# Questions tagged [associated-graded]

Questions about graded algebraic structures associated to a filtration.

19
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### Associated graded algebras and symmetric Frobenius algebras

Let $A$ be a filtered algebra and let $G$ be its associated graded algebra. As discussed in this question, if $G$ is Frobenius, then $A$ is also Frobenius.
If $G$ is a symmetric Frobenius algebra, ...

3
votes

0
answers

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### Semi-direct products and associated graded Lie algebras

Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...

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### Filtrations and Koszul algebras

I was looking at this question and asked my self the following:
Let $A$ be graded algebra, which is also an $\mathbb{N}_0$-filtered algebra. If its associated
graded algebra $\mathrm{gr}(A)$ is ...

2
votes

0
answers

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### Is there a coalgebraic definition of filtered algebras?

If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define
$$...

2
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0
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### Is the associated algebra of a quadratic filtered algebra again quadratic

Let $A$ be a filtered quadratic algebra. Let $G(A)$ be the associated graded algebra. Will $G(A)$ again be a quadratic algebra or can higher relations appear when passing to the graded seeting?
EDIT: ...

7
votes

1
answer

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### Properties of a filtered algebra that can be concluded from properties of its associated graded algebra

Let $F$ be a filtered algebra and let $G$ be its associated graded algebra. Some examples of properties of $F$ that can be concluded from properties of $G$:
(A) The dimension of $F$ is equal to the ...

4
votes

1
answer

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### Quotients and associated graded

$\DeclareMathOperator\gr{gr}$Let $A = \cup_{i=0}^\infty F_i A$ be a filtered commutative ring, $I \subseteq A$ an ideal. Then we have a canonical surjection
$$ \gr(A)/\gr(I) \to \gr(A/I).$$
Under what ...

1
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0
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### Surjection from finite rank free $R$-module to finitely generated $R$-module and basis associated to generator set

Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ ...

5
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### Does the associated graded functor take products of filtered k-coalgebras to graded k-coalgebras?

Let's suppose we have two noncommutative graded k-coalgebras $C_1$ and $C_2$ with respective admissible filtrations (i.e $F_{0}C_i=0$ and $\mathrm{colim}_k F_kC_i=C_i$), I would like to know if there ...

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### Standard reference/name for "initial ideals $\Leftrightarrow$ associated graded rings"

Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...

0
votes

1
answer

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### Adic filtration and integral closure

Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.
Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. ...

3
votes

0
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### Gelfand Kirillov dimension and associated graded algebras

Given a filtered module $M$ over a Lie algebra $\mathfrak{g}$, can one describe the Gelfand Kirillov dimension of $M$ in terms of Gelfand Kirillov dimension of its associated graded $grM$? If so, ...

3
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1
answer

203
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### Liftability of a submodule from an associated graded module

Let $k$ be a field, $A$ a $k$-algebra (probably noncommutative), and $M$ an $A$-module that's finite-dimensional as a vector space over $k$.
Let $Gr(M;k)$ denote the set of all $k$-subspaces of $M$, ...

8
votes

1
answer

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### Generators of associated graded algebra

Suppose that $A = \bigcup_{n=0}^{\infty} A_n$ is a filtered algebra over a field $k$. The associated graded algebra is $\mathrm{gr} A = \bigoplus_{n=0}^{\infty} A_n/A_{n-1}$, where we define $A_{-1} =...

5
votes

1
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### Associated graded of filtered module-algebra over a Hopf algebra

I ran across the following statement in a paper, and it seems fishy to me:
Lemma: If $A$ is any Hopf algebra, and if $U$ is an $\mathbb{N}_0$-filtered $A$-module algebra, then $U$ and $\mathrm{gr} (...

10
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2
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### An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal?

Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an augmentation ideal $I = \ker \epsilon$, which is a ...

9
votes

2
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### Associated graded and flatness

Let $M$ be a filtered module over a filtered algebra $A$, and suppose $gr(M)$ is flat over $gr(A)$, where $gr$ means the associated graded module and algebra, respectively.
What can one say in ...

9
votes

2
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### If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf?

Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll ...

33
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4
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### What is the universal property of associated graded?

Given a filtered vector space (or module over a ring) $0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V$, you can construct the associated graded vector space $\mathrm{gr}\left(V\right)=\oplus_{i}V_{i+...