Questions tagged [cohen-macaulay-rings]

The tag has no usage guidance.

62 questions
Filter by
Sorted by
Tagged with
100 views

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&...
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. ...
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 ...
153 views

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 ...
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 ...
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 ...
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 ...
104 views

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 ...
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 ?
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 ...
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 ...
131 views

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?
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 ...
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 ...
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 ...
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 ...
140 views

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 $A$-...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $R$-...
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 $A$....
196 views

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 ...
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 ...
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 ...
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 ...
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 $P\in {\rm Spec}(R)$