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Projective bundle pullback detects cohomology classes

PDF p. 8: Injective pullback in the splitting principle

Statement

Assume the Axiom of Choice and let $H^{*, *}$ be an additively oriented bigraded theory with the projective bundle contract on a category $C$ as specified in the contract. For a finite locally free $O_{X}$ -module $E$ of constant rank $r > 0$ , the pullback $π^{*} : H^{*, *} (X) \to H^{*, *} (P_{X} (E))$ along the projective bundle projection is injective. Under the contract's direct-sum isomorphism, $π^{*}$ is the map from the summand indexed by zero. Consequently, pullback along any finite composite of such projective-bundle projections is injective. This conclusion does not assert injectivity for arbitrary pullbacks.

Statement references

Bigraded cohomology and the projective bundle contract

Facts & Assumptions

Given: The Axiom of Choice, the category and theory in the statement, an object $X$ of $C$ , and a finite locally free module $E$ of constant rank $r > 0$ . A finite locally free module of rank $r$ is locally isomorphic to $O_{X}^{r}$ ; $P_{X} (E)$ uses the quotient convention.

Goal: Prove the stated injectivity and its preservation under finite iteration.

[F1]

The projective bundle contract says that, for each $(a, b) \in Z^{2}$ , the map $Φ_{a, b} : ⨁_{m = 0}^{r - 1} H^{a - 2 m, b - m} (X) \to H^{a, b} (P_{X} (E))$ sending $(x_{m})$ to $\sum_{m} π^{*} (x_{m}) ξ^{m}$ is an isomorphism, where $ξ$ is the first Chern class of the tautological quotient line bundle. Pullbacks preserve bidegrees and are contravariantly functorial; $H^{*, *}$ is the direct sum of its bidegree groups (Bigraded cohomology and the projective bundle contract).

Proof

technique · Identify pullback with a direct-summand inclusion and compose injective maps

1.1F1construct

Fix $(a, b)$ . Since $r > 0$ , the index zero occurs in the direct sum of [F1]. Let $j_{0}$ insert $x \in H^{a, b} (X)$ in that summand and put zero in all other summands. The coordinate projection onto that summand is a left inverse to $j_{0}$ , so $j_{0}$ is injective. By the displayed formula, $Φ_{a, b} j_{0} (x) = π^{*} (x) ξ^{0} = π^{*} (x)$ . As $Φ_{a, b}$ is an isomorphism, $π^{*} : H^{a, b} (X) \to H^{a, b} (P_{X} (E))$ is injective.

2.1F1step 1.1∎

A class in $H^{*, *} (X)$ has finitely many nonzero bidegree components, and pullback preserves bidegrees. If its pullback vanishes, each component pulls back to zero and is therefore zero by step 1.1. Thus $π^{*}$ is injective on the whole bigraded group. For a finite tower of projections satisfying the same positive-rank hypotheses at each stage, each pullback is injective by the same argument; their composite is injective, and functoriality identifies it with pullback along the composite projection. The empty composite gives the identity.

Depends on

Bigraded cohomology and the projective bundle contract

Used by · 1

Detection on a simultaneous flag tower

Dependency tree · 3 levels

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