Fibre-regular hypersurface cuts preserve flatness and produce finite image slices

Flatness identifies ${\mathfrak{m}}^{j}B\mathrm{/}{\mathfrak{m}}^{j+1}B$ with $({\mathfrak{m}}^j\mathrm{/}{\mathfrak{m}}^{j+1}){\otimes }_{A\mathrm{/}\mathfrak{m}}(B\mathrm{/}\mathfrak{m}B)$, by tensoring the inclusions of the ideals of $A$. Multiplication by $f$ is injective on every such graded piece, since it is injective on the second factor and the first is a vector space. If $fb=0$, injectivity first gives $b\in \mathfrak{m}B$, then successively $b\in {\mathfrak{m}}^{j}B$ for every $j$. Krull intersection in the local ring $B$ gives $b=0$. The exact sequence $0\to B\stackrel{f}{\to }B\to B\mathrm{/}fB\to 0$ and $A$-flatness of $B$ identify ${\mathrm{Tor}}_1^A(A\mathrm{/}\mathfrak{m},B\mathrm{/}fB)$ with the kernel of $f$ on $B\mathrm{/}\mathfrak{m}B$, which is zero. The finite-over-target criterion [F1] proves that $B\mathrm{/}fB$ is $A$-flat. No finiteness over $A$ is used.

If $u(Y_z)$ is already finite, take $F=X$. Otherwise its image is constructible by applying [F2] on a finite affine cover of $Y_z$, and contains infinitely many closed points: a nonempty locally closed positive-dimensional piece has infinitely many closed points, while a zero-dimensional finite-type scheme has only finitely many points. Here $\kappa (z)\mathrm{/}k$ is finite, so $Y_z$ is finite type over $k$. Choose a closed image point $x$ different from the images of the finitely many associated points of $Y_z$. Write $X=\mathrm{Spec}C$ and ${\mathfrak{p}}_i$ for their image primes. Since the maximal ideal ${\mathfrak{m}}_x$ is not contained in any ${\mathfrak{p}}_i$, prime avoidance supplies $h\in {\mathfrak{m}}_x\setminus {\bigcup }_i{\mathfrak{p}}_i$. Thus $V(h)$ meets $u(Y_z)$ at $x$ and avoids all associated points after pullback. At every point of the cut over $z$, its defining element is regular in the fibre local ring, and step 1.1 proves flatness of the cut over $Z$.

Repeat inside the affine closed subscheme $V(h)$ if its fibre image is still infinite, using the new fibre's associated points. Each cut has nonempty fibre and is proper: its defining element avoids the fibre's associated points, so cannot vanish identically on that fibre. Thus the defining ideals in the original Noetherian affine ring strictly increase at every repetition. The ascending chain condition forces termination. The last image is finite and nonempty, and step 1.1 preserves the required flatness at each stage. AC is inherited from the associated-prime, Krull-intersection and local-flatness suppliers.