LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Hilbert polynomial of a finite scheme is its length

Statement

Assume AC and DC. For a finite scheme $Z$ over a field $k$ , of length $d = \dim_{k} Γ (Z, O_{Z})$ , and any invertible sheaf $L$ on $Z$ , $χ (Z, L^{\otimes r}) = d$ for every integer $r$ . Thus its Hilbert polynomial for every polarization is the constant polynomial $d$ . Nonreduced schemes, non-rational closed points, and the empty scheme are included.

Facts & Assumptions

Given: The hypotheses in the statement and AC and DC, inherited from the scheme, cohomology, and finite-module suppliers (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

Affine quasi-coherent higher cohomology vanishes (Affine acyclicity of quasi-coherent sheaves). Nakayama's lemma is Assuming the Axiom of Choice, Nakayama's lemma.

Proof

1.1F1algebra

The algebra $C = Γ (Z, O_{Z})$ is finite-dimensional over $k$ , hence Artinian, and decomposes as a finite product of Artinian local rings. On each local factor an invertible module is free of rank one: lift a generator from its residue field, use Nakayama for surjectivity, and use the local rank-one trivialization to see that the map is an isomorphism. Consequently every power $L^{\otimes r}$ has a section module isomorphic, as a $C$ -module, to $C$ , and therefore of $k$ -dimension $d$ .

2.1F1step 1.1algebra∎

The finite scheme is affine, so [F1] makes all positive cohomology vanish. Euler characteristic is therefore $d$ for each $r$ , including negative powers and $r = 0$ . When $Z$ is empty all modules and dimensions are zero and the same argument gives polynomial zero.

Depends on

Used by

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Sources

Nitin Nitsure, Construction of Hilbert and Quot Schemes, Sections 2–5 (standard reference, not scraped)
Alexander Grothendieck, Les schémas de Hilbert, Bourbaki 221, Sections 2–3 (standard reference, not scraped)