Proof.
By Lemma 10.69.2 part (3) follows from part (1).
Assume R is Noetherian. Let \mathfrak p = R \cap \mathfrak q'. By Lemma 10.135.4 for example we see that f_1, \ldots , f_ c form a regular sequence in the local ring R[x_1, \ldots , x_ n]_{\mathfrak q'} \otimes _ R \kappa (\mathfrak p). Moreover, the local ring R[x_1, \ldots , x_ n]_{\mathfrak q'} is flat over R_{\mathfrak p}. Since R, and hence R[x_1, \ldots , x_ n]_{\mathfrak q'} is Noetherian we see from Lemma 10.99.3 that (1) and (2) hold.
Let R be general. Write R = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } R_\lambda as the filtered colimit of finite type \mathbf{Z}-subalgebras (compare with Section 10.127). We may assume that f_1, \ldots , f_ c \in R_\lambda [x_1, \ldots , x_ n] for all \lambda . Let R_0 \subset R be as in Lemma 10.136.11. Then we may assume R_0 \subset R_\lambda for all \lambda . It follows that S_\lambda = R_\lambda [x_1, \ldots , x_ n]/(f_1, \ldots , f_ c) is a relative global complete intersection (as base change of S_0 via R_0 \to R_\lambda , see Lemma 10.136.9). Denote \mathfrak p_\lambda , \mathfrak q_\lambda , \mathfrak q'_\lambda the prime of R_\lambda , S_\lambda , R_\lambda [x_1, \ldots , x_ n] induced by \mathfrak p, \mathfrak q, \mathfrak q'. With this notation, we have (1) and (2) for each \lambda . Since
R[x_1, \ldots , x_ n]_{\mathfrak q'}/(f_1, \ldots , f_ i) = \mathop{\mathrm{colim}}\nolimits R_\lambda [x_1, \ldots , x_ n]_{\mathfrak q_\lambda '}/(f_1, \ldots , f_ i)
we deduce flatness in (2) over R from Lemma 10.39.6. Since we have
\begin{align*} R[x_1, \ldots , x_ n]_{\mathfrak q'}/(f_1, \ldots , f_ i) \xrightarrow {f_{i + 1}} R[x_1, \ldots , x_ n]_{\mathfrak q'}/(f_1, \ldots , f_ i) \\ = \mathop{\mathrm{colim}}\nolimits \left( R_\lambda [x_1, \ldots , x_ n]_{\mathfrak q_\lambda '}/(f_1, \ldots , f_ i) \xrightarrow {f_{i + 1}} R_\lambda [x_1, \ldots , x_ n]_{\mathfrak q_\lambda '}/(f_1, \ldots , f_ i) \right) \end{align*}
and since filtered colimits are exact (Lemma 10.8.8) we conclude that we have (1).
\square
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