# Questions tagged [cohen-macaulay-rings]

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62
questions

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### For a Cohen-Macaulay module $M$ of dimension $t$ over a local CM ring of dimension $n$, is $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&...

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94 views

### $A \to B$ with $A$ regular imply that $B$ is CM

The answer to this question says the following:
"The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat.
...

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**1**answer

158 views

### Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...

**2**

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**1**answer

153 views

### On the functors $\text{Hom}_R(k,-)$ and $k \otimes_R ( -)$ for Artinian local Gorenstein ring $R$

Let $(R, \mathfrak m,k)$ be an Artinian local Gorenstein ring, hence $\text{Hom}_R(k, R)\cong k$, and so
$\text{Hom}_R(k, R^{\oplus n})\cong k^{\oplus n} \cong k \otimes_R R^{\oplus n} , \forall n \ge ...

**7**

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**1**answer

181 views

### Class group of hypersurfaces of finite representation type

Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...

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90 views

### Quotient of ideal generated by regular sequence is a perfect module

I have recently started reading Bruns-Herzog's 'Cohen Macaulay rings' and this is problem 1.4.27 in it.
We say that a module $M$ over a Noetherian ring $R$ is perfect if the projective dimension of $M$...

**10**

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**1**answer

258 views

### Fixed point scheme of finite group Cohen-Macaulay?

Let $X$ be a quasi-projective scheme over a field $k$.
Let $G$ be a finite group acting on $X$ whose order is invertible in $k$.
If $X$ is Cohen-Macaulay, can we conclude that the subscheme of fixed ...

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**1**answer

125 views

### Weighted blowup of a Cohen-Macaulay ring along a regular sequence is Cohen-Macaulay?

Let $(R, \mathfrak{m})$ be a Cohen-Macaulay ring, let $f_1, \dotsc, f_d \in \mathfrak{m}$ be a regular sequence, and let $n_1, \dotsc, n_d > 0$ be weights (feel free to assume that $n_1 = 1$ if it ...

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189 views

### Linkage and Cohen-Macaulay-ness

Suppose I have a reduced l.c.i. scheme with two irreducible components: $X = Y \cup Z$. I want to say that if $Y$ is Cohen-Macaulay then $Z$ is as well.
I think this follows from Eisenbund Theorem 21....

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120 views

### On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...

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80 views

### Can the conductor of a local, unramified, Cohen-Macaulay domain ever be contained in a parameter ideal?

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay domain which is analytically unramified (i.e. the $\mathfrak m$-adic completion of $R$ is reduced). Let $\bar R$ be the integral closure of $R$ in the ...

**3**

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**1**answer

191 views

### Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface

Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...

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94 views

### Dualizing complex description in Stacks project

The question is closely related to this one this one (more precisely the reference the comment by AGl earner) and is aimed to understand the proof of Lemma 20.2 from notes from Stacks notes from ...

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136 views

### Pushforward of a maximal Cohen-Macaulay sheaf

Let $X$ be an (affine) noetherian, Cohen-Macaulay scheme of dimension at least $2$. Choose a closed point $x \in X$. Denote by $U:=X\setminus \{x\}$ the open subset and $i:U \to X$ the natural ...

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104 views

### Is a Maximal Cohen-Macaulay sheaf on a Cohen-Macaulay scheme locally free?

Let $F$ be a Maximal Cohen-Macaulay module on a Cohen-Macaulay scheme $X$.
By definition $\mathrm{depth}_{\mathcal{O}_{X,x}}(F_x)=\mathrm{dim}(\mathcal{O}_{X,x})$ and $\mathrm{depth}(\mathcal{O}_{X,x})...

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342 views

### Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has these two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim_k ...

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321 views

### difference between Cohen Macaulay and locally Cohen Macaulay curve

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.
The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in ...

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**1**answer

246 views

### For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?

Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?

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139 views

### On Cohen-Macaulay schemes

I know that the irreducible components of a Cohen-Macaulay (CM) scheme don't need to be Cohen-Macaulay. But what happens if we assume that that the irreducible components are CM, what can we say about ...

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503 views

### Depth under localization over a Cohen-Macaulay ring

Let $A$ be a Cohen-Macaulay local ring, and let $M$ be an $A$-module that is (S$_2$) and has depth $\ge n$ for some fixed $n$. Let $\mathfrak{p} \subset A$ be a prime of height $\ge n$. Is it true ...

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131 views

### For isolated singularity algebra, is every maximal Cohen-Macaulay module locally projective?

Let $R$ be a Cohen-Macaulay noetherian local ring. Let $\Lambda$ be a noetherian $R$-algebra which is maximal Cohen-Macaulay as an $R$-module, where for every nonmaximal prime $\mathfrak{p}$, $\...

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151 views

### Cohen-Macaulay rings in GIT

I'm starting to learn about Cohen-Macaulay rings in my commutative Algebra course, and my teacher mentioned its use in Geometric Invariant Theory. Having some basic knowledge on the subject and ...

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243 views

### Hironaka decomposition over $\mathbb{Z}$?

Let $A=\bigoplus_{\ell\geq 0}A_\ell$ be a finitely generated graded $\mathbb{Z}$-algebra, with $A_0=\mathbb{Z}$, that is free as a $\mathbb{Z}$-module.
If $k$ is a field, then $A\otimes k$ is a ...

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188 views

### If a finite poset supports a Cohen-Macaulay ASL, how far can it be from Cohen-Macaulay?

By the fundamental work of De Concini, Eisenbud, and Procesi, an algebra with straightening law (ASL) must be Cohen-Macaulay if it is built on a Cohen-Macaulay poset. I would like to understand the ...

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407 views

### Dualizing sheaf on a Cohen-Macaulay scheme

I would like to fill a detail in the answer given by Francesco Polizzi to the question "Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?".
Let $X$ be a normal, Cohen-Macaulay scheme of ...

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938 views

### Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?

I am reading the paper Frobenius splitting of Hilbert schemes
of points on surfaces by Kumar and Thomsen. At the end of Lemma 11, they seem to imply that the dualizing sheaf on a Cohen-Macaulay scheme ...

**2**

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**1**answer

249 views

### Does the graded face poset of a BN-pair admit a EL-labeling?

Let $G$ be a finite group with a BN-pair of rank $n$. Let $B$ be the associated Borel subgroup.
Let $P$ be the poset of proper right cosets (i.e. $Kg$ with $K \in [B,G)$ and $g \in G$).
Let $\hat{P}:...

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201 views

### symbolic powers of prime ideals

Let $R$ be a CM ring and $P$ be a prime ideal. Let $P^{(n)}$ denote the $n$th symbolic power of $P$.
Is the sequence $\operatorname {depth} (R/P^{(n)})$ eventually constant?

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561 views

### The concept “opposite” of Cohen-Macaulayness

Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be a monomial ideal.
For a monomial ideal $J$, let $\#(J)$ be the smallest number of ...

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83 views

### Is the bounded coset poset of a boolean interval of finite groups, Cohen-Macaulay?

Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above ...

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303 views

### Cohen-Macaulay non-normal toric variety

Given a quasi-smooth toric variety $X$ in the sense of Gelfand-Kapranov-Zelevinsky,
i.e. a (not necessarily normal) toric variety $X$ whose normalization is a $\mathbb{Q}$-factorial toric variety and ...

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597 views

### Auslander-Reiten theory for Gorenstein algebras

In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra ...

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140 views

### Björner-Wachs theorem for posets admitting an EL-labeling

In the survey paper Poset Topology: Tools and Applications by Michelle Wachs, there is the following theorem on p46:
Theorem 3.2.4 (Björner and Wachs [40]). Suppose $P$ is a poset for
which $\...

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182 views

### when there is an injection $0 \to R \to K_R$?

Let $(R,m)$ be a Cohen-Macaulay local ring which possesses the canonical module $K_R$. Then $R$ is said to be an almost Gorenstein local ring, if there is an exact sequence $0 \to R \to K_R \to C \to ...

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162 views

### Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...

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88 views

### Upper bound for the minimum number of generators of the canonical module

Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the ...

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398 views

### Matrix factorizations as a dg-category?

Matrix factorizations (in the graded case) give a triangulated category. I would imagine that there should be an underlying dg-category. Is there such a definition, and if so, where can I find it in ...

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201 views

### if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is “Gorenstein”

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...

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467 views

### canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$.
So we ...

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**1**answer

395 views

### A criterion for complete intersection in terms of the Hilbert series?

Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of ...

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211 views

### Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay $R$-...

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233 views

### ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of $A$....

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**1**answer

196 views

### Does projective duality preserve arithmetic-Cohen-Macaulay-ness?

Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring $\...

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201 views

### A doubt from the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring”

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...

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314 views

### Maximal Cohen-Macaulay modules of type one

Does anybody know an example of a Noetherian local ring $(R,m)$ which admits a maximal Cohen-Macaulay module of type one, but the ring $R$ itself is not CM?
If $C$ is a maximal CM module then the ...

**7**

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**1**answer

628 views

### Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...

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156 views

### Producing local complete intersection curves in singular surfaces

Let $X$ be a smooth surface in $\mathbb{P}^3$ and $C$ a local complete intersection curve in $X$. Does there exist a hyperplane $H$ in $\mathbb{P}^3$ such that $C$ is a local complete intersection ...

**13**

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757 views

### When is the reduced subscheme of a Cohen-Macaulay scheme also Cohen-Macaulay?

Let $X$ be a Cohen-Macaulay scheme (I will be interested in the case when this is ${\rm Spec}(A/I)$ where $A$ is a polynomial ring over a field and $I$ is a homogeneous ideal). I would like to know ...

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136 views

### ideals of a noetherian ring $R$ Cohen-Macaulay as $R-$modules

When are (prime) ideals of a noetherian ring $R$ Cohen-Macaulay as $R-$modules?
That is, $depth_R(Ann_R(P))=dim_R(R/Ann_R(P)$ for each $P\in {\rm Spec}(R)$

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**1**answer

92 views

### What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?

http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...