Questions tagged [associated-graded]
Questions about graded algebraic structures associated to a filtration.
13
questions
1
vote
0answers
85 views
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 $B:=\{m_{1},\dots, m_{r}\}$ ...
2
votes
0answers
93 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
265 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
134 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
98 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
94 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
194 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
563 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
723 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
943 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 ...
8
votes
2answers
741 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 ...
28
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+...
