Let $R$ be a commutative (Noetherian) Cohen-Macaulay ring, and $f \in R[X]$ be monic. I guess that $R[X]/(f)$ is also Cohen-Macaulay. Is my hunch valid? Thanks for any help.
2026-03-25 09:27:39.1774430859
Is $R[X]/(f)$ Cohen-Macaulay if $R$ is so?
130 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in COMMUTATIVE-ALGEBRA
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- Extending a linear action to monomials of higher degree
- Tensor product commutes with infinite products
- Example of simple modules
- Describe explicitly a minimal free resolution
- Ideals of $k[[x,y]]$
- $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements
- There is no ring map $\mathbb C[x] \to \mathbb C[x]$ swapping the prime ideals $(x-1)$ and $(x)$
- Inclusions in tensor products
- Principal Ideal Ring which is not Integral
Related Questions in MODULES
- Idea to make tensor product of two module a module structure
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
- Example of simple modules
- $R$ a domain subset of a field $K$. $I\trianglelefteq R$, show $I$ is a projective $R$-module
- $S_3$ action on the splitting field of $\mathbb{Q}[x]/(x^3 - x - 1)$
- idempotent in quiver theory
- Isomorphism of irreducible R-modules
- projective module which is a submodule of a finitely generated free module
- Exercise 15.10 in Cox's Book (first part)
- direct sum of injective hull of two modules is equal to the injective hull of direct sum of those modules
Related Questions in NOETHERIAN
- In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element?
- Prove that the field $k(x)$ of rational functions over $k$ in the variable $x$ is not a finitely generated $k$-algebra.
- Ascending chain of proper submodules in a module all whose proper submodules are Noetherian
- Noetherian local domain of dimension one
- Dimension of Quotient of Noetherian local ring
- Is $\mathbb{Z}[\frac{1}{2}]$ Noetherian?
- Finitely generated modules over noetherian rings
- Simplicity of Noetherian $B$, $A \subseteq B\subseteq C$, where $A$ and $C$ are simple Noetherian domains
- Why noetherian ring satisfies the maximal condition?
- If M is a a left module over $M_n(D)$ where $D$ is a division ring, M is Noetherian iff Artinian
Related Questions in COHEN-MACAULAY
- Looking for easy example of 2-dimensional Noetherian domain which is not Cohen-Macaulay
- regular sequences: proving the geometric interpretation
- Completion and endomorphism ring of injective envelope
- Integral extension of a local ring is semilocal
- For a Cohen-Macaulay local ring grade and height are same
- Cohen-Macaulay ring without non-trivial idempotent is homomorphic image of Noetherian domain?
- Noetherian Catenary ring and Cohen-Macaulay ring
- Background of Commutative Algebra for Cohen-Macaulay orders and bibliography
- Is a quotient of ring of polynomials Cohen-Macaulay?
- Determining maximal Cohen-Macaulay modules over an invariant ring
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- 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?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- 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
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Proposition: If $S$ is CM, and $f \in S$ is a nonzerodivisor, then $S/(f)$ is CM.
Proof: Choosing a maximal ideal of $S/(f)$, we may assume $(S, m)$ is local. Then $\text{depth}(S/(f)) = \text{grade}(m/(f)) \ge \text{grade}(m) - 1 = \text{depth}(S) - 1$, while $\dim S/(f) \le \dim S - 1$.
Now, if $R$ is CM, then $R[x]$ is CM, and if $f$ is monic, then $f$ is a nonzerodivisor in $R[x]$.