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Chern coefficients of the projective relation

PDF p. 8: Definition of motivic Chern classes

Statement

Assume the Axiom of Choice and the data and axioms of Bigraded cohomology and the projective bundle contract. Let $E$ be a finite locally free module of constant rank $r > 0$ on an object $X$ of the specified category. Write $π : P_{X} (E) \to X$ for the projective bundle of one-dimensional quotients and $ξ = c_{1} (O_{P_{X} (E)} (1)) \in H^{2, 1} (P_{X} (E))$ for the first Chern class of its tautological quotient line bundle.

There are unique classes $c_{i} (E) \in H^{2 i, i} (X)$ for $1 \leq i \leq r$ such that $ξ^{r} + \sum_{i = 1}^{r} (- 1)^{i} π^{*} c_{i} (E) ξ^{r - i} = 0.$ They are natural under every morphism $f : Y \to X$ in the specified category: $f^{*} c_{i} (E) = c_{i} (f^{*} E)$ . For the rank-zero bundle, set its total Chern series $c_{t} (0) = 1$ separately; equivalently, $c_{0} (0) = 1$ and $c_{i} (0) = 0$ for $i > 0$ . No coefficient is recovered from its empty projective bundle.

Facts & Assumptions

[F1]

By Bigraded cohomology and the projective bundle contract, pullbacks are functorial degree-preserving ring homomorphisms, first Chern classes of line bundles are natural, and the projective bundle map $⨁_{m = 0}^{r - 1} H^{a - 2 m, b - m} (X) \to H^{a, b} (P_{X} (E))$ , $(x_{m}) \mapsto \sum_{m} π^{*} x_{m} ξ^{m}$ , is an isomorphism. The contract includes the quotient projective bundle and its base change, and assumes the Axiom of Choice.

Proof

Given: The theory and bundle in the statement, and a morphism $f : Y \to X$ in the specified category.

Goal: Establish existence, uniqueness, bidegrees and base-change naturality of the coefficients, with the separate rank-zero convention.

1.1F1given

Apply the isomorphism in [F1] in bidegree $(2 r, r)$ to $- ξ^{r}$ . It gives a unique tuple $a_{m} \in H^{2 (r - m), r - m} (X)$ , $0 \leq m < r$ , with $- ξ^{r} = \sum_{m = 0}^{r - 1} π^{*} a_{m} ξ^{m}$ . Set $c_{i} (E) = (- 1)^{i} a_{r - i}$ . Then $c_{i} (E)$ has bidegree $(2 i, i)$ and the displayed relation holds. Conversely, any tuple satisfying that relation gives the same expansion of $- ξ^{r}$ , so the isomorphism makes every coefficient unique.

2.1F1givenstep 1.1∎

Let $g : P_{Y} (f^{*} E) \to P_{X} (E)$ be the base-change morphism and $π_{Y}$ the projection. The quotient construction identifies $g^{*} O_{P_{X} (E)} (1)$ with $O_{P_{Y} (f^{*} E)} (1)$ ; hence naturality of $c_{1}$ gives $g^{*} ξ = ξ_{Y}$ . Pulling back the relation in step 1.1 and using $π \circ g = f \circ π_{Y}$ gives the same relation for $f^{*} E$ with coefficients $f^{*} c_{i} (E)$ . Their degrees are preserved, so uniqueness in step 1.1 applied over $Y$ yields $f^{*} c_{i} (E) = c_{i} (f^{*} E)$ . For rank zero, the separately prescribed series $1$ has the stated coefficients and is preserved by unital pullback; no projective-bundle isomorphism for rank zero is invoked. This proves the statement.

Depends on

Bigraded cohomology and the projective bundle contract

Used by · 4

Dependency tree · 2 levels

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