon} \newcommand{\<}{\leqslant} \newcommand{\>}{\geqslant} \newcommand{\ss}{\subset} \newcommand{\k}{\mathrm{k}} \newcommand{\gr}{\mathrm{gr}} \newcommand{\R}{\mathrm{R}} \...
Dmitry's user avatar
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Algebra is noetherian if associated graded algebra is noetherian

$ \newcommand{\cn}{\colon} \newcommand{\<}{\leqslant} \newcommand{\>}{\geqslant} \newcommand{\ss}{\subset} \newcommand{\gr}{\mathrm{gr}} $ Let $A$ be associative commutative algebra with unity ...
Dmitry's user avatar
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Dimension one graded algebra $R$ is a domain then $l(R_n)\leq 1$ for all $n$.

Let $R$ be a Noetherian positively graded $k$-algebra of dimension $1$, where $k$ is an algebraically closed field. If length $l(R_n)=H (R, n) > 1$ for some $n$, show $R$ is not a domain. I assume $...
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Definition of a Graded Algebra and $R_0$

Below we have a definition of a graded $k$-algebra where $k$ is a field. I have a few questions. First, looking it up, there seems to be some ambiguity as to what $R$ is a direct sum of the $R_n$ as. ...
Thomas Anton's user avatar
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$Z_2$ action yields decomposition into a direct sum

If I have a $Z_2$ (group with two elements) action on a $C^*$-algebra $A$, i.e. $A$ is graded by the definition of Ralf Meyer for example, then how may I decompose $A$ into a direct sum $A_0\oplus A_1$...
Roland's user avatar
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1 answer
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Difference between the graded symmetric algebra and the exterior algebra

The graded symmetric algebra is the tensor algebra modulo the supercommutator $xy-(-1)^{|x||y|}yx$ where $|x|$ is the grade of $x$. The Wikipedia article says that this generalizes both the symmetric ...
Lave Cave's user avatar
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Graded algebras and symbols

Take $A$ a commutative unitary $\mathbb{C}$-algebra and take $A_0\subset A_1\subset...$ a filtration on $A$. If $$GrA=\bigoplus \frac{A_n}{A_{n-1}}$$with $A_{-1}=0$, is the associated graded algebra. ...
wood's user avatar
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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
1 answer
127 views

Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. If $S_+$ is finitely generated, then $S$ is a finitely generated graded $A$-algebra?

Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. Suppose $S_+= (a_1, \dots, a_n)$ where $a_i$ are homogeneous of positive degree. I want to show that $S$ is a finitely generated graded $A$-...
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About the structure of a Hopf algebra on universal enveloping algebras of Lie algebras

We know that the universal enveloping algebra construction provides a functor from Lie algebras to cocommutative Hopf algebras which is left adjoint to the primitive functor. Furthermore, if we ...
Nil's user avatar
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Ideals in localisation of graded ring

Let $k$ be a field and $I$ a homogeneous ideal of $k[x_1,...,x_n]$. Consider the graded ring $R=k[x_1,...,x_n]/I$ with unique homogeneous maximal ideal $\mathfrak m=(x_1,...,x_n)R.$ Let $J$ be a ...
uno's user avatar
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3 votes
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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 ...
Eric Nathan Stucky's user avatar
1 vote
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Structure of commutative finite dimensional strongly $\mathbb{Z}_2$ graded algebra

Let $k$ be an algebraic closed field with $\mathrm{char}{k}=0$. Assume $A$ is a finite dimensional commutative algebra grading by $\mathbb{Z}_2$, "strongly" means that $A_1A_1=A_0$. Claim:$A\...
Mic's user avatar
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1 answer
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Sum and multiplication on graded algebras

Let $A$ be an algebra over a field $\mathbb{K}$ and $G$ a commutative group under addition. This algebra is called a $G$-graded if there exists a family $\{A_{g}\}_{g\in G}$ of vector subspaces of $\...
JustWannaKnow's user avatar
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$\Omega^*(M)$ as a graded algebra over the ring of smooth functions. Is there a notion of basis?

In Introduction to Smooth Manifolds by John M. Lee p.360, he defines $\Omega^*(M) =\displaystyle\bigoplus_{k=0}^n\Omega^k(M)$, the differential forms on a manifold $M$, as being an associative, ...
horned-sphere's user avatar
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Is there an equivalence between a graded vector space and its degree-shifted counterpart?

I am studying graded vector spaces and I have a simple question. Let me denote by $\mathbb Z_2=\mathbb Z\text{mod}2=\{-1,0,+1\}$. Now let's perform a shift by an integer $k$ and get the set $\mathbb ...
Cyclops's user avatar
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Graded Leibniz algebras induced by some filtrations

Let consider a filtration of Leibniz algebra $L$ as $L_0=L \supseteq L_1 \supseteq \dots L_n \supseteq \dots$, with the property that $L_i L_j \subseteq L_{i+j}$. If we consider the grading $gr L = \...
Nil's user avatar
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Graded commutator defines graded derivation?

${}$ I'm having difficulties understanding what to do on a problem for my exercise class. The problem is as follows: ${}$ Consider a $\mathbb{Z}_2$ graded algebra $\mathcal{A} = \mathcal{A}_0 \oplus \...
Octavius's user avatar
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What is the (co)homology of a free (graded) Lie algebra?

In characteristic $0$, what is the Chevalley-Eilenberg (co)homology of a free (graded) Lie algebra? Not the definition, but $H^i =$ ??
Jim Stasheff's user avatar
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2 answers
144 views

subring not graded

Definition 1. A pair $(A, \{A_{i}\}_{i\in\mathbb{Z}})$ is a graded ring if $A$ is a commutative ring with unit, and $\{A_{i}\}_{i\in\mathbb{Z}}$ is a family of $\mathbb{Z}$-submodules of $A$, such ...
K. Y.'s user avatar
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What is the proof of graded Jacobi identity?

$\newcommand{\parcir}[2]{\frac{\partial^R #1}{\partial #2}}$ $\newcommand{\parcil}[2]{\frac{\partial^L #1}{\partial #2}}$ $\newcommand{\vprcir}[2]{\frac{\overleftarrow{\partial} #1}{\partial #2}}$ $\...
Angel Octavio Parada Flores's user avatar
2 votes
1 answer
145 views

If the associated graded algebra of a filtered algebra is commutative, is the condition below satisfied by the filtration?

Suppose we're given a filtered algebra $A$ over a field $k$ with filtration $F_{\bullet}A$ over the subspaces of $A$: $$\{0\}\subseteq F_{0}A\subseteq\cdots\subseteq F_{i}A\subseteq \cdots\subseteq A,$...
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local versus graded free resolutions

I'm currently trying to learn about syzygies. Most material is written in the context of graded rings and/or graded modules but I'm interested in a specific question about local rings. Hence I need to ...
amateur's user avatar
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Where does the dualization of maps sign come from in graded vector spaces?

In the book Rational Homotopy Theory of Félix, Halperin and Thomas they state the following. Given linear maps $f:V\rightarrow W$ and $g:W'\rightarrow V'$ between graded vector spaces then we have $\...
Ivan Burbano's user avatar
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