Exercise 111.12.2. Give an example of a Noetherian local ring $(R, \mathfrak m, \kappa )$ of depth $\geq 1$ and a prime ideal $\mathfrak p$ such that

$\text{depth}_\mathfrak m(R) \geq 1$,

$\text{depth}_\mathfrak p(R_\mathfrak p) = 0$, and

$\dim (R_\mathfrak p) \geq 1$.

If we don't ask for (3) then the exercise is too easy. Why?

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