# Questions tagged [associated-graded]

Questions about graded algebraic structures associated to a filtration.

10 questions
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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 ...
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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 $ux \in R$. ...
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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, ...
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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 $M$, ...
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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 ...
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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 ...
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+...