The flat-local Cohen--Macaulay fibre criterion

Noetherian local maximal ideals are finitely generated. A zero maximal ideal gives a field of dimension zero; otherwise [L0] bounds its height by a positive finite number of generators. Thus $R,S,F$ all have finite dimension. Choose $d$ parameters $x_i$ in $R$ and $e$ lifts $y_j$ in $S$ of parameters of $F$, using [L1]. The ideal $J=(x_i,y_j)S$ has radical $\mathfrak{n}$: a power of $\mathfrak{m}$ lies in $(x_i)R$, and a power of $\mathfrak{n}\mathrm{/}\mathfrak{m}S$ lies in the ideal of the images of $y_j$ in $F$. Taking radicals successively gives $\sqrt{J}=\mathfrak{n}$. By the minimal-generator bound [L1] for $S$, $\dim S\le d+e$.

Choose a maximal chain of primes ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d=\mathfrak{m}$ in $R$ and a maximal fibre chain ${\mathfrak{q}}_0\mathrm{/}\mathfrak{m}S⊊\cdots ⊊{\mathfrak{q}}_e\mathrm{/}\mathfrak{m}S=\mathfrak{n}\mathrm{/}\mathfrak{m}S$ in $F$. All ${\mathfrak{q}}_j$ contract to $\mathfrak{m}$. Starting at ${\mathfrak{q}}_0$, repeatedly apply going down [L2] along the base chain, under the declared AC, obtaining $d$ strict prime inclusions below ${\mathfrak{q}}_0$. Concatenate them with the $e$ strict fibre inclusions to get a chain of length $d+e$ in $S$. Hence $\dim S\ge d+e$, and step 1.1 gives $\dim S=\dim R+\dim F$.

By [L3] and step 2.1, $\dim S-\mathrm{depth}S=(\dim R-\mathrm{depth}R)+(\dim F-\mathrm{depth}F)$. Each summand is nonnegative by [L4]. Their sum vanishes exactly when both vanish, which by [L4] is the claimed Cohen–Macaulay equivalence.