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Exterior-square line factors on a flag space

PDF p. 17: Roots of exterior square

Statement

Let $S$ be a scheme and $E$ a rank-three vector bundle with a filtration $0 = F_{0} \subset F_{1} \subset F_{2} \subset F_{3} = E$ whose successive quotients $L_{i} = F_{i} / F_{i - 1}$ are line bundles and whose inclusions are locally split. Then $Λ^{2} E$ has a locally split filtration with successive line-bundle quotients $L_{1} \otimes L_{2}$ , $L_{1} \otimes L_{3}$ , and $L_{2} \otimes L_{3}$ , in that order. There is an isomorphism $\det E ≅ L_{1} \otimes L_{2} \otimes L_{3}$ .

Assume the Axiom of Choice and that $S$ is locally noetherian. In the vector-bundle Grothendieck ring $K^{0} (S)$ , write $[L_{i}]$ for the class of $L_{i}$ . Then $[Λ^{2} E] = [L_{1}] [L_{2}] + [L_{1}] [L_{3}] + [L_{2}] [L_{3}]$ and $[\det E] = [L_{1}] [L_{2}] [L_{3}]$ . If $c_{1}$ is any homomorphism from the Picard group of $S$ to an abelian group, and $ℓ_{i} = c_{1} (L_{i})$ , the first Chern classes of the three exterior-square line factors are respectively $ℓ_{1} + ℓ_{2}$ , $ℓ_{1} + ℓ_{3}$ , and $ℓ_{2} + ℓ_{3}$ .

A vector bundle is a finite locally free $O_{S}$ -module, and a line bundle has rank one. Locally split means that every inclusion in the filtration is locally the inclusion of a direct summand. The Picard group consists of line-bundle isomorphism classes under tensor product. The stated homomorphism condition means $c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)$ ; these line-factor classes are the exterior-square roots relative to the chosen flag. In particular, under the Axiom of Choice, the flag-tower supplier provides such a filtration after pulling a rank-three bundle back to its flag tower.

Facts & Assumptions

Given: A scheme $S$ , a rank-three bundle $E$ , and the locally split line-quotient filtration in the statement. For the $K^{0}$ assertions, also assume the Axiom of Choice and that $S$ is locally noetherian. For the root assertion, also assume the stated additive map $c_{1}$ .

Goal: Construct the exterior-square filtration and determinant isomorphism and derive the displayed class formulas.

[A1]

The Axiom of Choice is used only through the flag-tower supplier [F2] and the Grothendieck-ring supplier [F3]; the filtration and determinant constructions for a given flag require no additional Choice assumption. The Axiom of Choice

[F1]

Exterior powers and determinant are functorial for bundle maps; the exterior power of a free module has the basis of increasing wedges. In particular the determinant of a rank-three bundle is its third exterior power. Exterior powers and determinant of an algebraic bundle Exterior Algebra Basis Monomials

[F2]

Under the Axiom of Choice, the flag tower of a rank-three bundle carries a locally split filtration with line-bundle quotients. Flag towers with line-bundle quotients

[F3]

Under the Axiom of Choice, for a locally noetherian scheme $S$ , $K^{0} (S)$ is generated by vector-bundle classes with the relations $[B] = [A] + [C]$ for short exact sequences $0 \to A \to B \to C \to 0$ of vector bundles; multiplication is induced by tensor product. Grothendieck groups of coherent sheaves and of vector bundles on a scheme

Proof

technique · Construct the filtration by wedge images and identify its quotients on bases adapted to the flag

1.1givenF1construct

Let $G_{1}$ be the image of the wedge map $F_{1} \otimes F_{2} \to Λ^{2} E$ , and let $G_{2}$ be the image of $F_{1} \otimes E \to Λ^{2} E$ . These are subsheaves with $G_{1} \subseteq G_{2} \subseteq Λ^{2} E$ . Locally the two short exact sequences defining the flag split, and the three line quotients are trivial. Thus there is a local basis $e_{1}, e_{2}, e_{3}$ of $E$ with $F_{1}$ generated by $e_{1}$ and $F_{2}$ by $e_{1}, e_{2}$ . By [F1], $G_{1}$ is generated by $e_{1} \land e_{2}$ , $G_{2}$ by $e_{1} \land e_{2}, e_{1} \land e_{3}$ , and $Λ^{2} E$ by these two wedges and $e_{2} \land e_{3}$ . Consequently these subsheaves are locally direct summands and the successive quotients are line bundles.

2.1F1step 1.1construct

Wedge multiplication induces maps $L_{1} \otimes L_{2} \to G_{1}$ , $L_{1} \otimes L_{3} \to G_{2} / G_{1}$ , and $L_{2} \otimes L_{3} \to Λ^{2} E / G_{2}$ . They are well defined: changing a lift by a vector in the preceding flag term changes its wedge by zero or by a section of the indicated preceding $G$ term. On an adapted basis as in step 1.1, the maps send the tensor products of the quotient basis vectors to the corresponding wedge generators. They are therefore local isomorphisms, hence global isomorphisms. This proves the claimed filtration without choosing a global splitting.

2.2F1step 1.1construct

The map $L_{1} \otimes L_{2} \otimes L_{3} \to Λ^{3} E$ sending local quotient sections, with arbitrary lifts to their flag terms, to the wedge of those lifts is well defined: changing a lift adds a wedge containing two vectors from $F_{1}$ or three from $F_{2}$ , which vanishes locally on an adapted basis. On that basis it sends the tensor product of quotient generators to $e_{1} \land e_{2} \land e_{3}$ , a generator by [F1]. It is thus an isomorphism to $\det E = Λ^{3} E$ .

3.1A1F3step 2.1step 2.2assume-hyp

Under the additional $K^{0}$ hypotheses, apply [F3] to the short exact sequences $0 \to G_{1} \to G_{2} \to G_{2} / G_{1} \to 0$ and $0 \to G_{2} \to Λ^{2} E \to Λ^{2} E / G_{2} \to 0$ . Using step 2.1 and the tensor-product multiplication gives the displayed formula for $[Λ^{2} E]$ . Step 2.2 and the same multiplication give the determinant formula.

4.1A1F2step 2.1step 2.2step 3.1assume-hyp∎

Apply the assumed homomorphism $c_{1}$ to the three tensor-product line bundles identified in step 2.1. Additivity gives the three pairwise sums. Finally, [F2] supplies the required flag after pullback to the flag tower when the Axiom of Choice is assumed. All stated conclusions follow.

Depends on

Used by · 1

Chern classes of the alternating exterior class

Dependency tree · 3 levels

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