I have a question which is probably caused by some confusion I have with extensions of local fields. Let us fix a finite extension of number fields $L/K$ and fix further $\mathfrak p$ denote a non trivial prime ideal of $\mathcal O _K$. Now, for every prime $\mathfrak P$ of $\mathcal O _L$ dividing $\mathfrak p$ we get an embedding $\sigma_\mathfrak P : K_\mathfrak p \to L_\mathfrak P$.
Is it now true or false that for every $i\geq 1$ we have $\bigcap_{\mathfrak P | \mathfrak p} \ \sigma_\mathfrak P ^{-1}(\mathfrak P ^ i \mathcal O _{L_\mathfrak P}) \subset \mathfrak p ^i \mathcal O _{K_\mathfrak p} \ \ \ ?$
The question occurred during my struggles to understand relations between ray class fields of different number fields.
Thank you very much in advance!