The flat locus of a finitely presented algebra is open

Fix a prime $\mathfrak{q}\subseteq B$ at which $M_{\mathfrak{q}}$ is $R$-flat, and put $\mathfrak{p}=\mathfrak{q}\cap R$. If $M_{\mathfrak{q}}=0$, finite presentation and [F7] give $g\mathrm{\notin }\mathfrak{q}$ with $M_g=0$, hence $M_g$ is flat. Otherwise use [F2] to express the finitely presented pair $(R\to B,M)$ at these contracted primes as a directed Noetherian local approximation. Eventual flatness in [F2] makes the module flat over the base at a sufficiently late Noetherian stage. It therefore suffices to prove the neighbourhood assertion there: a principal neighbourhood at that stage pulls back to one at $\mathfrak{q}$, and its flat module remains flat after base change and localization.

Assume henceforth that $R$ is Noetherian and that $M_{\mathfrak{q}}$ is $R_{\mathfrak{p}}$-flat. Present $B$ as a quotient of $P=R[T_1,\dots ,T_n]$ and regard $M$ as a finite $P$-module; its support lies in the closed subscheme $\mathrm{Spec}B$ of $\mathrm{Spec}P$. If the chosen presentation has $n=0$, adjoin a dummy variable $T_1$ and the relation $T_1=0$, so $n\ge 1$. Let $\mathfrak{Q}$ be the contraction of $\mathfrak{q}$ to $P$. Since $P$ is Noetherian, construct a free resolution $\cdots \to F_1\to F_0\to M\to 0$ with each $F_j$ finite free over $P$; only its first $n$ terms will be used. Set $K_n=\ker (F_{n-1}\to F_{n-2})$, with $F_{-1}=M$ when $n=1$. Then $K_n$ is a finite $P$-module.

At $\mathfrak{Q}$, the map $R_{\mathfrak{p}}\to P_{\mathfrak{Q}}$ is flat and $M_{\mathfrak{q}}$ is $R_{\mathfrak{p}}$-flat. Apply [F9] to the truncated free resolution: $(K_n{)}_{\mathfrak{Q}}$ is $R_{\mathfrak{p}}$-flat, and tensoring the sequence $0\to (K_n{)}_{\mathfrak{Q}}\to (F_{n-1}{)}_{\mathfrak{Q}}\to \cdots \to (F_0{)}_{\mathfrak{Q}}\to M_{\mathfrak{q}}\to 0$ with $\kappa (\mathfrak{p})$ preserves exactness. The closed-fibre ring $P_{\mathfrak{Q}}\mathrm{/}\mathfrak{p}{P}_{\mathfrak{Q}}$ is a localization of $\kappa (\mathfrak{p})[T_1,\dots ,T_n]$, so [F3] gives global dimension at most $n$. The displayed reduced resolution makes $(K_n{)}_{\mathfrak{Q}}\mathrm{/}\mathfrak{p}(K_n{)}_{\mathfrak{Q}}$ a finite projective, hence free, module over this local fibre ring. If $(K_n{)}_{\mathfrak{Q}}=0$ it is already free of rank zero; otherwise [F4] makes it free over $P_{\mathfrak{Q}}$.

The finite $P$-module $K_n$ is finitely presented because $P$ is Noetherian. Lift a basis of the free localization $(K_n{)}_{\mathfrak{Q}}$ to finitely many sections of $K_n$ after one principal localization. Their map from a finite free module has finite kernel and cokernel, both zero at $\mathfrak{Q}$, so [F7] lets us shrink to a principal $D(g)$ containing $\mathfrak{Q}$ on which $K_n$ is free. The truncated finite complex $0\to (K_n{)}_g\to (F_{n-1}{)}_g\to \cdots \to (F_0{)}_g$ is therefore a finite free complex on $P_g$. Its fibre at $\mathfrak{Q}$ is exact in positive degrees by step 2.1. The family $R\to P_g$ is flat and finite type, and every nonempty fibre is a principal open in affine $n$-space over a residue field, hence regular, Cohen–Macaulay and equidimensional of dimension $n$. By [F5], after another principal shrink around $\mathfrak{Q}$, this complex is exact in positive degrees on every local fibre.

At each point of that final neighbourhood, the terms of the truncated complex are finite over the local target ring and flat over the local base ring. Apply [F6] to its exact local fibre complex: its final cokernel $M$ is flat over the local base. Thus $M$ is $R$-flat at every point of a principal neighbourhood of $\mathfrak{Q}$ in $\mathrm{Spec}P$. Intersecting with $\mathrm{Spec}B$ gives a neighbourhood of $\mathfrak{q}$ where $M$ is $R$-flat. Step 1.1 transports this neighbourhood from the Noetherian stage to the original pair. Since every flat point has such a neighbourhood, the module flat locus is open; taking $M=B$ gives the algebra claim. AC is inherited through [F2]–[F6], as recorded in [F8].