Intereting Posts

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Why sequential continuity from $E$ to $E'$ implies continuity?
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Geometric multiplicity of an eigenvalue
Embedding, immersion
What's the solution of the functional equation
Help with calculating infinite sum $\sum_{n=0}^{\infty}\frac1{1+n^2}$
Calclute the probability?
When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)?
Let $A, B$ be sets. Show that $\mathcal P(A ∩ B) = \mathcal P(A) ∩ \mathcal P(B)$.
Is $\pmb{\eta}\cdot\pmb{\omega_1} = (\pmb{\eta} + \pmb{1})\cdot\pmb{\omega_1}$?
The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid.
Is it true that, in a Dedekind domain, all maximal ideals are prime?

Any 1-dimensional Noetherian domain is Cohen-Macaulay (C-M).

For the $2$-dimensional case, a condition of being integrally closed is necessary to be added for a Noetherian domain to be C-M, which I could not prove it.

Would anybody be so kind as to solve this?

- Prove that $ k/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay.
- Why is this ring not Cohen-Macaulay?
- Does maximal Cohen-Macaulay modules localize?
- Examples of Cohen-Macaulay integral domains
- Are the rings $k]$ and $k]$ Gorenstein? (Matsumura, Exercise 18.8)
- Cohen-Macaulay ring and saturated ideal

I also search for a non C-M $2$-dimensional Noetherian domain which is not integrally closed.

Thanks for any cooperation!

- A commutative ring whose all proper ideals are prime is a field.
- When is the ring of continuous functions Noetherian?
- If $R$ is a Noetherian ring then $R]$ is also Noetherian
- $M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$
- (Ir)reducibility criteria for homogeneous polynomials
- Linear Algebra Basis Trick
- Are bimodules over a commutative ring always modules?
- Maximal ideals and the projective Nullstellensatz
- ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?
- Flatness and intersection of ideals

Let $R$ be a noetherian integral domain of dimension $2$. If $R$ is integrally closed, then $R$ is Cohen-Macaulay.

From Serre’s normality criterion we have that $R$ satisfies $(R_1)$ and $(S_2)$.

$(R_1)$ gives that all the localizations of $R$ at height one primes are regular, and therefore Cohen-Macaulay. (In fact, we don’t need to use $(R_1)$ in order to prove that $R_{\mathfrak p}$ is Cohen-Macaulay for prime ideals $\mathfrak p$ of height one.)

Now let $\mathfrak p$ be a height two prime ideal of $R$. From $(S_2)$ we get that $\operatorname{depth}R_{\mathfrak p}\ge2=\dim R_{\mathfrak p}$, so $R_{\mathfrak p}$ is Cohen-Macaulay.

$k[x^4,x^3y,xy^3,y^4]$ is $2$-dimensional, *not* Cohen-Macaulay and *not* integrally closed.

- Given any nine integers show that it is possible to choose, from among them, four integers a, b, c, d such that a + b − c − d is divisible by 20.
- How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?
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- sequentially continuous on a non first-countable
- Product Identity Multiple Angle or $\sin(nx)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(\frac{k\pi}n+x\right)$
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- Does $\lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \rfloor\right)=a?$
- Normal, Non-Metrizable Spaces
- Help in understanding Derivation of Posterior in Gaussian Process
- How does knowing a function as even or odd help in integration ??
- Is the max of two differentiable functions differentiable?
- For what functions is $y'' = y$?