All Questions
4 questions
7
votes
0
answers
213
views
Existence of $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal
Let $(R,\mathfrak m)$ be a Noetherian local domain of dimension at least $2$.
Then, must there exist $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal of $R$? What if we also ...
- 105
4
votes
1
answer
116
views
Cohen-Macaulayness and regularity of $A/p$
This question claimed (and proved) that if $p$ is a prime ideal of $A=k[x_1,\ldots,x_n]$
with $\operatorname{ht}(p) \in \{0,1,n-1,n\}$, then $A/p$ is Cohen-Macaulay.
Now, let $A$ be a (Noetherian) UFD ...
- 6,937
2
votes
0
answers
49
views
Local Cohen-Macaulay rings
Case one:
Let $(R,m)$ be a Noetherian local ring of Krull dimension $d$, $\dim(R)=d$.
Let $I$ be an ideal of $R$.
Assume that $\operatorname{depth}(I,R)=d$, namely, the maximal length of a regular ...
- 6,937
0
votes
0
answers
269
views
Flat morphism of local rings
Let $(A,m_A)$ and $(B,m_m)$ be two Noetherian local rings, $A \subseteq B$
and $B$ is a finitely generated $A$-algebra.
Step 1: Assume that:
(1) $A$ is regular.
(2) $A \subseteq B$ is flat.
Question ...
- 6,937