I would like to solve an exercise in Bourbaki's Algèbre Commutative (chapter 8, §5) stating that $A[T]$ is regular whenever $A$ is. We argue as follows (the book gives the idea of the argument):
$A[T]$ is Noetherian by the Hilbert basis theorem
"by localizing we may reduce to the case where $A$ has finite global dimension"
Then $\operatorname{gldim}A[T]=\operatorname{gldim}A+1$ is finite, hence $A[T]$ regular.
I don't quite understand how one can do the second step. I mean if we take a prime ideal $p$ of $A[T]$, I think $A[T]_p$ need not be isomorphic to $A_p[T]$ (which doesnt even make sense unless $p$ comes from a prime ideal of $A$). Can we express $A[T]_p$ as a polynomial ring?