I have a question about a step in the proof of 5.3.17 in Liu's "Algebraic Geometry" (page 201):
Why is $B = A[T] / (P(T))$ flat over $A$?
Does anybody see a clever base change to a flat module?
My attempt was to consider the flat map $k(y) \to k(y)[T]/P(T)$ but the can quotient map $A \to A/M_y =k(y)$ doesn't provide the base change in "correct" direction, namely if $R \to S$ is flat and $R \to R'$ is a ring map then $R' \to R' \otimes_R S$ is flat,
so $A \to k(y)$ provides a map "in a wrong direction" for this criterion.
Can anybody see another flat map $R \to S$ with $R \to A$ such that $B = A \otimes_R S$?