In reading on the stacks project I came across a result I don't quite follow:
"Assume M is finitely presented and flat, i.e., (1) holds. We will prove that (7) holds. Pick any prime p and x1,…,xr∈M which map to a basis of M⊗Rκ(p). By Nakayama's Lemma 10.18.1 these elements generate $M_g$ for some g∈R, g∉p. The corresponding surjection φ:R⊕rg→M⊕rg has the following two properties: (a) Ker(φ) is a finite $R_g$-module (see Lemma 10.5.3) and (b) Ker(φ)⊗κ(p)=0 by flatness of $M_g$ over $R_g$ (see Lemma 10.36.11). "
(The cited lemma states that $-\otimes_{R_g} M_g$ is exact, since $M_g$ is flat due to the exactness of localisation. )
The part I'm uncertain about is: The surjection φ:$R_g^n→M_g^n$ has the following property: Ker(φ)⊗κ(p)=0 by flatness of $M_g$ over $R_g$.
Why?
(Here I assume $\kappa (p) := R/p$).
Since $0 \to \ker(\phi) \to R_g^n \to M_g^n \to 0$ is exact and $M_g^n$ is flat, it follows that this sequence stays exact when tensoring with any module (by symmetry of Tor, for example).