Intereting Posts

Is there an irrational number $a$ such that $a^a$ is rational?
How can I prove that every maximal ideal of $B= \mathbb{Z} $ is a principal?
Finding number of functions from a set to itself such that $f(f(x)) = x$
Orthonormal basis in $L^2(\Omega)$ for bounded $\Omega$
Basic question about mod
Inequality between $\ell^p$-norms
Every path has a simple “subpath”
Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?
Symmetric Matrices with trace zero
Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities?
If $n$ is a positive integer greater than 1 such that $3n+1$ is perfect square, then show that $n+1$ is the sum of three perfect squares.
How many strings of $8$ English letters are there (repetition allowed)?
The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
Irreducible polynomials have distinct roots?
Is “integrability” equivalent to “having antiderivative”?

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.
- Does maximal Cohen-Macaulay modules localize?
- Cohen-Macaulay ring and saturated ideal
- Why is this ring not Cohen-Macaulay?
- Associated ideals of a principal ideal generated by a nonzero divisor
- showing Cohen-Macaulay property is preserved under a ring extension

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

Thanks for any cooperation!

- Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$?
- Points and maximal ideals in polynomial rings
- Geometrical interpretation of $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$, $X_i$ algebraic sets in $\mathbb{A}^n$
- One-dimensional Noetherian UFD is a PID
- rational functions on projective n space
- Why isn't $\mathbb{C}/(xz-y)$ a flat $\mathbb{C}$-module
- Using Nakayama's Lemma to prove isomorphism theorem for finitely generated free modules
- Associated primes of a sum of modules
- Does $\text{End}_R(I)=R$ always hold when $R$ is an integrally closed domain?
- Definition of a finitely generated $k$ - algebra

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.

- My favorite proof of the generalized AM-GM inequality: where it came from?
- Relating the Künneth Formula to the Leray-Hirsch Theorem
- What is the difference between matrix theory and linear algebra?
- The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions
- Proving Holder's inequality using Jensen's inequality
- Equivalence of continuity definitions
- proof of l'Hôpital's rule
- Quadirlogarithm value $\operatorname{Li}_4 \left( \frac{1}{2}\right)$
- Are the functions $\sin^n(x)$ linearly independent?
- Is semidirect product unique?
- Does Euclidean geometry require a complete metric space?
- How far do I need to drive to find an empty parking spot?
- Topologist's Sine Curve not a regular submanifold of $\mathbb{R^2}$?
- Holomorphic function of a matrix
- Understanding Integration techniques?