TITLE: Flatness from constancy of dimension of fibers
QUESTION [9 upvotes]: Let $M$ be a finitely presented module over the ring $R$. Suppose that for all primes $P\subset R$ the $k(P)$ vector space  $M\otimes _Rk(P)$ has a dimension $d(P)$ independent of $P$.
 Can I conclude that $M$ is flat over $R$?
I am asking because I want to better understand the criterion for a family of projective schemes to be flat over some base in terms of invariance of their Hilbert polynomial.
I checked some examples. 
For example let  $M$ be the ideal  of a point $P$  of an affine curve $C$.
Here $d(P)$ is the dimension of the Zariski tangent space of $C$at $P$ so that the proposed criterion for flatness holds.

REPLY [14 votes]: Yes if $R$ is reduced (no nilpotent elements). This is (for instance) Lemma 1 p. 51 in Mumford's Abelian Varieties (2nd edition). No in general: just take $R=k[\varepsilon ]/(\varepsilon ^2)$, $M=k$.