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Generic flatness for finite type morphisms over Noetherian integral bases

Statement

Assume the Axiom of Choice (AC). Let $f : X \to S$ be a morphism of finite type (Locally finite type and finite type morphisms) whose base $S$ is a Noetherian integral scheme (Locally Noetherian and Noetherian schemes, Integral schemes). Then there exists a dense open subscheme $U \subseteq S$ such that the restriction $f^{- 1} (U) \to U$ is flat.

The open set produced is a finite union of nonempty principal opens and may meet parts of $S$ over which the fibre of $f$ is empty; no nonemptiness of fibres is used.

Facts & Assumptions

Given: The data and hypotheses displayed in the Statement, with the conventions fixed there.

[F1]

Assume AC. If $A$ is a Noetherian domain, $B$ a finitely generated $A$ -algebra and $M$ a finitely generated $B$ -module, then there is a nonzero $a \in A$ with $M_{a}$ free over $A_{a}$ (Generic freeness over a Noetherian domain).

[F2]

An integral scheme is nonempty and every nonempty affine open of it is the spectrum of a domain (Integral schemes).

[F3]

A morphism is of finite type when it is locally of finite type and quasi-compact; equivalently it is affine-locally given by finite-type ring maps, and inverse images of affine opens are quasi-compact (Locally finite type and finite type morphisms, Quasi-compact and quasi-separated morphisms).

[F4]

On affine charts $U = Spec B \subseteq X$ over $V = Spec A \subseteq S$ the morphism is flat at every point of $U$ if and only if $B$ is flat over $A$ (Affine-local flatness).

[F5]

For an affine chart $Spec B$ over $Spec A$ and $g \in A$ , the inverse image of $D (g)$ is $Spec (B \otimes_{A} A_{g}) = Spec B_{g}$ : fibre products of affine schemes are computed by tensor products (Affine fibre products are spectra of tensor products).

[F6]

Free modules are flat (Under the stated choice boundary, free modules are projective and hence flat).

Proof

technique · direct

1.1F2F3

Since $S$ is integral and Noetherian, affine opens form a basis and $S$ is quasi-compact; choose finitely many affine opens $Spec A_{1}, \dots, Spec A_{r}$ covering $S$ , so that each $A_{i}$ is a Noetherian domain by [F2].

1.2F3

For each $i$ the open $f^{- 1} (Spec A_{i})$ is quasi-compact by [F3]; choose a finite affine open cover $Spec B_{i 1}, \dots, Spec B_{i s_{i}}$ of it, so the induced ring maps $A_{i} \to B_{i j}$ are of finite type.

1.3F1F6

Apply [F1] to the Noetherian domain $A_{i}$ , the finitely generated $A_{i}$ -algebra $B_{i j}$ and the finitely generated $B_{i j}$ -module $M = B_{i j}$ : there is $0 \neq g_{i j} \in A_{i}$ with $(B_{i j})_{g_{i j}}$ free, hence flat, over $(A_{i})_{g_{i j}}$ .

2.1F2step 1.3

For each $i$ put $g_{i} = \prod_{j = 1}^{s_{i}} g_{i j} \in A_{i}$ , with the empty product $g_{i} = 1$ when $s_{i} = 0$ , and set $U_{i} = D (g_{i}) \subseteq Spec A_{i}$ . Since $A_{i}$ is a domain and every $g_{i j}$ is nonzero, $g_{i} \neq 0$ , so $U_{i}$ is a nonempty principal open. Put $U = U_{1} \cup \dots \cup U_{r} \subseteq S$ . Each $U_{i}$ is a nonempty open subset of the irreducible space $S$ , hence is dense in $S$ ; therefore $U$ is a dense open subset of $S$ and is a finite union of nonempty principal opens.

2.2F4F5F6step 1.3algebra

Fix $i$ and $j$ . The inverse image of $U_{i} = D (g_{i})$ inside the source chart $Spec B_{i j}$ is $Spec (B_{i j})_{g_{i}}$ by [F5]. Since $g_{i}$ is a multiple of $g_{i j}$ , this is the base change $(B_{i j})_{g_{i j}} \otimes_{(A_{i})_{g_{i j}}} (A_{i})_{g_{i}}$ ; the free $(A_{i})_{g_{i j}}$ -basis from step 1.3 remains a free basis after this base change. Thus $(B_{i j})_{g_{i}}$ is free, hence flat by [F6], over $(A_{i})_{g_{i}} = Γ (U_{i}, O_{S})$ . By [F4], $f$ is flat on this restricted source chart.

3.1F4step 1.2step 2.1step 2.2

For each $i$ , the source charts $Spec B_{i j}$ from step 1.2 cover the whole inverse image $f^{- 1} (Spec A_{i})$ , so their restrictions $Spec (B_{i j})_{g_{i}}$ cover the whole inverse image $f^{- 1} (U_{i})$ . If $s_{i} = 0$ , that inverse image is empty and flatness there is vacuous. Step 2.2 makes every nonempty restricted chart flat over $U_{i}$ . As the $U_{i}$ cover $U$ , flatness is local on source and target by [F4], and $f^{- 1} (U) \to U$ is flat.

4.1

The construction uses only the algebra maps $A_{i} \to B_{i j}$ ; no fibre of $f$ is assumed nonempty, and where $B_{i j}$ is the zero ring the lemma [F1] still supplies a nonzero $g_{i j}$ with $(B_{i j})_{g_{i j}} = 0$ free over $(A_{i})_{g_{i j}}$ , so charts with empty fibres are included in $U$ without harm. The Axiom of Choice is used exactly through [F1], as the Statement declares. [F1, step 1.3] $□$

Generic flatness for finite type morphisms over Noetherian integral bases

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources