Given a unital associative commutative algebra $A$ with over a ring/field K, with a filtration, i.e. collection of vector subspaces $0=F_0 \subset F_1 \subset ... \subset F_n = A$ such that $F_m \cdot F_n \subset F_{m+n}$, is it true that the associated graded algebra $\mathcal{G}(A)$ is isomorphic to $A$ as an algebra? Does the answer depend on whether K is ring/field?
2025-01-13 11:50:08.1736769008
Filtration on commutative algebra
66 Views Asked by Filip https://math.techqa.club/user/filip/detail AtRelated Questions in LINEAR-ALGEBRA
- Proving a set S is linearly dependent or independent
- An identity regarding linear operators and their adjoint between Hilbert spaces
- Show that $f(0)=f(-1)$ is a subspace
- Find the Jordan Normal From of a Matrix $A$
- Show CA=CB iff A=B
- Set of linear transformations which always produce a basis (generalising beyond $\mathbb{R}^2$)
- Linear Algebra minimal Polynomial
- Non-singularity of a matrix
- Finding a subspace such that a bilinear form is an inner product.
- Is the row space of a matrix (order n by m, m < n) of full column rank equal to $\mathbb{R}^m$?
Related Questions in ABSTRACT-ALGEBRA
- Projective Indecomposable modules of quiver algebra
- Binary relations for Cobb-Douglas
- Relations among these polynomials
- Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads
- Page 99 of Hindry's Arithmetics, follows from exact sequence that $\text{N}(IJ) = \text{N}(J)\text{card}(J/IJ)$?
- How to write the identity permutation as a product of transpositions
- Is $H$ a subgroup?
- $x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$
- Having some problems with understanding conics and graphing (eccentricity)
- Is this Cayley Diagram contradictory?
Related Questions in GRADED-RINGS
- Coalgebra differential on reduced symmetric algebra
- Associated graded ring isomorphic to polynomial ring implies regularity
- Is there an ideal, which is homogenous over $\overline F$, generated by elements of $F[X]$, but not by homogenous elements of $F[X]$?
- Description of Blow up via Ress Algebra
- For graded algebras over a field, does finite Krull dimension imply finite generation?
- Question about the classification theorem for finitely generated graded $F[t]$-modules
- $A$-module is free if and only if equation involving Hilbert-Poincaré series holds.
- Can a graded $k$-algebra have torsion over $k[\theta]$ for $\theta$ a non-zerodivisor?
- Principal open sets in graded rings
- Bijective correspondence of rational points in projective space
Related Questions in GRADED-ALGEBRAS
- Filtration on commutative algebra
- Banach space decomposition
- Equality of localization of homogeneous ideal by a variable $x_i$.
- On the $k$-vector space dimension of graded pieces of an Artinian $k$-algebra $k[x,y]/J$
- Definition of Homomorphism of $I$-Graded Vector Spaces
- Definition of Graded Algebra
- Hilbert-Samuel multiplicity of standard graded $k$-algebra which is an integral domain and $k$ is algebraically closed
- On an analogy of the highest generating degree and reduction of ideals
- Computation of Associated Graded Module
- Reason to apply the Koszul sign rule everywhere in graded contexts
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity