My question refers to a step of in the proof of Corollary 8.5.17 in Bosch's "Commutative Algebra and Algebraic Geometry"; see page 395
See the red tagged line below:
We consider the exact sequence
$$0 \to I/J \to R[t_1, ..., t_n]/J \to R[t_1, ..., t_n]/I \to 0$$
and we tensor it with $k(s) = \mathcal{O}_{S,s}/m_s$.
Why does it stay exact? Indeed, by assumption $\mathcal{O}_{S,s}$ is flat so it's ok to tensor it with $\mathcal{O}_{S,s}$ but what about $k(s)$? Why does it conserve the exactness?
In addition: Lemma 5.2.9: