# 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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### Finitely generated torsion-free indecomposable modules over one-dimensional complete local domains are isomorphic to ideal?

Let $R$ be a complete local domain of dimension $1$. Let $M$ be a finitely generated torsion-free indecomposable $R$-module. Then, must $M$ be isomorphic to an ideal of $R$? Also clearly, $R$ embeds ...
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### Going mod an element which annihilates a prime ideal (Associated primes)

Suppose $R$ is a Cohen-Macaulay local ring and $P=ann(x)$, an associated prime of $R$. Now consider the ring $S=R/(x)$. Will $PS$ (extension of $P$ to $S$) still consists of zero divisors of $S$? Or ...
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### Tensor product of field with Noetherian ring is Cohen-Macaulay

Let $R$ be a Noetherian ring, free as $\mathbb{Z}$-module. Suppose $R \otimes K$ is Cohen-Macaulay for some field $K$ of characteristic $p > 0$. Show that $R \otimes L$ is Cohen-Macaulay for every ...
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### Homomorphisms between regular local rings with regular system of parameters

Let $(R,\mathfrak{m})$ be a regular local ring with regular system of parameters $x_1, . . . , x_n$. Let $f : R \rightarrow S$ be a homomorphism of local rings, $f_i = f(x_i)$ for $i = 1, . . . , n$, ...
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### When $R/I \cong S/J$, where $R$ is Cohen-Macaulay, $S$ is regular local and $ht(J)=\mu(J)$ [closed]

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring. Let $I\subseteq \mathfrak m$ be an ideal of $R$. If $R/I \cong S/J$ for some regular local ring $S$ and ideal $J$ of $S$ such that $ht(J)=\mu(J)$, ...
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### Cohen-Macaulay property of graded ideals

Let $R=K[x_1,\dots,x_n]$ and $I$ be a graded ideal of $R$. My question is the following: if $R/I$ is a Cohen-Macaulay ring then $I$ is a Cohen-Macaulay ideal?
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### When is a Noetherian Standard graded algebra over a field Cohen-Macaulay? Any counter-example when it is not the case?

Let $R$ be a Noetherian standard graded algebra with $R_0 = k$, a field, then it is finitely generated over $R_0$ by $R_1$ and is the homomorphic image of some $k[x_1, \ldots, x_n]$, hence isomorphic ...
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### On local rings $(R, \mathfrak m)$ such that $\text{Spec}(R)$ is disjoint union of $\text{Ass}(R)$ and $\{\mathfrak m \}$

Let $(R, \mathfrak m)$ be a Noetherian local ring such that $\mathfrak m \notin \text{Ass}(R)$ and $\text{Spec}(R)=\text{Ass}(R)\cup \{\mathfrak m \}$. Then, must $R$ be Cohen-Macaulay? Of course the ...
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### The definition of Cohen-Macaulay algebra

Let $x=(x_1,\ ...\ ,x_d)$ a system of parameters of local ring $(R,m)$. We know that an $R$-algebra $B$ is called Cohen-Macaulay algebra if $x_{i+1}$ is a non-zero divisor in $(x_1,\ ...\ ,x_i)B$ for ...
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### Existence of $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal

Let $(R,\mathfrak m)$ be a Noetherian local domain of dimension at least $2$. Then, must there exist $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal of $R$? What if we also ...
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### On maximal Cohen-Macaulay property of a special kind of ideal

Let $R$ be a local Cohen-Macaulay domain of dimension at least $2$. Let $M$ be a maximal Cohen-Macaulay $R$-module such that localization of $M$ at every height $1$ prime ideal of $R$ is free. ...
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### Non Cohen-Macaulay rings of dimension 2

I am working in the context of Noetherian commutative rings with unit. It is well-known that for a field $K$, the ring $R=K[x^4,x^3y,xy^3,y^4]$ is a 2-dimensional standard graded $K$-algebra which not ...
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### Regulare sequences and tensor product of modules

Let $(R,\mathfrak m)$ be a Noetherian local ring of Krull dimension $d$ and $M$,$N$ two maximal Cohen-Macaulay $R$-modules,that is Cohen-Macaulay with their dimensions equal to dimension of $R$. ...
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### If $R/I$ is Cohen-Macaulay, is $R/I^2$ also Cohen-Macaulay?

Let $R$ be a local ring and $I \subset R$ an ideal such that $R/I$ is cohen-macaulay. Is $R/I^2$ also cohen-macaulay? I've seen that it always works for principal ideals here: Is a quotient of ring of ...
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### Is $(\omega:_{Q(R)}\omega)=R$ for a canonical ideal $\omega$ of a one-dimensional local Cohen-Macaulay ring?

Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring of dimension $1$ admitting a canonical ideal $\omega\subseteq R$. Then $\omega$ contains a non-zero divisor. Let $Q(R)$ be the total ring of ...
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### For complete local domain of dimension $1$, when does vanishing of $\text{Ext}^1_R(\omega_R, R)$ forces $R$ to be Gorenstein?

Let $R$ be a complete local domain of dimension $1$ with canonical module $\omega_R$. If $\text{Ext}^1_R(\omega_R, R)=0$, then does it follow that $R$ is Gorenstein ? Is it true at least for some more ...
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### When $R/\operatorname{Ann}_R(M)$ is a Cohen-Macaulay ring?

Let $(R,\frak m)$ denote a commutative Noetherian local ring and $M$ a finitely generated $R$-module. We say $R$ is Cohen-Macaulay, provided $\operatorname{dim}R=\operatorname{depth}R$. Similarly, $M$ ...
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### Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring?

Let $M$ be a finitely generated module over a local Cohen-Macaulay ring $(R,\mathfrak m)$. If $x\in M$ is annihilated by a non-zero-divisor $r\in \mathfrak m$ , then is it true that $\mathfrak m^n x=0$...
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### Bruns & Herzog 1.4.9: Rank of projective modules

I'm reading Cohen-Macaulay Rings by Bruns & Herzog. The lemma below which is used in the proof of the Proposition 1.4.9 and whose proof is left to the reader is difficult for me to prove: Lemma. ...
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### Partial order on the set of non-split short exact sequences ending at a MCM module

Let $R$ be a Henselian Cohen-Macaulay (CM) local ring with maximal ideal $\mathfrak m_R$. For an indecomposable finitely generated maximal Cohen-Macaulay (MCM) $R$-module $M$, the set $\mathfrak S(M)$ ...
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### Example of non-reduced Noetherian dimension 1 Cohen-Macaulay ring.

As the name suggested, is there a non-reduced Noetherian dimension 1 Cohen-Macaulay ring? I know that all reduced Noetherian ring with dimension 1 is Cohen-Macaulay, however it seems difficult for me ...
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