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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.

A ring is called Cohen-Macaulay if its depth is equal to its dimension. More generally, a commutative ring is called Cohen-Macaulay if it 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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232 questions
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What is a geometric interpretation of regular sequences in various instances?

This question arose from my attempts to understand the inclusion Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay There are many related questions here and in mathoverflow and they have been really helpful but I…
Theo Douvropoulos
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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 that the induced homomorphism $R \to…
Mellon
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Does maximal Cohen-Macaulay modules localize?

Let $A$ be a Noetherian local ring and $M$ a finitely generated $A$-module such that $$\operatorname{depth}M= \dim M=\dim A.$$ I can prove that $$\operatorname{depth}M_{\mathfrak{p}}= \dim M_{\mathfrak{p}}\leq\dim…
ashpool
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The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is Cohen-Macaulay". The definition of Cohen-Macaulay…
dadexix86
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Example that M* is not reflexive

Let $R$ be a noetherian ring. Set $(-)^\ast={\rm Hom}_R(-,R)$. For each $R$-module $N$, let $\pi_N:N\rightarrow N^{\ast\ast}$ be the map which maps $n\in N$ to $(f\mapsto f(n))$. $N$ is called reflexive if $\pi_N$ is isomorphism. Question: Does…
Jian
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Cohen-Macaulay rings and Normal rings

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
martia
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$f_\ast \mathcal{O}_X$ locally free iff $X$ Cohen-Macaulay

I am reading Theorem 27.5 here which states the following: Let $f : X \to Y$ be a finite morphism of Noetherian schemes with $Y$ non-singular. Then $X$ is Cohen-Macaulay iff $f_\ast\mathcal{O}_X$ is locally free. Now implicit in the proof is…
user38268
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Associated ideals of a principal ideal generated by a nonzero divisor

Let $R$ be a Noetherian ring and $b\in R$ a nonzerodivisor. Krull's Principal Ideal Theorem implies that every minimal prime ideal over the ideal $(b)$ has codimension $1$. However, could it be possible that one of the associated ideals of $(b)$ is…
Jiangwei Xue
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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 assume $R$ is normal? My thoughts: If $R$ is a…
feder
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What has projectiveness to do with Cohen-Macaulay rings?

I read in Jacob Lurie's lecture notes that if $R=k[x_{1},\dots,x_{s}]/p$, and $R'=k[y_{1},\dots,y_{t}]$ injects into $R$ via Noether normalization such that $R$ is finite over $R'$, then $R$ is Cohen-Macaulay if and only if $R$ is a projective $R'$…
Bombyx mori
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Reduced one-dimensional Noetherian ring is Cohen-Macaulay

If $(R,m)$ is a local Noetherian reduced ring of Krull dimension $1$ then $R$ is Cohen-Macaulay, since in a reduced Noetherian ring the set of zero divisors is the (finite) union $U$ of minimal prime ideals, so there exists an element $x\in m$ which…
karparvar
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Cohen-Macaulay but not regular

In the Wiki page it is claimed that $K[[t^2,t^3]]$ is a $1$-dimensional Cohen-Macaulay ring which is not regular. Is there anybody who kindly explain to me the above assertion? Thanks in advance!
karparvar
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Injective Maximal Cohen-Macaulay modules

Let $R$ be a Gorenstein (not necessarily commutative) ring and let $I$ be an injective finitely generated module over $R$. Is it true that if $\operatorname{Ext}_R^i(I, R)=0$ for $i > 0$, then $I$ is projective?
Alex
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Examples of Cohen-Macaulay rings.

I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay: $k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$. Also I am looking for a ring which is not Cohen-Macaulay. Can anyone help me?
user142710
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showing Cohen-Macaulay property is preserved under a ring extension

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, with $R_0$ local Artinian. Assume also that $R$ is finitely generated over $R_0$ by elements of degree $1$. Let $M$ be a Cohen-Macaulay $R$-module. Let $Y$ be an indeterminate over $R_0$ and define…
Manos
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