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Quotient and line projectivizations agree after dualizing

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

Statement

Assume AC. For a rank- $r > 0$ finite locally free sheaf $E$ on $X$ , let $Q_{X} (E) = {Proj}_{X} Sym (E)$ denote the quotient projectivization, and let $P_{X}^{l i n} (E^{\lor})$ denote EM’s projectivization of rank-one subbundles in $E^{\lor}$ . There is a canonical base-change-compatible $X$ -isomorphism $α : Q_{X} (E) ≃ P_{X}^{l i n} (E^{\lor})$ identifying the quotient $O_{Q} (1)$ with the dual of the subbundle $O_{P^{l i n}} (- 1)$ . In particular $α^{*} O_{P^{l i n}} (1) ≃ O_{Q} (1)$ , with no sign change of $c_{1}$ . Here subbundle means an inclusion whose cokernel is locally free.

Facts & Assumptions

[A1]

Assume the Axiom of Choice, as formulated in The Axiom of Choice. It is used in step 2.1 to invoke the AC-qualified line-quotient representation in [F1]; that published supplier uses it to select trivializing affine data.

[F1]

Projective bundle represents line quotients gives the natural representation of quotient lines. External EM multiplicative motivic complexes and actual Chern map records the source line convention and tautological dual.

[F2]

Finite locally free sheaves are canonically isomorphic to their double duals, checked on finite free local frames. A surjection onto a free rank-one sheaf locally splits: lift its basis section; elementary row operations show that its kernel is finite free of rank one less.

Proof

Given: AC, $E$ as stated and a test scheme $g : T \to X$ .

Goal: Construct the isomorphism and identify its actual tautological line.

1.1F1F2construct

Dualizing a quotient $g^{*} E ↠ L$ gives $L^{\lor} ↪ g^{*} (E^{\lor})$ with locally free cokernel by the local splitting in [F2]. Conversely, dualizing a rank-one subbundle $S ↪ g^{*} E^{\lor}$ gives $g^{*} E ↠ S^{\lor}$ , using the local splitting of its locally free cokernel and double duality. The two operations are inverse on isomorphism classes.

2.1A1F1F2step 1.1

These bijections commute with pullback, because all sheaves are finite locally free and the sequences locally split. Under [A1], the representing properties in [F1] give inverse morphisms $α$ . Applying the construction to the universal quotient identifies $S = L^{\lor}$ , hence $S^{\lor} = L$ ; this is exactly the asserted identification of the two bundles named $O (1)$ .

3.1step 2.1∎

For any base change the same correspondence applies, so the isomorphism and tautological identification commute with it. The Chern class is applied to the identical line bundle $S^{\lor} = L$ , not to its inverse, so no negative sign is introduced.

Depends on

Used by · 3

Dependency tree · 3 levels

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