Fibrewise regular sequences persist openly in equidimensional Cohen-Macaulay fibres

If $Z=\mathrm{\varnothing }$, the assertion is immediate. For $i=0$, the empty sequence is regular at every point, so the locus is all of $Z$. Assume $i>0$, and fix a point $\mathfrak{q}\in Z$ where the displayed fibre sequence is regular. Put $\mathfrak{p}=\mathfrak{q}\cap R$ and $A=S{\otimes }_R\kappa (\mathfrak{p})$, with corresponding prime $q$ of $A$. The local fibre ring $A_q$ is Cohen–Macaulay of dimension $h$, and [F1] makes $A_q\mathrm{/}(f_1,\dots ,f_i)A_q$ of dimension $h-i$. In particular $i\le h\le d$.

The residue field $\kappa (q)$ is the same for the point of the quotient fibre. By [F2], the local scheme dimension of the original fibre at $q$ is $h+{\mathrm{trdeg}}_{\kappa (\mathfrak{p})}\kappa (q)=d$. Applying the same formula to the quotient fibre and using step 1.1 gives its local scheme dimension at $q$ as $h-i+{\mathrm{trdeg}}_{\kappa (\mathfrak{p})}\kappa (q)=d-i$.

Apply [F3] to the finite-type map $R\to C=S\mathrm{/}(f_1,\dots ,f_i)$ at the point $\mathfrak{q}\mathrm{/}I$, with $n=d-i$. It gives an open neighbourhood $W\subseteq \mathrm{Spec}C=Z$ of $\mathfrak{q}$ on which every quotient fibre has local scheme dimension at most $d-i$.

Let ${\mathfrak{q}}^{\mathrm{\prime }}\in W$ over ${\mathfrak{p}}^{\mathrm{\prime }}$ and put $A^{\mathrm{\prime }}=S{\otimes }_R\kappa ({\mathfrak{p}}^{\mathrm{\prime }})$. By the meaning of local scheme dimension, there is a principal open $D(g)\subseteq \mathrm{Spec}(A^{\mathrm{\prime }}\mathrm{/}(f_1,\dots ,f_i))$ containing the point corresponding to ${\mathfrak{q}}^{\mathrm{\prime }}$ whose dimension is at most $d-i$. Lift $g$ to $A^{\mathrm{\prime }}$. The nonempty ring $A_g^{\mathrm{\prime }}$ is Cohen–Macaulay and equidimensional of dimension $d$ by [F4], and $A_g^{\mathrm{\prime }}\mathrm{/}(f_1,\dots ,f_i)$ is the coordinate ring of that chosen principal open, of dimension at most $d-i$. Apply [F4] to $A_g^{\mathrm{\prime }}$; the tuple is regular at its prime corresponding to ${\mathfrak{q}}^{\mathrm{\prime }}$, which is exactly the local-fibre regularity in the Statement.

Every ${\mathfrak{q}}^{\mathrm{\prime }}\in W$ therefore lies in the regular-sequence locus, while $\mathfrak{q}$ was an arbitrary point of that locus. Such neighbourhoods make it open in $Z$. AC is inherited through [F1]–[F4]; the principal-open selections are finite at each point.