Fibres of standard smooth algebras are regular of relative dimension

Set $B:=K[x_1,\dots ,x_n]\mathrm{/}({\bar{f}}_1,\dots ,{\bar{f}}_c)$ and $\bar{h}$ for the image of $h$ in $B$. By [F2] applied to $R\to \kappa$ the fibre is $F\cong (\kappa [x_1,\dots ,x_n]\mathrm{/}({\bar{f}}_1,\dots ,{\bar{f}}_c){)}_{\bar{g}}$, and applying [F2] again to the ring map $\kappa \to K$ shows that $F_K\cong B_{\bar{g}}$, with the image of $\bar{h}$ in $B_{\bar{g}}$ a unit; the relative dimension $n-c$ is unchanged throughout.

Primes of $B_{\bar{g}}$ correspond under [F8] to the primes $P\subseteq B$ with $\bar{g}\mathrm{\notin }P$, and these in turn correspond to the primes $Q^{\mathrm{\prime }}\subseteq K[x_1,\dots ,x_n]$ with $({\bar{f}}_1,\dots ,{\bar{f}}_c)\subseteq Q^{\mathrm{\prime }}$ and $\bar{g}\mathrm{\notin }{Q}^{\mathrm{\prime }}$. For such a $Q^{\mathrm{\prime }}$ one has $\bar{h}\mathrm{\notin }{Q}^{\mathrm{\prime }}$: since the image of $\bar{h}$ in $B_{\bar{g}}$ is a unit, there are $v\in B$ and $N\ge 0$ with $\bar{h}v={\bar{g}}^N$ in $B_{\bar{g}}$, so by [F7] there is $M\ge 0$ with ${\bar{g}}^M(\bar{h}v-{\bar{g}}^N)=0$ in $B$; if $\bar{h}$ lay in $Q^{\mathrm{\prime }}$, hence in $P=Q^{\mathrm{\prime }}\mathrm{/}({\bar{f}}_1,\dots ,{\bar{f}}_c)$, the contradiction ${\bar{g}}^{M+N}\in P$ with $\bar{g}\mathrm{\notin }P$ would follow.

Let $Q$ be a prime of $B_{\bar{g}}$, let $P\subseteq B$ be the prime it contracts to and let $Q^{\mathrm{\prime }}\subseteq K[x_1,\dots ,x_n]$ be the corresponding prime; then $(F_K{)}_Q=B_P=(K[x_1,\dots ,x_n{]}_{Q^{\mathrm{\prime }}})\mathrm{/}({\bar{f}}_1,\dots ,{\bar{f}}_c)$. Indeed $B_{\bar{g}}=(K[x_1,\dots ,x_n]\mathrm{/}({\bar{f}}_1,\dots ,{\bar{f}}_c){)}_{\bar{g}}$, localising further at $Q$ gives $B_P$ by [F9], and [F10] identifies $B_P$ with the quotient of $K[x_1,\dots ,x_n{]}_{Q^{\mathrm{\prime }}}$ by the ideal generated by the ${\bar{f}}_j$.

Regularity of local rings. In the situation of step 3.1 the elements ${\bar{f}}_1,\dots ,{\bar{f}}_c$ lie in the prime $Q^{\mathrm{\prime }}$ and, by step 2.1, have leading minor $\bar{h}\mathrm{\notin }{Q}^{\mathrm{\prime }}$; so [F3] applies over the field $K$ and shows that $(F_K{)}_Q=(K[x_1,\dots ,x_n{]}_{Q^{\mathrm{\prime }}})\mathrm{/}({\bar{f}}_1,\dots ,{\bar{f}}_c)$ is a regular local ring with $\dim (F_K{)}_Q=\dim K[x_1,\dots ,x_n{]}_{Q^{\mathrm{\prime }}}-c=\mathrm{ht}(Q^{\mathrm{\prime }})-c$, the last equality by [F15]. This proves clause 1, including for $c=0$, where the empty leading minor is $1$ and the quotient is $K[x_1,\dots ,x_n{]}_{Q^{\mathrm{\prime }}}$.

Height of the ambient prime at a component. Let $\mathfrak{P}$ be a minimal prime of $F_K=B_{\bar{g}}$, let $P$ be its contraction to $B$, and let $Q^{\mathrm{\prime }}$ be the corresponding prime of $K[x_1,\dots ,x_n]$. The prime $P$ is minimal in $B$: a prime strictly below it avoids $\bar{g}$ and would localize to a prime strictly below $\mathfrak{P}$ by [F8]. The local ring $(F_K{)}_{\mathfrak{P}}$ has dimension zero by [F5, F15], so clause 1, already proved in step 4.1, gives $0=\mathrm{ht}(Q^{\mathrm{\prime }})-c$. Thus $\mathrm{ht}(Q^{\mathrm{\prime }})=c$, also when $c=0$.

Component dimension. Put $A:=K[x_1,\dots ,x_n]\mathrm{/}{Q}^{\mathrm{\prime }}$, a finite-type $K$-domain, and let $g_A$ be the image of $\bar{g}$. Then $g_A\mathrm{\ne }0$, and [F10] gives $F_K\mathrm{/}\mathfrak{P}\cong A_{g_A}$. The localization is again a finite-type $K$-domain (adjoin $z$ with $z{g}_A=1$), and has the same fraction field as $A$. Applying [F16] to both rings gives $\dim A_{g_A}=\dim A$. Now [F11, F12] and step 5.1 give $\dim A=n-\mathrm{ht}(Q^{\mathrm{\prime }})=n-c$. By [F13, F14], the component $V(\mathfrak{P})$ is homeomorphic to $\mathrm{Spec}(F_K\mathrm{/}\mathfrak{P})$, so has dimension $n-c$.

Both clauses hold: every local ring of $F_K$ is regular local of dimension $\mathrm{ht}(Q^{\mathrm{\prime }})-c$ by step 4.1, and every irreducible component has dimension $n-c$ by step 6.1; if $F_K$ is the zero ring there are no primes and no components, so both clauses are vacuous. ∎