# Questions tagged [cohen-macaulay]

A ring is called Cohen-Macaulay if its depth is equal to its dimension. More generally, a commutative ring is called Cohen-Macaulay if is Noetherian and all of its localizations at prime ideals are Cohen-Macaulay. In geometric terms, a scheme is called Cohen-Macaulay if it is locally Noetherian and its local ring at every point is Cohen–Macaulay.

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### Existence of ideal in Cohen-Macaulay ring, going modulo which still gives Cohen-Macaulay [closed]

Let $R$ be a local Cohen-Macaulay ring of dimension $\le 2$. Does there necessarily exist an ideal $J$ of $R$ such that $\sqrt J$ is a minimal prime ideal of $R$ and $R/J$ is Cohen-Macaulay ?
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### Determining maximal Cohen-Macaulay modules over an invariant ring

Suppose that $G$ is a finite small (i.e. reflection-free) subgroup of $\text{GL}(n,\mathbb{C})$ acting on $S := \mathbb{C}[x_1, \dots, x_n]$. Set $R := S^G$. By 5.20 Corollary of this, the maximal ...
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### Is a quotient of ring of polynomials Cohen-Macaulay? [closed]

Let $R = \mathbb{R}[x_1 , \ldots , x_n]$ be polynomial ring and $I \subset R$ be a principal ideal with $I = \langle f \rangle$. I know that $R$ is a CM ring. So my question is that: Is the quotient ...
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### Background of Commutative Algebra for Cohen-Macaulay orders and bibliography

In this semester, I'm doing a project in Algebra, and I would like take some advices and suggestions. To be more precise, I will study the Cohen-Macaulay Orders and modules in relation to the Krull ...
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### Noetherian Catenary ring and Cohen-Macaulay ring

Let $A$ be a Noetherian ring. $A$ is called catenary if for any two prime ideals $p$ and $q$ in $A$, $p\subset q$, every saturated chain of prime ideals starting at $p$ and ending at $q$ have same ...
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### Cohen-Macaulay ring without non-trivial idempotent is homomorphic image of Noetherian domain?

Let $R$ be a Cohen-Macaulay ring with no non-trivial idempotent element. Then is it true that there is a Noetherian domain $S$ such that $R\cong S/I$ for some ideal $I$ of $S$ ? If this is not true ...
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### For a Cohen-Macaulay local ring grade and height are same

In Matsumura's book ' Commutative ring theory' Theorem 17.4, page 135 its been proved that in a Noetheriam local ring $(A,\mathfrak m)$ for any proper ideal $I$, grade $I=$ ht$I$, where grade of an ...
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### Integral extension of a local ring is semilocal

Let $S\subseteq R$ be commutative rings with $1$. It is given that $S$ is local and $R$ is integral over $S$. I need to show that $R$ is semilocal that is $R$ has finitely many maximal ideals. It is ...
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### Completion and endomorphism ring of injective envelope

Let $(R, m ,k)$ be a commutative Noetherian local ring. We denote by $E$ the injective envelope of $k$ and by $R^～$ the $m$-adic completion of $R$. For any module $M$ over $R$, we let $M^*=Hom_R(M,E)$....
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### regular sequences: proving the geometric interpretation

I found the following discussion about the geometric interpretation of regular sequences very helpful: What is a geometric interpretation of regular sequences in various instances? However I tried to ...
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### Looking for easy example of 2-dimensional Noetherian domain which is not Cohen-Macaulay

I am looking for an easy example , with proof, of a 2-dimensional Noetherian domain which is not Cohen-Macaulay . I know and it is easy to prove that such an example doesn't exist in dimension 1. ...
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### Regular local ring if every maximal Cohen-Macaulay module is free

I have a problem like this "Let $R$ be a Cohen-Macaulay local ring, $\dim R=d$. Given that every maximal Cohen-Macaulay $R$-module is free, prove that $R$ is a regular local ring." My lecturer gave ...
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### n-th Syzygies over a Cohen-Macaulay ring are Maximal Cohen-Macaulay for n big enough

I am trying to proof the following: Let $R$ be a Cohen-Macaulay local ring with $\operatorname{dim}(R)=d$ and let $$0\to M\to F_{n-1}\to F_{n-2}\to\cdots\to F_1\to F_0$$ be an exact ...
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### A question on finitely generated $k$ algebra

According to this wikipidea link https://en.m.wikipedia.org/wiki/Cohen–Macaulay_ring : Let $R$ be a local ring which is finitely generated as a module over some regular local ring $A$ contained in $R$....
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### nonsplit exact sequence for non free module

Let $R$ be a Henselian Cohen-Macaulay (esp. noetherian) local ring and let $M$ be an indecomposable Cohen-Macaulay (esp. finitely generated) $R$-module which is not free. Why can I always find a ...
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### h-vector of an unmixed ideal

We know that: if a ring is Cohen-Macaulay, then it is unmixed. But the converse is not true. if a ring is Cohen-Macaulay, then its $h$-vector is positive. The converse is not true. Then I expect to ...
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### Structure of Cohen-Macaulay local algebra

I have the following question: Let $(A,\mathfrak m)$ be a Cohen-Macaulay local $k$-algebra, where $A/\mathfrak m=k$. Then there is a homomorphism $R=k[X_1,\cdots ,X_n]\rightarrow S$ so that $S$ is ...
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### Equivalent definition of maximal Cohen-Macaulay modules over a Gorenstein local ring

$\newcommand{\Ext}{\mathop{\rm Ext}\nolimits} \newcommand{\depth}{\mathop{\rm depth}\nolimits} \newcommand{\dim}{\mathop{\rm dim}\nolimits}$ A module $M$ is a maximal Cohen-Macaulay ...
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### The natural inclusion $R\varphi\to\mathrm{Hom}_R(C,C')$ induces the above isomorphism modulo $x$.

This is a part in proof of Theorem 3.3.4 in Cohen-Macaulay Rings, Bruns and Herzog. I don't understand it. Here is the Theorem: Theorem 3.3.4. Let $(R,m,k)$ be a Cohen-Macaulay local ring and let $C$ ...
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### Cohen-Macaulay Analytic Rings

Let $X \subset \mathbb{C}^n$ be an affine Cohen-Macaulay variety. I would like to know whether for every point $p \in X$, the local ring of germs of analytic functions on $X$ at $p$ is also Cohen-...
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### Is $R=K[\![x^3,x^2y,xy^2,y^3]\!]$ a Gorenstein ring? [duplicate]

Let $K$ be a field and $R=K[\![x^3,x^2y,xy^2,y^3]\!]$ the ring of formal power series. Is $R$ a Gorenstein ring? $R$ is Cohen-Macaulay of dimension 2. So, I have to check if $Ext^2_{K}(K,R)=K.$
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### Points in linear general position in a section of a curve in $\mathbb P^{n}(\mathbb C)$

Given a natural number $p\in \mathbb N \setminus{0}$, let $X\subseteq\mathbb P^{g+p+1}(\mathbb C)$ be a curve of genus $p$ and degree $2g+p+1$, and $Y\subseteq \mathbb P^{g+p}$ a general hyperplane ...
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### Subring of a Cohen-Macaulay ring [closed]

I try to find a obvious of this example: Subring of a Cohen-Macaulay ring is not Cohen-Macaulay. However, I got stuck. Could you help me say it in more detail? Thanks in advanced.
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### Embedded Components and Divisors

I'm trying to understand a comment in the introduction to chapter 18 of Eisenbud's Commutative algebra with a View Toward Algebraic Geometry: '..is the unimxedness theorem which explains, for example,...
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### Cohen-Macaulay and connected implies equidimensional?

I'm asking for a reality check. It seems to me that since Cohen-Macaulay rings are locally equidimensional, such a ring is either equidimensional or else disconnected (with different dimensions ...
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### Does a dualizing sheaf $\omega_X$ give rise to a dualizing module?

Let $X = \text{Proj } R$ be a projective equidimensional Cohen-Macaulay scheme, where $R$ is a finitely generated graded Cohen-Macaulay $\mathbb{C}$-algebra and $\mathcal{O}_X(1)$ is ample. Suppose ...
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### When is the tensor product of Cohen-Macaulay modules Cohen-Macaulay?

Let $M_1$ and $M_2$ be Cohen-Macaulay modules over a ring $R$. When is $M_1 \otimes_R M_2$ a Cohen-Macaulay module over $R$? That is the finite question I have. Motivation: I know that when $M$ is a ...
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### When is the Chow pull-back of a cycle its preimage?

Given a morphism (not necessarily flat) $f \colon X \rightarrow Y$ between smooth varieties over $\mathbb{C}$ say, we have a pull-back homomophism of graded rings $f^* \colon A^*(Y) \rightarrow A^*(X)$...
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### translation of “Der kanonische Modul …”

Do you know a note that is the translation of the following in English? J. HERZOG et al., "Der kanonische Modul eines Cohen-Macaulay-Rings," Lecture Notes in Mathematics No. 238, Springer-Verlag,...
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### Connection between saturated ideals an CM algebras.

Let $I$ be an homogenous ideal of the polynomial ring $K[x_1,\dots,x_n]$. Is there any relations between $I$ being saturated and $R/I$ being a Cohen-Macaulay?
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### When a graded ring is Cohen-Macaulay?

I am trying to solve exercise 19.10 from Eisenbud's Commutative Algebra. I want to show that if $R=k[x_0,...,x_n]/I$ is a graded ring, then $R$ is Cohen-Macaulay iff $R_{\mathfrak p}$ is Cohen-...
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### Confusion with arithmetically Cohen-Macaulay varieties

I'm a bit struck about this fact; I think it's really a silly question, but I'm not completely sure about it. Let $X\subseteq \mathbf{P}^m$ be a projective variety; choose the best hypotheses ...
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### How to check Cohen-Macaulayness?

Let $R=k[x,y,z]$. Consider the ideal $I=(x^2z^2,xyz,y^2z^4,y^4z^3,x^3y^5,x^4y^3)$. Is $R/I$ Cohen-Macaulay ? By definition it seems tough to solve this problem. Is there any other way to check this?
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### Cohen-Macaulayness inherits to quotient (Matsumura, CRT, Exercise 17.4)

A well-known theorem in commutative algebra states the fact that if $R$ is a Cohen-Macaulay ring, and $a_1,...,a_r$ is an $R$-sequence, then $R/I$ is Cohen-Macaulay, where $I=(a_1,...,a_r)$. Now, ...
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### A local Cohen-Macaulay ring [closed]

It may be a simple question, but I am stuck at: If $k$ is a field why $k[x^3,x^2y,xy^2,y^3]$ is Cohen-Macaulay when localized at the maximal ideal $(x^3,x^2y,xy^2,y^3)$? Any help? Thanks!
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### Systems of Parameters are exactly $R$-sequences
If $(R,m)$ is a local Cohen-Macaulay ring, it is well-known that each system of parameters is an $R$-sequence. Is any $R$-sequence (in a Cohen-Macaulay ring) a system of parameters? I am aware ...