# Questions tagged [graded-algebras]

The tag has no usage guidance.

27 questions
1answer
18 views

### 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{...
0answers
48 views

### 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 ...
1answer
74 views

### 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 ...
0answers
54 views

### Pushout of cdgas and the quotient by an acyclic ideal.

Let $A$ be a cdga over $\mathbb{R}$, $B\subset A$ a dg-subalgebra and $I\subset B$ a dg-ideal. Consider the pushout of cdga's $\require{AMScd}$ \begin{CD} B @>\pi>> B/I\\ @V i V V @VV ...
1answer
33 views

### 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),...
1answer
43 views

### 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 ...
0answers
36 views

### 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 ...
1answer
31 views

### 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 ...
1answer
43 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 $\Lambda_\mathbb{Z}[x]$ ...
1answer
25 views

1answer
63 views

0answers
35 views

### 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 ...
0answers
41 views

### 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 ...
0answers
23 views

### Can you have supersymmetry inside $E_8$?

$E_8$ which is of size 248 has an SO(16) sub-algebra (120 generators, M) as well as 128 spinor generators (Q). The M's can be thought of as the even part of the algebra and the Q's as the odd part. ...
1answer
38 views

### 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 ...
1answer
65 views

### 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, ...
1answer
78 views

### 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 ...
2answers
76 views

### 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 ?? ...
0answers
97 views

### 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 ...
1answer
75 views

### What is the notation $\mathcal{A}_\infty$ for a sheaf of graded $\mathcal{O}_X$-algebras mean?

Today I encountered Hartshorne's condition $(\dagger)$ for quasi-coherent sheaves of algebras. This states that gives a graded $\mathcal{O}_X$-module $\mathcal{A}$ which has the structure of a graded ...
0answers
54 views

### The associated graded of a filtered coalgebra

Given a coalgebra $C$ with a filtration $F$ such that $\Delta(F^n C)\subset \sum_{i=0}^n F^i C\otimes F^{n-i} C$, how does the coproduct manifest in the associated graded? Do we get something to the ...
0answers
30 views

### Morphisms between the graded algebras of forms of vector bundles?

For a vector bundle $\pi_A:A\longrightarrow M$ and a non-negative integer $k$ let us define \Omega^k(A):=\left\{\begin{array}{clc} C^\infty(M) &\textrm{if}& k=0\\ \displaystyle\mathsf{Hom}_{...