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Let $P$ be a prime ideal in an integral domain $A$. When is the following property satisfied? $$ fg \in P^2 \Rightarrow \text{either} \, f \in P \, \text{and} \, g \in P \text{, or} \, f \in P^2 \text{, or} \, g \in P^2$$

I am unsure if it always holds or if some condition (deformations? symbolic powers?) must be imposed.

In other words I am asking for when the conormal sheaf of $A/P$ in $A$ is torsion-free.

ofiz
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    If $P^2$ is $P$-primary you get what you want. A counterexample is the usual example of a prime ideal whose square is not primary: https://math.stackexchange.com/questions/384781 – user26857 Dec 06 '20 at 22:30
  • If $P$ is maximal then it works fine (no need that $A$ is an integral domain). – reuns Dec 06 '20 at 23:48
  • Thank you! I should have been more careful during my algebra lectures. – ofiz Dec 07 '20 at 00:26

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