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Finite clopen common rank strata

PDF p. 8: Computations after the splitting principle

Statement

Assume AC. Let $X$ be a quasi-compact quasi-separated scheme and $E_{1}, \dots, E_{m}$ a finite family of finite locally free sheaves, with no fixed-rank assumption. The nonempty common rank loci $X_{α}$ , where $α = (r_{1}, \dots, r_{m})$ and $rank E_{i} = r_{i}$ , form a finite partition by open-and-closed qcqs subschemes. Their disjoint union is canonically isomorphic to $X$ . Each $E_{i} ∣_{X_{α}}$ has constant rank $r_{i}$ . The simultaneous flag towers $q_{α} : Y_{α} \to X_{α}$ form a qcqs finite disjoint union $q : Y = ∐_{α} Y_{α} \to X$ . Rank-zero bundles have empty filtrations and require no projectivization; if every rank on a stratum is zero the tower is the identity there. The empty scheme has the empty partition and identity tower.

Facts & Assumptions

Given: AC, $X$ and its finite family of bundles. A rank is the cardinality of a local free basis; qcqs means quasi-compact and quasi-separated.

Goal: Establish finiteness, category membership and the stated componentwise construction.

[F1]

The flag construction gives finite positive-constant-rank projective towers for constant-rank bundles, preserves qcqs, and uses the identity for rank zero. Flag towers with line-bundle quotients

[F2]

AC is retained from the flag construction. The Axiom of Choice

Proof

1.1given

Each rank function is locally constant because local free trivializations have a fixed finite rank. A common rank locus is the finite intersection of its rank fibers, hence open; its complement is the union of the other such open loci, hence also open. Thus every common rank locus is open and closed, and they partition $X$ .

2.1step 1.1construct

These pairwise disjoint loci form an open cover of $X$ . Quasi-compactness gives a finite subcover; disjointness means every nonempty locus occurs in that subcover, so there are finitely many. Each is quasi-compact as a closed subset of $X$ and quasi-separated as an open subscheme of the quasi-separated $X$ . The open immersions identify their disjoint union with $X$ , since they cover and have empty pairwise intersections, with the same restricted structure sheaves.

3.1F1F2step 2.1construct

On each locus apply [F1] to the constant-rank restrictions and concatenate the finitely many towers. Each $Y_{α}$ is qcqs. A finite disjoint union of these schemes is quasi-compact, and intersections of affine opens from different components are empty while those within a component are quasi-compact; hence it is quasi-separated. The maps $q_{α}$ assemble to $q$ .

4.1F1step 1.1step 2.1step 3.1∎

The zero-rank and empty cases use the identities stipulated in [F1]. No bound on the dimension of $X$ , no noetherianity, connectedness or integrality, and no global rank constancy was used. This proves the claims.

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