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.
232 questions
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How is $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ related to $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$
Let $X \subset \mathbb{P}^N$ be a scheme of dimension $n$ and $I$ its ideal in $S = k[x_0, \ldots, x_N]$, and $R = S/I$. Suppose $R$ is Cohen-Macaulay, then $X$ is Cohen-Macaulay. Define:
The sheaf $\...
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Algebraically Independent Sequence of two polynomials is a regular sequence
let us fix an algebraically closed field $\kappa$ and the ring of polynomials $R \doteq \kappa[x_0, \ldots, x_3]$, and let $\sigma = (f,g)$ be a sequence of two homogeneous polynomials in $R$. If $\...
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If $k[x_0,\ldots,x_n]/I$ homogeneous is Cohen-Macaulay/Gorenstein then so is $X \subset \mathbb{P}^n$, and converse except possibly at the origin
Let $X \subset \mathbb{P}^n$ be a closed subscheme, $I$ be its (saturated) ideal in $k[x_0,\ldots,x_n]$, and $R = k[x_0, \ldots, x_n]/I$ is its homogeneous coordinate ring. I think the following are ...
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Cohen-Macaulay local ring modulo its nilradical
Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring with nilradical $N$. Must $R/N$ also be Cohen-Macaulay?
The answer is obviously yes when $\dim R=0$. The answer is also yes when $\dim R=1$ as ...
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Alternat definition for Cohen Macaulay and homogenous system of parameters.
I have a question about Cohen-Macaulay rings. I am a newcomer to commutative algebra in general. I am reading up on invariant theory for a seminar at my university and that is where I encountered ...
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Implication of a module being isomorphic to one of its Ext
Let $R$ be a Gorenstein ring. What I have is $R=\mathbb Z/n[x, y, z, x^{-1}, y^{-1}, z^{-1}]$. There is this finitely generated $R$-module $M$ which I wish to show is Cohen-Macaulay. What I have ...
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Is $\dim (M/xM) = \dim M - 1$ for some $x \in m$ implies $x$ is an $M$-regular element ? where $m$ is the maximal ideal of a local ring $R$.
Let $R$ be a Noetherian local ring with maximal ideal $m$, and let $M$ be a finitely generated $R$-module. It is known that if $x \in m$ is an $M$-regular element, then the dimension of $M/xM$ is ...
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Projective dimension of Hom(M,R) for a Cohen-Macaulay ring R
I want to check whether the projective dinesion of $Hom(M,R)$ is finite or not, where $M$ is a finitely generated R module and $R$ is a Cohen-Macaulay ring of dimension at most $2$.
This has a ...
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Comparison between height of an ideals from a Cohen-Macaulay ring to another Cohen-Macaulay ring under a special surjective ring homomorphism.
Let $f: R \rightarrow S$ be a surjective ring homomorphism, where $R$ and $S$ are Cohen-Macaulay rings. Suppose that the kernel of $f$ is generated by a regular sequence. We are asked to determine ...
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Cohen-Macaulay nonlocal rings implications
I mostly work with local or graded commutative rings, so I have some elementary questions when it comes to nonlocal commutative rings (associative + has 1).
So a commutative ring is Cohen-Macaulay if ...
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Is the normalization $S_2$ as an $R$-module?
Let $R$ be a Noetherian commutative ring for which the normalization $\bar{R}$ is a finite $R$-module. Then is $\bar{R}$ $S_2$ as an $R$-module?
We know that $\bar{R}$ is $S_2$ as a ring (i.e. as a ...
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Reference request for self-duality/Gorenstein claim?
I am reading Mark Green's "Generic Initial Ideals". On page 134, he gives an example of an impossible candidate minimal free resolution $E_\bullet$ of an ideal $K\subseteq S:=\mathbb C[x_1,...
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Are two-periodic two-generated ideals of Gorenstein local domains self-dual?
Let $I$ be a non-principal ideal of a Gorenstein local domain $R$ such that there is an exact sequence $0\to I \to R^{\oplus 2}\to R^{\oplus 2}\to I\to 0$.
Then, must it be true that $I\cong \text{Hom}...
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The generators of an ideal and regular sequence.
The problem comes from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry", 18.13. He uses the following fact:
Let $R$ be a Cohen-Macaulay ring. If $I = (x_1, \dots, x_n)$...
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Are determinantal ideals Cohen-Macaulay?
Let $R=K[X_{ij}:i=1,\dots,m,j=1,\dots,n]$. The ideal in $R$ generated by all the $t$-minors of the $m\times n$ matrix
$$ X=\begin{pmatrix}
X_{11} & X_{12} & \dots & X_{1n}\\
X_{21} &...
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