This is in reference to Intuition for when modules are flat and the MO https://mathoverflow.net/questions/259574/flatness-from-constancy-of-dimension-of-fibers.
My question is as follows. Assume $M$ is a flat module over some commutative ring $R$. For any prime ideal $p \subset R$, we have a $k(p)$-vector space $M \otimes_R k(p)$, where $k(p)$ is the residue field $k(p) = R_p / p R_p$. Is the dimension of this space independent from $p$, knowing that $M$ is flat (and perhaps further assumptions)?
The motivation is that I am trying to understand flatness in algebraic geometry. If we have a flat morphism of (say affine) schemes $$ \pi : X= \text{Spec } M \rightarrow \text{Spec } R = Y$$
then the fibres over any point $p \in Spec R$ are given by $$ X \times_Y \text{Spec } k(p) = \text{Spec }(M \otimes_R k(p)) .$$
Saying that $\pi$ is flat is identical to saying that $M$ is flat over $R$, thus my interest in $$ \dim_{k(p)} M \otimes_R k(p) .$$