Flag towers with line-bundle quotients

Set $X_0=X$ and $K_0=E$. For $j=1,\dots ,r$, provided $K_{j-1}$ has rank $r-j+1$, put $X_j={\mathbb{P}}_{X_{j-1}}(K_{j-1})$ with projection $p_j$, and define $K_j$ as the kernel of the tautological surjection $p_j^{\ast }{K}_{j-1}\twoheadrightarrow L_j:={\mathcal{O}}_{X_j}(1)$. By [F1], $L_j$ is a line bundle, and by [F2] $K_j$ is a vector bundle of rank $r-j$ and the resulting short exact sequence is locally split. Induction constructs all $r$ stages. The rank-zero bundle $K_r$ is zero. For $r=0$ no stage is taken.

Put $Y=X_r$ and $q=p_1\circ \cdots \circ p_r$, with the identity map when $r=0$. Pull every sequence of step 1.1 to $Y$ along the composite of the subsequent projections. A locally split exact sequence remains split after pullback on the inverse images of its splitting neighbourhoods; consequently these pullbacks are exact and their kernels embed as locally direct summands. Write ${\bar{K}}_j$ for the pullback of $K_j$ to $Y$, and ${\bar{L}}_j$ for the pullback of $L_j$. Then ${\bar{K}}_0=q^{\ast }E$, ${\bar{K}}_r=0$, and $0⟶{\bar{K}}_j⟶{\bar{K}}_{j-1}⟶{\bar{L}}_j⟶0$ is locally split, with ${\bar{L}}_j$ a line bundle. Set $F_i={\bar{K}}_{r-i}$. This gives the required filtration, with $F_i\mathrm{/}{F}_{i-1}\cong {\bar{L}}_{r-i+1}$. Pullback preserves local freeness because it takes a local free basis to a local free basis.

We prove the topological assertion for one projective-bundle stage $\pi :P={\mathbb{P}}_S(G)\to S$ with $S$ quasi-compact and quasi-separated and $G$ of positive constant rank. Choose a finite affine cover $U_1,\dots ,U_m$ of $S$ on which $G$ is free: refine a trivializing cover by affine opens and use quasi-compactness. By [F1] and [F3], each ${\pi }^{-1}(U_i)$ is covered by finitely many standard affine charts, so $P$ is quasi-compact.

Let $A$ and $B$ be two of those affine charts, lying over $U_i$ and $U_j$. The intersection $U_i\cap U_j$ is quasi-compact because $S$ is quasi-separated; cover it by finitely many affine opens $W_{\nu }$. The inverse images of $W_{\nu }$ in $A$ and $B$ are affine: each chart is affine over its base open, and the inverse image of an affine open under an affine morphism is affine. They are affine opens of $P_{W_{\nu }}\cong {\mathbb{P}}_{W_{\nu }}^{n-1}$. Their intersection is affine, since the diagonal of this projective space over the affine scheme $W_{\nu }$ is closed by [F3]: the intersection is its inverse image in the affine fibre product of the two affine opens, hence is a closed subscheme of an affine scheme. These finitely many intersections cover $A\cap B$, so $A\cap B$ is quasi-compact. A scheme with a finite affine open cover whose pairwise intersections are quasi-compact is quasi-separated: any quasi-compact open is covered by finitely many principal affine opens inside the cover, and intersections of such opens are quasi-compact as principal opens in the quasi-compact pairwise chart intersections. Thus $P$ is quasi-separated.

Applying steps 3.1–4.1 successively to the finitely many stages in step 1.1 shows that $Y$ is quasi-compact and quasi-separated whenever $X$ is. The rank-zero case is immediate because $Y=X$.

For a finite family of constant-rank bundles, first construct the tower for the first bundle, then pull the second bundle back to its top and construct its tower, and continue. Pullback of each previously constructed locally split filtration remains a locally split filtration with line-bundle quotients, by the argument of step 2.1. The concatenation is finite, and step 5.1 applies at every stage when the initial scheme is quasi-compact and quasi-separated.

For a finite family with locally varying ranks on a quasi-compact quasi-separated $X$, each rank function is locally constant: a local free chart has a fixed number of basis elements. Hence every common rank stratum is open, its complement is the union of the other open strata, and it is therefore also closed. The common strata form an open cover of $X$; quasi-compactness makes the set of nonempty strata finite. Each is quasi-compact as a closed subset of $X$ and quasi-separated as an open subscheme: intersections of its quasi-compact opens are intersections in $X$. Apply step 6.1 on each stratum and take the finite disjoint union of the resulting towers and maps. The filtrations have constant length on each stratum, and their pullbacks give the claimed locally varying filtrations. The finite disjoint union is quasi-compact and quasi-separated, since intersections between distinct components are empty and the same properties hold within each component.

Steps 1.1–2.1 construct the flag tower and its line quotients; steps 5.1 and 6.1–7.1 prove the remaining claims. Only local splittings have been used, so no global direct-sum decomposition is claimed.