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The following counterexample might be interesting too. There exists an unmixed d-dimensional commutative Noetherian local domain $(R,\mathfrak{m})$ such that its completion is not unmixed. Then $H^d_\ …

By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed …

[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the canoni …

According to this reference an elliptic curve is $F$-pure if and only if the $F$-pure threshold of its defining ideal is $1$. Does there exist an $F$-pure local ring $R=A/\mathfrak{a}$ such that $\tex …

Let $G$ be a group of order $p$ acting linearly on $A=\mathbb{Z}/p\mathbb{Z}[x_1,\ldots,x_r]$. Does there exist a formula for the Hilbert-Samuel multiplicity of the unique homogeneous maximal ideal of …

Assume that $(R,\mathfrak{m})$ is a commutative local ring of equal characteristic zero. So $R$ contains the field of rationals. The well known $\mathfrak{m}$-adic completion of $R$ provides a comple …

Suppose that the ring homomorphism $R\rightarrow S$ is faithfully flat ($R$ and $S$ are Noetherian commutative rings). Let $A$ be an Artinian $R$-module. Do we have $0:_S(A\bigotimes_R S)=(0:_RA)S$. …

Given a perfect field $F$ of prime characteristic the ring of Witt Vectors $W(F)$ is a discrete valuation ring. For example, $W(\mathbb{F}_p)$ is the ring of $p$-adic integers. Is it possible to embed …

It is well known that a local commutative unital ring $R$ is Gorenstein if and only if every parameter ideal is irreducible. Why the irreducibility of parameter ideals in a Gorenstein local ring is …

For the concept of the grade of an arbitrary module with respect to a finitely generated ideal see the section 9.1 (entitled as "Grade and acyclicity") of the book entitled as "Cohen-Macaulay Rings" w …

I strongly guess that, you mean, the maps of your direct system, $$R/I_t\rightarrow R/I_{t+1},$$ whose colimit is, $$R/I^\infty,$$ are the multiplication by $x_0\ldots x_{n-1}$. So I think that your …

Is the canonical module of a Buchsbaum ring a Buchsbaum module?

Suppose that $(A,\mathfrak{m})$ is a complete intersection and $\mathbf{x}$ is a minimal basis for $\mathfrak{m}$. Consider the Koszul homologies $H_\bullet(\mathbf{x},A)$. It is well-known that $\tex …

Not, necessarily. For example, in $K[x,y]$, $x^3,xy^2$ is not a $d$-sequence, but it generates an almost complete intersection. However, in some cases, the answer is positive. For example, see (5) of …

Let ,$R=\mathbb{Z}/3\mathbb{Z}[x,y,z,w]/(x^4+y^3+z^4)$. Then, $y\in (x,z)^F\subseteq (x,z)^*\subseteq \overline{(x,z)}$, and thence $(x,z)$ is not integrally closed, however $x,z$ is a regular sequenc …