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Projective bundle closure and naturality of the actual decomposition

PDF p. 2: Motivic projective bundle property; PDF p. 8: Orientation for the splitting principle

Statement

Assume the Axiom of Choice. The category of qcqs $k$ -schemes is closed under the positive-constant-rank projective bundles and their base changes required by the bigraded projective bundle contract. For every finite locally free $E$ of constant rank $r > 0$ , the decomposition isomorphisms in every integer bidegree $(a, b)$ , given by $(x_{m})_{m} \mapsto \sum_{m = 0}^{r - 1} π^{*} (x_{m}) c_{1} (O (1))^{m}$ , are natural under base change $f : Y \to X$ , using $f^{*} E$ and the induced map $\overset{ˉ}{f} : Q_{Y} (f^{*} E) \to Q_{X} (E)$ .

Statement references

Bigraded cohomology and the projective bundle contract

The exact projective bundle map in every integer bidegree

Facts & Assumptions

[A1]

Assume the Axiom of Choice, as formulated in The Axiom of Choice. It discharges the projective-bundle construction and dual-convention hypotheses in step 1.1, and the all-bidegrees decomposition hypothesis in step 2.1.

[F1]

Projective bundle in the quotient convention gives local projective-space charts and arbitrary base change. Quotient and line projectivizations agree after dualizing identifies their tautological quotient with the source convention.

[F2]

Natural unital bigraded motivic ring with negative zero extension gives natural ring pullbacks; The actual EM first Chern class is additive and natural gives naturality of $c_{1}$ ; The exact projective bundle map in every integer bidegree gives the decomposition.

Proof

Given: $X, Y, f, E$ as stated, and AC.

Goal: Check the scheme-category hypotheses and map-level naturality.

1.1A1F1given

Choose a finite affine cover of $X$ trivializing $E$ , possible by quasi-compactness, refining finite affine trivializations. Above each member the projective bundle has finitely many standard affine charts. Hence its total space is quasi-compact. For intersections of two such charts, cover the quasi-compact intersection of their base affines by finitely many affine opens. On each, the intersection is a homogeneous principal chart $D_{+} (l_{1} l_{2})$ in a relative projective space, hence affine: $l_{1}, l_{2}$ are the linear coordinates for the two trivializations. Thus chart intersections are quasi-compact, proving quasi-separatedness. The same argument applies to $Y$ and $f^{*} E$ .

2.1A1F1F2step 1.1∎

Under [A1], the decomposition of [F2] applies to both $X$ and $Y$ . By base change, ${\overset{ˉ}{f}}^{*} O_{Q} (1) = O_{Q_{Y} (f^{*} E)} (1)$ and the projection square commutes. Therefore ${\overset{ˉ}{f}}^{*} ξ_{X} = ξ_{Y}$ by [F2]. For each summand, ${\overset{ˉ}{f}}^{*} (π_{X}^{*} x ξ_{X}^{m}) = π_{Y}^{*} (f^{*} x) ξ_{Y}^{m}$ by functoriality and multiplicativity. Summing proves the decomposition square commutes; it also holds on the zero negative-weight pieces.

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Used by · 1

Motivic projective bundle formula

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