Questions tagged [associated-graded]

Questions about graded algebraic structures associated to a filtration.

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2
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0answers
72 views

Derivations of algebras graded by a group

Let $A$ be an algebra. A derivation of $A$ is a linear map $d:A \rightarrow A$ such that $d(ab)=d(a)b + a d(b)$ for $a, b \in A$. If $A$ is a $\mathbb{Z}-$graded algebra, where $\mathbb{Z}$ is the ...
5
votes
0answers
214 views

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 ...
7
votes
0answers
122 views

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
1answer
90 views

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
0answers
90 views

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
votes
1answer
193 views

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 $M$, ...
7
votes
1answer
1k views

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
1answer
544 views

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
votes
2answers
661 views

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
2answers
868 views

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 ...
7
votes
2answers
665 views

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 give my definitions before ...
25
votes
4answers
3k views

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+...