Questions tagged [graded-algebras]

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

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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 $\...
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Reference for: a nontrivially graded integral domain is never quasi-local

Let $\Gamma$ be a torsionless grading monoid and $R=\bigoplus_{\alpha\in\Gamma}R_\alpha$ be a $\Gamma$-graded integral domain. I'm interested in the following result: if $R$ is nontrivially graded,...
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Graded Rings , divided polynomial algebra

I just read about the notion of a divided polynomial algebra, which is defined as follows: Consider the elements $y^{(i)}=y^i/i!, i\geq 0$ in the polynomial ring $\mathbb{Q}[y]$. They satisfy $y^{(i)}...
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Hilbert functions for affine vs projective varieties and filtered vs graded algebras.

I'm somewhat confused about how to define Hilbert functions for affine varieties and for filtered rings in a compatible way. I'm familiar with how they are defined for projective varieties: Let $X$ be ...
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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 $\bigwedge(x_1,\dots, x_n)$ ...
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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 ...
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How to extend to a unique derivation on the graded tensor algebra $TV$?

If $V$ is a finitely generated graded module and $TV$ its graded tensor algebra then, Any degree $k$ linear map $V \to TV$ extends to a unique derivation of $TV$. I found this in page 45 of the book "...
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Why is this sequence of graded algebras wrong?

I'm studying Hilbert sequences, and when trying to understand the proof of $h_A(t)= \frac{1}{(1-t)^m}$ when $A= k[x_1,\dots,x_m]$ ($m \geq 2)$, I'm kinda stuck. I can prove it using the standard ...
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Suspension Operator on Graded Algebras

Given a morphism of GAs $f:A\to B$ of degree $-k$, that is, $f(A_n)\subset B_{n-k}$, I want to understand the sign conventions of commutation with the suspension operator. That is, if $s:A\to A$ is ...
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Symbolic Rees Algebra of an ideal in a Noetherian excellent ring

For an ideal $I$ in a commutative Noetherian ring $R$ and integer $n\ge 0$, the $n$-th symbolic power of $I$ is define as $I^{(n)}:=\cap_{P\in Ass(R/I)} \phi_P^{-1} (I^nR_P)$ , where $\phi_P : R\to ...
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On an analogy of the highest generating degree and reduction of ideals

Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$ . Let $J \subseteq \mathfrak m$ be a homogenous ideal with $\sqrt J=\mathfrak m$ i.e. $\mathfrak m^n \subseteq J$ for some integer $n\ge 1$. Let $a\ge ...
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Hilbert-Samuel multiplicity of standard graded $k$-algebra which is an integral domain and $k$ is algebraically closed

Let $R=\oplus_{i\ge 0} R_i $ be a graded domain such that $R_0=k$ is an algebraically closed field, $R$ is finitely generated $k$-algebra and $R=k[R_1]$. Let $d=\dim R>0$. Let $\mathfrak m=\oplus_{...
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Definition of Graded Algebra

I attempt to understand the definition of graded algebra from An Introduction to Manifolds by Loring Tu (page no. 30). Below I quote the definition supplied in the book. An algebra $A$ over a ...
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Definition of Homomorphism of $I$-Graded Vector Spaces

I am trying to understand the definition of homomorphism of $I$-graded vector spaces from the Wikipedia page, where $I$ is any set. According to this web page, a homomorphism of two $I$-graded vector ...
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On the $k$-vector space dimension of graded pieces of an Artinian $k$-algebra $k[x,y]/J$

Let $R=k[x,y]$ be a polynomial ring in two variables over an infinite field $k$. Let $\mathfrak m=(x,y)$. Let $J$ be a homogeneous ideal whose radical is $\mathfrak m$. Consider the standard grading ...
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Equality of localization of homogeneous ideal by a variable $x_i$.

Consider the polynomial ring $S = k[x_0,\cdots, x_n]$ of $n+1$ many variables, where $k$ is a field. Let $I$ and $J$ be homogeneous ideals in $S$. Consider the localization at the variable $x_i$: $$I_{...
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Banach space decomposition

A graded $C^*$ algebra is a $C^*$ algbebra $A$ equipped with an order two $*$ automorhism $\phi_A$. $A$ can be decomposed into two eigenspaces for $\phi_A$,$A=A_0\oplus A_1$,where $A_0=\{a\in A:\...
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On the Krull dimension of a quotient algebra

Given a graded $\mathbb{C}$-algebra $R$, which is finitely generated and of Krull dimension $n$. Let $\phi_1,\ldots,\phi_n$ homogeneous elements in $R$ such that $R$ is finitely generated as a module ...
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Filtration on commutative algebra

Given a unital associative commutative algebra $A$ with over a ring/field K, with a filtration, i.e. collection of vector subspaces $0=F_0 \subset F_1 \subset ... \subset F_n = A$ such that $F_m \cdot ...
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Definition of $\mathfrak{g}$-differential graded algebra

I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003). In page $45$, definition $5.2$, author introduce the notion of $\mathfrak{g}$-...
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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 ...
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From Sullivan algebra to Poincare duality algebra

I have a Poincare duality algebra $H^*$ and let $V$ be a graded vector space,with $V=V^2\oplus V^3\oplus..$ generating the minimal Sullivan algebra $(\wedge V,d)$. My question is how I must ...
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What (mathematical) field does a (physics) superfield belong?

A superfield, $\phi(x,\theta)$, is define by a Grassman-even (i.e. commuting) function of a set of commuting variables ($x^\mu$) and a set of Grassman variables ($\theta^\alpha$ where $\theta^\alpha\...
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Degree part of exterior algebra

I have a graded vector space $V$ and exterior algebra $\bigwedge V$.Suppose further that $V^0=V^1=0$. I don't understand why $(\wedge V)^2=V^2,(\wedge V)^3=V^3$ and $(\wedge V)^4=P^2V^2$. notation: $...
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How can show the exisrence of normal vector for a graded algebra

For an algebra $A$ of dimension $n$ equuiped with an inner product, for subalgebras $Ã$ of codimention 1, we can find a unit normal vector? Can one write $A=Ã\oplus Ā$? Is there a vector in $Ā$ ...
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Action of the multiplicative group induced by a grading

I found in several different fonts, (in the first section of this (https://arxiv.org/abs/alg-geom/9405004) paper, or even in this answer (https://mathoverflow.net/questions/212960/intuition-behind-...
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What is meaning of even (odd) operator?

Let $ V^{\bullet}=\bigoplus_{i\in \mathbb{Z}} V^{i} $ is graded vector space. What is the meaning of even(odd) operator on this graded vector space? It related to shifts the grading by even(odd) ...
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Prove that reduced symmetric algebra is graded

I've tried to prove that the reduced symmetric algebra is graded as an algebra and a coalgebra. $V$ is a vector space on a field of characteristic $p$. $T(V)$ is the tensor algebra. $s(V):=\frac{T(V)}...
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How can we define a $\mathbb{Z}$-grading on sl2

Let $sl_{2}$ be $2 \times 2$ traceless matrices over field $K$ of characteristic $0$. How can we define a $\mathbb{Z}$-grading on $sl_{2}$? Let consider $h ,e ,f$ as follows respectively: \begin{...
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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 ...
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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 ...
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Induced map between associated graded algebra?

I have just learned how one can take an filtered algebra and get an associated graded algebra, (here's the wikipedia article for reference). I have also seen in various places, (such as this question),...
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naturally graded algebras

I found a good explanation on graded algebras Understanding of graded algebra But I am confused about difference between "naturally graded" and "graded". May you please clarify the notion of ...
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Categorical product of non-unital associative differential graded coalgebras

Given two non-unital associative dg coalgebras $D$ and $C$, I want to give an explicit construction of the product $C\prod D$, this may follow from the dual construction (coproduct of non-unital ...
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Rational homotopy type of a Lie group

I have to show that Lie groups have the rational homotopy type of a wedge of spheres. Unfortunately, my knowledge of Sullivan models is a bit shaky. The proof I come up with shows that they have the ...
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59 views

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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Graded $C^*$ algebra homomorphism

I am pretty sure I have some definition wrong. But I do not see where. Here is the context: Consider the $C^*$ algebra of continuously compactly supported functions $\Bbb R$ into $\Bbb C$. $$C_0(\...
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The twisted tensor product $BA\otimes_{\tau} A$ as the non-unital Hochschild complex

The twisting universal morphism $\tau: BA\rightarrow A$ induces a differential $\partial_{\tau}$ on $BA\otimes_{\tau}A$, we have: $$\partial_{\tau}(x\otimes y)=\partial x\otimes y+(-1)^{\lvert x\rvert}...
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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=\...
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quasi isomorphism of two dg algebras

I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is ...
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Irreducible highest weight representations as a graded algebra

Let $L$ be a semisimple Lie algebra and let $V(\lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $\lambda$. How can we view the sum \begin{align*} \oplus_{n\in\mathbb{N}...
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Trying to understand more about polynomials in noncommuting variables.

I'm working on a project and I will need to look at some theory regarding polynomials in noncommuting variables. I have a 1st ed. copy of Rotman's Advanced Modern Algebra, which I've been getting some ...
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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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When a homogeneous ideal is written as a product of two ideals, then each of two ideals is homogeneous?

I know that in a graded domain, if a homogeneous element is written as a product of two elements, then each of two elements is also homogeneous. That is, the set of all the homogeneous elements of ...
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Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a graded (non-commutative) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) Noetherian as a ring. In the ...
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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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Direct sum and the inclusion property

Let $R$ be a ring and if $R= \bigoplus R_i$ as additive subgroups for each $i \in I$ where $I $ is a finite group Is that implies $R_i R_j $ must be contained in $R_p$ for any $p \in I$ ?? In ...
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On graded fields

Tom Marly here noted that if R is a graded field then $R$ is concentrated in degree 0, i.e., $R=R_0$ and $R_n =0$ for all $n \neq 0$. Is this proposition mentioned in any book or paper ?? ...
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Is the algebra dual to a graded coalgebra graded?

Given a graded coalgebra $C = \bigoplus_{n\geq 0} C_n $ with coproduct $$\Delta : C_n \to \bigoplus_{i=0}^n C_i\otimes C_{n-i} $$ must we have that the dual $C^* = \bigotimes_{n\geq 0}C_n^*$ is a ...