Finite relative integral-closure charts for classical quasi-finite morphisms

Cover $f^{-1}U$ by finitely many affine opens $W_i$. Cover their pairwise intersections by finitely many affine opens, using [F1]. The sheaf axiom realizes $M_U$ as the kernel of the difference map from the finite product of the coordinate rings of $W_i$ to the finite product of the overlap-chart rings. After tensoring by any flat $A$-algebra $E$, exactness and commutation with finite products identify that kernel with the sections on the base-changed affine cover. This proves the flat-base-change assertion. In particular $M_{D(a)}=(M_U{)}_a$. By [F2], $C_{D(a)}=(C_U{)}_a$. The identifications are canonical restrictions and preserve multiplication.

Let $S_i$ be the coordinate ring of $W_i$ and $\mathfrak{q}$ a minimal prime of $S_i$; set $D=S_i\mathrm{/}\mathfrak{q}$ and $A_0=A\mathrm{/}\ker (A\to D)$. The corresponding irreducible component maps dominantly onto its image closure, with finite fibres because it is a closed subvariety of $W_i$. By [F3] its function field $L=\mathrm{Frac}D$ is a finite extension of $\mathrm{Frac}{A}_0$: the general fibre is zero-dimensional, so the relative transcendence degree is zero, and both fields are finitely generated. Choose a polynomial subring $R\subseteq A_0$ for which $A_0$ is module-finite by [F4]. Then $L\mathrm{/}\mathrm{Frac}R$ is finite. The integral closure $B$ of $R$ in $L$ is finite over $R$. Every element integral over $A_0$ is integral over $R$ by [F2]; hence the integral closure of $A_0$ in $D$ embeds as an $R$-submodule of $B$, and is finite over $R$ by [F4]. Its $R$-generators are also $A_0$-generators, since it is an $A_0$-module. It is therefore finite over $A$.

The reduced ring $S_i$ injects into the finite product of its minimal-prime domains $D$. Restricting an element integral over $A$ into each of these domains gives an element of the finite module found in step 1.2. Thus ${\mathrm{Int}}_A(S_i)$ is a submodule of a finite $A$-module and is finite by [F4]. The restriction map $M_U\hookrightarrow {\prod }_i{S}_i$ is injective by the sheaf axiom. Consequently $C_U$ is a submodule of the finite product ${\prod }_i{\mathrm{Int}}_A(S_i)$ and is finite. It is reduced because it is a subring of the ring of actual functions on $f^{-1}U$. If the inverse image is empty, $M_U=C_U=0$ and all assertions hold.

On every principal open of an affine target chart, step 1.1 identifies the finite algebra restrictions. On an overlap of target charts use a finite principal-open refinement; the identifications agree because each comes from restriction of actual functions. They satisfy the cocycle condition and glue their finite affine spaces and sheaves to $N$. Finiteness is affine local by its defining finite-module charts, so $N\to Y$ is finite. On $f^{-1}U$, every element of $C_U$ is a global regular function and evaluation is a regular map to the affine space of that algebra: choose finitely many algebra generators, evaluate them, and note that their defining relations vanish. The maps agree on the same refinements and give $j:X\to N$. Thus neither the charts nor this map depend on the covers chosen in the proof.