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Line-quotient classes across clopen rank strata

PDF p. 8: Computations after the splitting principle

Statement

Assume AC. For the simultaneous componentwise flag tower $q : Y = ∐_{α} Y_{α} \to X$ of a finite bundle family on qcqs $X$ , let $j_{α} : Y_{α} ↪ Y$ be its clopen inclusions. If $E_{i}$ has rank $r_{i, α}$ on the stratum $X_{α}$ , its line-quotient filtration on $Y_{α}$ gives $[q_{α}^{*} (E_{i} ∣_{X_{α}})] = \sum_{s = 1}^{r_{i, α}} [L_{i, α, s}] .$ Here $K_{v b}^{0}$ denotes the group generated by isomorphism classes of finite locally free sheaves with the relation $[B] = [A] + [C]$ for every short exact sequence $0 \to A \to B \to C \to 0$ . Extension by zero along a clopen inclusion preserves finite local freeness and gives the global equality $[q^{*} E_{i}] = \sum_{α} \sum_{s = 1}^{r_{i, α}} [j_{α *} L_{i, α, s}] in K_{v b}^{0} (Y) .$ Each $j_{α *} L$ has rank one on $Y_{α}$ and zero elsewhere, so it is not a line bundle on all of $Y$ when the complement is nonempty. If all original ranks are constant, the ordinary simultaneous tower instead has global line quotients and gives a global sum of line-bundle classes. For varying ranks such an unqualified global line-bundle sum is false: on $X = Spec k ∐ Spec k$ , the bundle $0 ∐ O$ cannot be a finite sum of global line-bundle classes, nor can its pullback along a surjective componentwise flag tower.

Facts & Assumptions

Given: AC and the family, partition and tower in the Statement; an empty sum is zero.

Goal: Prove the valid class identities and the precise obstruction to a global line sum.

[F1]

The partition is finite, clopen and qcqs. Finite clopen common rank strata

[F2]

On each constant-rank stratum, the tower supplies locally split filtrations with line-bundle quotients; for constant ranks it supplies them globally. Flag towers with line-bundle quotients

[F3]

The defining relations of $K_{v b}^{0}$ are those stated above; they apply to locally split filtrations. AC is inherited from the tower. The Axiom of Choice

Proof

1.1F2F3algebra

For a stratum filtration $0 = F_{0} \subset \dots \subset F_{r} = q_{α}^{*} E_{i}$ with quotients $L_{s}$ , each short exact sequence $0 \to F_{s - 1} \to F_{s} \to L_{s} \to 0$ gives $[F_{s}] = [F_{s - 1}] + [L_{s}]$ . Starting at $[F_{0}] = 0$ and adding the finitely many relations proves the componentwise identity, including $r = 0$ . The same argument proves the global line-quotient sum when ranks are constant and [F2] gives a global filtration.

1.2F1F3construct

A sheaf extended by zero from $Y_{α}$ is the given sheaf on that open and zero on its open complement. If the sheaf is finite locally free there, these descriptions are local free charts of its extension: the ranks are its original ranks on the stratum and zero elsewhere. Extension is exact, since on both pieces it is exact, and stalkwise exactness detects exactness of sheaves. Thus it induces a homomorphism on $K_{v b}^{0}$ .

2.1F1F3step 1.1step 1.2

A finite locally free sheaf $G$ on $Y$ is canonically isomorphic to $⨁_{α} j_{α *} (G ∣_{Y_{α}})$ : the natural map is an isomorphism on every stratum, hence on every stalk. This is a finite direct sum by [F1], so its class is the sum of the classes of those extensions. Apply it to $q^{*} E_{i}$ , and then push forward the equalities of step 1.1 by the exact extensions of step 1.2. This proves the stated global equality with supported, possibly varying-rank bundles.

3.1F2F3step 2.1algebra∎

Rank at a point defines a homomorphism $K_{v b}^{0} (Y) \to Z$ : a short exact sequence of vector bundles splits locally by lifting a basis of its locally free quotient, so its fiber dimensions are additive. A sum of $N$ global line-bundle classes therefore has rank $N$ at every point. The displayed two-component bundle has ranks zero and one, so its class cannot equal such a sum. Any surjective componentwise tower has nonempty preimages of both components and preserves these two ranks, giving the same contradiction upstairs. This refutes the unqualified varying-rank line-sum assertion while retaining all valid identities above.

Depends on

Used by · 1

Detection on a simultaneous flag tower

Dependency tree · 3 levels

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