CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Module-finite affine maps have finite fibres

Statement

Every module-finite morphism between affine classical algebraic sets is quasi-finite.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

For affine classical algebraic sets $X, Y$ , call a morphism $f : X \to Y$ module-finite if $k [X]$ , via pullback, is a finitely generated $k [Y]$ -module. This is the affine module criterion. Empty affine sets are allowed, with zero coordinate ring; the definition does not assert a global affine-preimage criterion for arbitrary varieties. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Module-finite affine maps for the quasi-finite comparison).

[F2]

A morphism $f : X \to Y$ of classical varieties is quasi-finite if every closed-point fibre $X_{y}$ is a finite set; empty fibres are allowed. Classical morphisms here are of finite type: for an affine target chart and an affine source chart above it, any finite set of $k$ -algebra generators of the source ring also generates it over the target ring. The inverse image has a finite affine cover because it is an open of a Noetherian variety. Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Quasi-finite classical morphisms).

[F3]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Reduced closed-point fibres and their dimension).

[F4]

Assume the Axiom of Choice. Let $k$ be an algebraically closed field. 1. The assignments $X ⟼ I (X), J ⟼ V (J)$ induce mutually inverse inclusion-reversing correspondences between affine algebraic sets $X \subseteq A_{k}^{n}$ and radical ideals $J \subseteq k [x_{1}, \dots, x_{n}]$ . 2. Under this correspondence, nonempty irreducible affine algebraic sets correspond exactly to prime ideals. (Affine algebraic sets correspond to radical ideals, and irreducible ones to prime ideals).

Proof

1.1

Put $A = k [Y]$ and $B = k [X]$ , a finite $A$ -module. For $y \in Y$ , the quotient $D = B / m_{y} B$ is finite-dimensional over $A / m_{y} = k$ . Its reduced quotient is the coordinate ring of the fibre. Thus it suffices to bound the number of distinct maximal ideals of $D$ .

F1F3

2.1

For any finite family of distinct maximal ideals $n_{1}, \dots, n_{s}$ , pairwise comaximality supplies, for every $i \neq j$ , an element of $n_{j}$ congruent to $1$ modulo $n_{i}$ . Multiplying these elements for fixed $i$ produces $e_{i}$ with residues $1$ at $i$ and $0$ at all other indices. The $e_{i}$ are linearly independent over $k$ , by reduction modulo each $n_{i}$ . Hence $s \leq \dim_{k} D$ , so there can only be finitely many maximal ideals. If $D = 0$ there are none.

step 1.1

3.1

Each fibre point gives a distinct evaluation maximal ideal of $D$ , and the affine point/ideal correspondence accounts for these points. Thus every fibre is finite, including the empty fibre, and the morphism is quasi-finite by definition. Empty source or target causes no exception.

F2F4step 2.1∎

Depends on

Used by

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Sources

Milne Proposition 8.28 and Lemma 8.29, p.185 (standard reference, not scraped)