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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On $\operatorname{Hom}_R(S,\omega_R)$ where $S$ is regular domain and finite extension of $R$

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring admitting a canonical module $\omega_R$. Let $R \subseteq S$ be a module finite extension such that $S$ is a regular integral domain i.e. $S$ is an ...
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Checking a ring is not Cohen-Macaulay

While reading a book, I found an example that said that the ring $K[w,x,y,z]/(wy,wz,xy,xz)$ is not Cohen-Macaulay. In order to check this, it is stated to take the quotient by the ideal generated by ...
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Second syzygy modules over rings , satisfying $(S_1)$ and $(G_0)$ , are reflexive

Let $(R,\mathfrak m,k)$ be a Noetherian local ring such that $R_P$ is Gorenstein for every minimal prime ideal $P$ of $R$ and $\text{depth }R_P\ge 1$ whenever $ht (P)\ge 1$. If $0\ne M$ is a finitely ...
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Proving a duality between Ext and Tor for maximal Cohen-Macaulay modules over Gorenstein ring

Let $(R,\mathfrak m, k)$ be a local complete Gorenstein ring of dimension $d$. Let $M,N$ are maximal Cohen-Macaulay modules (i.e. have depth equal to $d$) that are locally free on the punctured ...
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If $I$ is generated by a regular sequence, $\bigoplus_{n \geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring

Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal generated by a regular sequence. I want to show that: $\bigoplus_{n \geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring over $R/I$ in as many ...
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Divisor class group of $2$-dimensional excellent, local normal rational singularity

Let $(R, \mathfrak m)$ be an excellent, local normal domain of dimension $2$ (hence Cohen-Macaulay) with an algebraically closed residue field $k=R/\mathfrak m$. Assume that $IJ$ is an integrally ...
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Divisor class group of $2$-dimensional surface isolated singularity

For $n\ge 2$, consider the $2$-dimensional Noetherian local ring $A_n:=k[[x,y,z]]/(x^2+y^2+z^n)$ , where $k$ is an algebraically closed field of Characteristic zero. I can show that each $A_n$ is an ...
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AF+BG theorem and Cohen Macaulay property

I tried to solve the following exercise in Vakil's notes Question 1: According to the hint, I should try to show the intersection of affine cones is CM (since one dimensional scheme is CM iff it has ...
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Cancelling the canonical module in tensor products

Let $R$ be Cohen-Macaulay local ring with the canonical module $\omega_R$ and let $M$ and $N$ be two finitely generated $R$-modules. Assume that $$\omega_R \otimes_R M= \omega_R \otimes_R N$$ Can ...
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On the colon ideal of a torsion-free module inside it's reflexive hull

Let $(R,\mathfrak m)$ be a Noetherian local complete domain of dimension $1$, with fraction field $K$. (all these assumptions on $R$ imply in particular that for every finitely generated $R$-algebra ...
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Well behaved colon between powers of ideals when the associated graded ring is Cohen-Macaulay.

I'm reading a paper: A formula for the core of an ideal, by Claudia Polini and Bernd Ulrich and I'm in trouble with the following problem: Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal of $R$...
On a particular kind of simplicial complex with maximum facet size of $3$.
Let $\Delta$ be an abstract simplicial complex on $n$ vertices such that $\max \{|F| : F$ is a face of $\Delta \}=3$. Let $f_2$ be the number of faces of size (cardinality) $3$ and $f_1$ be the ...