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Relative projective-line cohomology shift

Statement

Assume the Axiom of Choice. Let $π : E \to S$ be a Zariski locally trivial $P^{1}$ -bundle of complex schemes, let $L$ be an invertible sheaf of constant geometric fibre degree $n \geq - 1$ , and write $K_{π} = ω_{E / S}$ . After fixing the invariant apolarity normalization of Relative projective-line cohomology and apolarity, for every $i \geq 0$ there is an isomorphism natural in $(E / S, L)$ and under restriction of $S$ , $H^{i} (E, L) ≅ H^{i + 1} (E, L \otimes K_{π}^{\otimes (n + 1)}) .$

Facts & Assumptions

Given: $π$ , $E$ , $S$ , $L$ , $n$ and $K_{π}$ as in the statement.

[F1]

The relative projective-line calculation gives $R^{q} π_{*} L = 0$ for $q > 0$ , $R^{q} π_{*} (L \otimes K_{π}^{\otimes (n + 1)}) = 0$ for $q \neq 1$ , and a base-restriction-compatible isomorphism $a : π_{*} L \tilde{\to} R^{1} π_{*} (L \otimes K_{π}^{\otimes (n + 1)})$ . For $n = - 1$ both displayed possibly nonzero sheaves vanish. (Relative projective-line cohomology and apolarity)

[F2]

For a morphism $f : X \to Y$ and an abelian sheaf $F$ there is a natural Leray spectral sequence $E_{2}^{p, q} = H^{p} (Y, R^{q} f_{*} F) \Rightarrow H^{p + q} (X, F)$ ; if all but one row $q = q_{0}$ vanish, its edge isomorphisms are $H^{p} (Y, R^{q_{0}} f_{*} F) ≅ H^{p + q_{0}} (X, F)$ . (Leray spectral sequence for sheaf cohomology)

[F3]

The Axiom of Choice is The Axiom of Choice.

Proof

1.1F1F2

Apply [F2] to $π$ and $L$ . By [F1] every $E_{2}$ row except $q = 0$ vanishes. There are therefore no possible incoming or outgoing differentials and the filtration of each abutment has one graded piece. Its edge map is the natural isomorphism $H^{i} (E, L) \tilde{\to} H^{i} (S, π_{*} L) (i \geq 0) .$

2.1F1F2step 1.1

Put $L^{'} = L \otimes K_{π}^{\otimes (n + 1)}$ . For $L^{'}$ the only possibly nonzero Leray row is $q = 1$ by [F1]. The same one-row argument yields $H^{i} (S, R^{1} π_{*} L^{'}) \tilde{\to} H^{i + 1} (E, L^{'})$ for every $i \geq 0$ . This isomorphism is the edge map with its degree-one shift, not a choice of a splitting of a multistep filtration.

3.1F1F2F3step 1.1step 2.1∎

Compose the isomorphism of 1.1, the map $H^{i} (S, a)$ of [F1], and the isomorphism of 2.1. This gives the displayed isomorphism. Every map in the composite is induced by a natural map of sheaves or by a one-row Leray edge map, so the composite commutes with restriction of $S$ and with isomorphisms of the bundle and line bundle that preserve the fixed apolarity normalization. If $n = - 1$ , [F1] makes both Leray rows zero, so both cohomology groups are zero and the same composite is the unique map $0 \to 0$ . AC is inherited through [F1] and [F2]. The completed sheaf-level apolarity isomorphism [F1] is used in the two Leray collapses and the final comparison.

Depends on

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Sources

The Stacks Project, Cohomology of Schemes (standard reference, not scraped)
Jacob Lurie, A Proof of the Borel–Weil–Bott Theorem (standard reference, not scraped)