TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Serre duality for locally free sheaves on a smooth projective variety

Statement

Assume the Axiom of Choice. Let $X$ be a smooth projective $k$ -scheme of pure dimension $n$ and let $E$ be a finite locally free $O_{X}$ -module. With $ω_{X} = ⋀^{n} Ω_{X / k}^{1}$ , there is a normalized trace $t_{X} : H^{n} (X, ω_{X}) \to k$ independent of a projective embedding, such that for every $0 \leq q \leq n$ the cup product, contraction and trace give a functorial perfect pairing of finite-dimensional $k$ -vector spaces $H^{q} (X, E) \times H^{n - q} (X, E^{\lor} \otimes ω_{X}) ⟶ H^{n} (X, ω_{X}) \overset{t_{X}}{\to} k .$ Outside $0 \leq q \leq n$ the relevant cohomology groups vanish.

Facts & Assumptions

Given: $X, k, n, E$ as in the statement.

[F1]

Projective-space coherent Serre duality gives, for a closed embedding $i : X ↪ P_{k}^{N}$ , a perfect natural Yoneda pairing between $H^{q} (P^{N}, i_{*} E)$ and ${Ext}_{P^{N}}^{N - q} (i_{*} E, ω_{P^{N}})$ . (Serre duality for coherent sheaves on projective space)

[F2]

If $c = N - n$ , the regular-immersion Ext collapse gives a natural isomorphism ${Ext}_{P^{N}}^{c + r} (i_{*} E, ω_{P^{N}}) ≅ H^{r} (X, E^{\lor} \otimes ω_{X})$ ; the determinant identification of its local Koszul generator is the conormal adjunction formula. (Local-to-global Ext collapse for a regular immersion, Adjunction for a smooth closed subvariety)

[F3]

The Gysin trace obtained from [F1]–[F2] by taking $E = O_{X}$ and evaluating at $1$ is independent of the projective embedding. The comparison respects cup/evaluation for all finite locally free $E$ . (Embedding compatibility of smooth-projective Gysin traces)

[F4]

Cup product is natural in its sheaf arguments; the dualizing line is $ω_{X} = ⋀^{n} Ω_{X / k}^{1}$ , and for projective space the normalization sends the ordered Laurent generator to $1$ . (Cup product in sheaf cohomology, Dualizing line bundle and trace datum of a smooth projective variety)

[F5]

The Axiom of Choice is The Axiom of Choice.

[F6]

On a separated Noetherian scheme of dimension at most $n$ , quasi-coherent cohomology vanishes in degrees above $n$ . (Dimension bound for quasi-coherent cohomology on a Noetherian scheme)

Proof

1.1F1F2F3F4

Choose a closed projective embedding $i : X ↪ P_{k}^{N}$ and put $c = N - n$ . The trace $t_{i} : H^{n} (X, ω_{X}) \to k$ is the image under [F2] of the projective-space Yoneda functional of [F1] evaluated at $1 \in H^{0} (X, O_{X})$ . By [F3] it is independent of $i$ ; write it $t_{X}$ . The normalization is the Laurent normalization of [F4], carried through the conormal determinant order of [F2]. If $X = \emptyset$ , every group displayed is zero and $t_{X} = 0$ .

2.1F1F2step 1.1

For $0 \leq q \leq n$ , [F1] is a perfect pairing of $H^{q} (P^{N}, i_{*} E) = H^{q} (X, E)$ with ${Ext}_{P^{N}}^{N - q} (i_{*} E, ω_{P^{N}})$ . Since $N - q = c + (n - q)$ , [F2] identifies the second vector space with $H^{n - q} (X, E^{\lor} \otimes ω_{X})$ . Therefore the transported pairing is perfect and both groups are finite-dimensional. This argument works componentwise and includes $n = 0$ : then the only degree is $q = 0$ and the same ambient perfectness applies.

3.1F1F2F3F4step 1.1step 2.1

Identify the transported pairing. The sign-normalized regular-immersion collapse in [F2] identifies the Yoneda product and evaluation of [F1] with the cup product, contraction $E \otimes E^{\lor} \to O_{X}$ , and the embedding trace $t_{i}$ , by the compatibility assertion of [F3]. This applies in every degree $0 \leq q \leq n$ and is natural in $E$ ; the Koszul determinant and shift signs are part of that normalized comparison. By 1.1, $t_{i} = t_{X}$ , so the perfect transported pairing of 2.1 is exactly the pairing displayed in the statement.

4.1F1F2F3F4F5F6step 1.1step 2.1step 3.1∎

Naturality in $E$ follows from the naturality of [F1]–[F3], and equivalently from cup product and contraction: a map $E \to E^{'}$ acts covariantly on the first factor and dually on the second. Naturality under an isomorphism of $X$ follows from [F3] and the functorial differential determinant. Since $X$ is projective over a field, it is separated and Noetherian of dimension $n$ , so [F6] makes cohomology above degree $n$ vanish; negative degrees vanish by definition of right derived cohomology. AC is inherited through [F1]–[F4] and [F6].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

76 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Duality for Schemes (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry Classes 53–54 (standard reference, not scraped)