= Localization of an integral extension at a prime need not be integral
An <integral extension> $A\subset B$ need not give an <integral extension> $A_{P\cap A}\subset B_P$. For example, take $\mathbb Z\subset\mathbb Z[i]$ and $P=(2+i)$, whose contraction is $(5)$. The element $1/(2-i)$ belongs to $\mathbb Z[i]_P$ but is not integral over $\mathbb Z_{(5)}$: multiplying a hypothetical monic equation by a power of $2-i$ and reducing modulo the other Gaussian <prime ideal> $(2-i)$ gives $1=0$ in $\mathbb F_5$. Localizing both rings by the same <multiplicative subset> does preserve integrality.
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