1
$\begingroup$

Let $A$ be a non-neotherian UFD such that $A$ satisfies the following condition $(\sharp)$$\colon$

$(\sharp)$ For any prime ideal ${\frak P}$ of $A$ such that ${\mathrm{ht}}({\frak P}) < \infty$, the localisation $A_{\frak P}$ at ${\frak P}$ is a regular local ring.

Q. Is A necessarily coherent?

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ The question as written is confusing. 1) A regular local ring is noetherian by definition. Do you use a different definition of regularity? 2) What else than ${\rm ht}(\mathfrak{P})$ could the Krull dimension of $A_{\mathfrak{P}}$ possibly be? $\endgroup$
    – Fred Rohrer
    Commented Feb 15, 2017 at 19:17

0

Reset to default

You must log in to answer this question.