Parabolic and Levi Borel models for quiver extension and flag stacks

Write the ordered decomposition as $V_i={⨁}_{r=1}^s{V}_i^r$ and put $d_i^r=\dim V_i^r$. At each vertex, the partial flag variety is the closed incidence locus in ${\prod }_{r=1}^{s-1}\mathrm{Gr}({\sum }_{j\le r}{d}_i^j,V_i)$: adjacent inclusions are the zero loci of the tautological subbundle maps to the next quotient bundle. It is projective by the relative Plucker construction of [F1]. On the open locus of flags transverse to an opposite coordinate block flag, successive elimination gives a unique block lower unitriangular adapted matrix; its entries are regular ratios of the invertible pivot minors. Its columns give an algebraic section of $G\to G\mathrm{/}P$. Permuting the coordinate pivots gives finitely many such charts covering all flags. The incidence locus of pairs consisting of a representation and a preserved flag is closed in ${\mathrm{Rep}}_Q(\mathbf{d})\times G\mathrm{/}P$, since each arrow's map from a flag subbundle to the corresponding quotient must vanish. The adapted-frame sections identify it locally with the product of the flag chart and the upper-block vector space $Y$, with transition action $P$; hence it is $S=G{\times }_{P}Y$, a separated quasi-projective finite-type variety. The space $G\mathrm{/}L$ is the open locus of tuples of subspaces of ranks $d_i^r$ whose direct sum is $V_i$, in the corresponding product of Grassmannians. Taking the normalized pivot frame in each summand gives algebraic local sections of $G\to G\mathrm{/}L$. The incidence locus in ${\mathrm{Rep}}_Q(\mathbf{d})\times G\mathrm{/}L$ requiring preservation of each cumulative sum is closed there; these sections identify it with $T=G{\times }_{L}Y$. Thus $T$ too is separated, quasi-projective and finite type. The quotient equivalences $[S\mathrm{/}G]\simeq [Y\mathrm{/}P]$ and $[T\mathrm{/}G]\simeq [Y\mathrm{/}L]$ extend and reduce principal bundles; the section charts show that these constructions are inverse, including their equivariant maps. Zero ranks and repeated flag stages impose no extra condition; omit their factors. Empty products are points.

We need a graded identification B_([T/G]) = B_([Y/L]) (2) with the same intrinsic convention of [F2] on both sides. On a G-frame model U, T x_G U = Y x_L U. To compare this with a usual L-frame model V, use R=(Y x U x V)/L. The projections to (Y x U)/L and (Y x V)/L are locally trivial fibrations with fibres V and U. The first quotient is locally split because an L-torsor is a tuple of frames of vector bundles, trivialized on their finite algebraic frame charts; for the second quotient the same fact holds directly. The spaces and their fibrations are algebraic, separated and finite type. Restricted to L, U is still a free action because L embeds in G; U has precisely the same ordinary cohomology gap as before. The acyclic-fibre lemma therefore gives the bounded-Hom comparisons on both projections. To see their coefficient identification, pull back to the common principal L-bundle Y x U x V. Both pullbacks are the given coefficient on Y; their descent data are identical L-actions. The locally split Cech descent theorem gives the canonical comparison. On a triple product both identifications are the identity on that same principal bundle, so the cocycle follows. This explicitly verifies the extra model hypotheses; it does not infer stabilization from freeness.

The projections from R are smooth of relative dimensions dim V and dim U. Their smooth BM pullbacks shift by twice those dimensions. The intrinsic index of [F2] on the first quotient is i+2 dim U, on the second i+2 dim V, and on R it is i+2 dim U+2 dim V. Thus the comparison has intrinsic degree zero. The same principal-bundle identity shows compatibility with external products and every equivariant morphism of Y. Proper maps remain proper on the associated quotients; smooth maps remain smooth, and ordinary base change identifies their operations on the common model. Thus (2) is a natural comparison of operations, not just a vector-space isomorphism. Exactly this argument handles any two induction presentations used in flags, with their common frame products. There is no parabolic category silently substituted for the proved GL-product category: parabolics are represented by the actual induced varieties S with acting group G.

Steps 1.1–3.1 prove the induced-variety descriptions, the canonical degree-preserving comparison, its product-resolution cocycle and naturality for the stated operations. AC enters only through [A1].