Questions tagged [filtered-algebras]

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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 ...
Didier de Montblazon's user avatar
2 votes
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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 $$...
Chris's user avatar
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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: ...
Didier de Montblazon's user avatar
3 votes
1 answer
344 views

What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)

I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
Didier de Montblazon's user avatar
7 votes
1 answer
248 views

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 ...
Jake Wetlock's user avatar
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1 vote
0 answers
181 views

generators for a graded algebra

I have a graded commutative algebra over a field $K$ and I also know that it is not generated by the degree 1 elements. Somehow I could guess a set of generators, is there a criterion to check that ...
jack's user avatar
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2 votes
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Can one bound the minimal degree of a solution to an inhomogeneous linear equation?

Suppose $A = \cup_{i\in \mathbb N} A_{\leq i}$ is a filtered algebra and $M = \cup_{i\in \mathbb N} M_{\leq i}$ a filtered $A$-module. In my case, I know $A$ to be Noetherian and $M$ to be finitely ...
Theo Johnson-Freyd's user avatar
1 vote
1 answer
148 views

hamilton type Lie algebras

If n be positive integer and for an n-tuple of positive integers m=(m1,...,mn) then p(n,m) is graded and filtered subalgebra of W(n,m).p(n,m) is called non-alternating hamilton lie algebra over GF(2). ...
user118746's user avatar
7 votes
1 answer
506 views

A geometric characterization of Rees algebras in categories without Choice

Before asking my question, a caveat: The category theorist in me would like me to ask this question in more generality, but I will restrict my scope since what I'm really after is some geometric ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
2k 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} =...
MTS's user avatar
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8 votes
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Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them

I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I have in ...
Giovanni Moreno's user avatar
5 votes
1 answer
590 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} (...
MTS's user avatar
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4 votes
0 answers
205 views

Kahler differentials and the m-adic filtration

Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...
Ian Shipman's user avatar
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