Questions tagged [associated-graded]
Questions about graded algebraic structures associated to a filtration.
13
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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 ...
- 1,238
1
vote
0
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114
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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 $B:=\{m_{1},\dots, m_{r}\}$ ...
- 522
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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 ...
- 2,806
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1
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99
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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 $ux \in R$. ...
- 2,921
3
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99
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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, ...
- 31
3
votes
1
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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 $M$, ...
- 26.8k
7
votes
1
answer
1k
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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} =...
- 8,139
5
votes
1
answer
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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} (...
- 8,139
10
votes
2
answers
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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 ...
- 49.4k
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 ...
- 5,993
8
votes
2
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814
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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 ...
- 49.4k
29
votes
4
answers
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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+...
