Is $M=\mathbb{C}[\partial_x,\partial_y]$ an injective $\mathbb{C}[x,y]$ module? The action is given by differentiation (swapping the role of $x$ and $\partial_x$). It is injective in the one variable case as the not finitely generated $M$ is always divisible and we are in the PID case. But I don’t know the answer when the number of variable is greater than 1.
2025-01-13 05:48:28.1736747308
Is the delta module injective?
53 Views Asked by FunctionOfX https://math.techqa.club/user/functionofx/detail AtRelated Questions in MODULES
- Proper essential extensions
- If module $M = N + mM$, then why is $m(M/N) = M/N$?
- On the invariance of basis number
- Geometric intuition for coherent rings, modules, and sheaves
- Finitely generated iff direct limits of subobjects are bounded by subobjects
- Finding highest weight of a $gl(3)$ submodule of a $gl(4)$-module
- A simple Congruence Equation
- On the $R$-linear independence of free $R$-module
- Do all submodules of $R^n$ have a complement in $R^n$?
- Formulas give irreducible representation, $SL(2, \mathbb{C})$.
Related Questions in HOMOLOGICAL-ALGEBRA
- Are there right-deformations for abelian sheaves?
- Obstruction to be conjugated by an automorphism for subgroups of an abelian group
- Galois cohomologies of an elliptic curve
- Generalization for Leray Hirsch theorem for Principal $G$-bundle
- Projective resolutions of modules in a short exact sequence (Dummit & Foote Proposition 17.1.7)
- Quasi isomorphism and pullback
- Can we produce a long exact sequence in cohomology from more than just short exact sequences?
- Affine smooth variety has only trivial first-order deformations
- What is an additive functor?
- Compute $\text{Tor}^R_n(M,M)$ in the following specific case.
Related Questions in INJECTIVE-MODULE
- Proper essential extensions
- Injectives realized as limits in some appropriate category
- Direct sums of modules versus products
- If $A$ and $B$ are summands of injective module $Q$ then $A+B$ is a sumand of $Q$.
- For this function, is f is injective?
- Show by an explicit example that the function $h$ given is not injective and is surjective
- Can you prove that over a division ring , every module is both projective and injective?
- $\mathrm{Ann}(I \cap J) = \mathrm{Ann}(I) +\mathrm{Ann}(J)$ if $R$ is self-injective
- Is it true that taking injective hull commutes with the tensor product?
- Prove that every $\mathbb{Z}/6\mathbb{Z}$-module is projective and injective. Find a $\mathbb{Z}/4\mathbb{Z}$-module that is neither.
Related Questions in D-MODULES
- Characteristic Variety of the Principal Symbol solves PDE system?
- Bernstein polynomial of $x_1^2+\cdots+x_n^2$
- Why does this identity hold for D-modules?
- Cyclic vector of local system
- Relationship between the Fourier transformation of functions and that of D-modules
- Need help with Fourier transforms. Is the following true?
- Localization of a Weyl-Algebra module
- “D-module” or “$D$-module”?
- Algebraic cohomology $R\Gamma_{[Z]}(\cdot)$ for locally closed analytic set $Z$
- Beilinson-Bernstein localization, equivariant modules
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