Adapted frames identify the universal quotient-projective stack

For $(V,q:V\twoheadrightarrow L)$ let $Q$ be the sheaf of adapted frames $(a:{\mathcal{O}}_T^n\stackrel{\sim }{\to }V,\mathrm{\ell }:{\mathcal{O}}_T\stackrel{\sim }{\to }L)$ satisfying $qa=\mathrm{\ell }{q}_0$. It has right action $(a,\mathrm{\ell })(g,\lambda )=(ag,\mathrm{\ell }\lambda )$. Full faithfulness in [F3] applied to fppf covers makes this adapted-frame presheaf a sheaf in the topology of [F5]. To see that it has Zariski local sections, trivialize $V$ and $L$. Write $q=(a_1,\dots ,a_n)$ in those coordinates. At each point at least one coefficient is a unit on a neighbourhood, since the row is surjective. After permuting this coefficient to position $n$, use the vectors $e_j-(a_j\mathrm{/}{a}_n)e_n$ for $j<n$ and $a_n^{-1}{e}_n$ for the last vector. Their images are $0,\dots ,0,1$, and the change-of-basis matrix is invertible with inverse obtained by the reverse elementary operations. This constructs an adapted frame. Any two adapted frames differ by a unique $(g,\lambda )$ with $q_{0}g=\lambda q_0$, hence by a unique section of $H_n$. On each such neighbourhood $Q$ is therefore $H_n\times T$, with its translation action; these identifications describe a right torsor and commute with pullback. Conversely, trivialize any right $H_n$-torsor fppf locally, using the assumed Choice to select the trivializations on a covering family within the fixed size bounds. By [F3] this family is an fpqc covering. More explicitly, over each affine open of $T$ its restriction has a finite covering refinement by affine opens in its members; this is an fpqc refinement by source reference 2. No finite covering of the whole $T$ is required. On the overlaps its transition matrices and characters act on the standard quotient ${\mathcal{O}}^n\twoheadrightarrow \mathcal{O}$. Their cocycle identities descend these two modules and their quotient by [F3]; finite local freeness descends, their ranks remain $n$ and one, and faithful flatness reflects surjectivity. Denote the resulting pair by $Q{\times }^{H_n}(F^n\twoheadrightarrow F)$; this notation means the descended pair, so representability of a separate orbit quotient is not assumed. The constructions on maps also descend by full faithfulness. Starting instead with arbitrary descent data of bundle-quotient pairs on any fppf cover, the same [F3] descends both modules and the quotient map, their finite local freeness and surjectivity. Descent of morphisms and their inverses is fully faithful. Thus $P_n$ is an fppf stack in the precise sense of [F5]; forgetting the line and quotient gives the same conclusion for $B_n$. Evaluation $[(a,\mathrm{\ell }),x]\mapsto a(x)$, and its line analogue, identify the pair reconstructed from adapted frames with $(V,q,L)$. Conversely a point of a torsor gives an adapted frame of its descended pair; this equivariant map is an isomorphism on a trivializing cover and therefore an isomorphism of torsor sheaves. These two isomorphisms are natural and give inverse equivalences on groupoids. The torsor construction also identifies the projection and line with their stated representations. By [F1] its bundle-quotient groupoid is precisely $\mathbb{P}(U_n)$. The constructions commute with restriction and their natural inverse isomorphisms identify the descent groupoids as well, so $B{H}_n$ satisfies the same stack condition. This proves the stack equivalence, not just a bijection on object classes.

For two pairs $(V,q,L)$ and $(V^{\mathrm{\prime }},q^{\mathrm{\prime }},L^{\mathrm{\prime }})$ over $T$, describe their isomorphisms locally where all four bundles are free. Introduce matrices $f,g$ for maps $V\to V^{\mathrm{\prime }}$ and $V^{\mathrm{\prime }}\to V$ and scalars $u,v$ for maps $L\to L^{\mathrm{\prime }}$ and $L^{\mathrm{\prime }}\to L$. Impose $gf=1$, $fg=1$, $uv=vu=1$ and $q^{\mathrm{\prime }}f=uq$. These are polynomial equations over the local base ring. Their affine coordinate algebra represents exactly an isomorphism of pairs; the inverse variables are uniquely determined, so they add no extra choices. On overlaps, changing bundle frames changes these variables by invertible linear coordinate changes. Thus these finitely presented algebra sheaves glue, with their cocycle identity inherited from the changes of frames, to a quasi-coherent algebra on $T$. By [F4] its relative spectrum is affine over $T$ and represents the isomorphism sheaf. This proves that every pullback of the diagonal of $P_n$ is affine. For $n=1$ the same adapted-frame argument has no kernel vectors and all formulas reduce to the invertible line quotient; the empty base causes no exception. Every construction used only pullback, descent or intrinsic evaluation, hence commutes with base change and restricts to the smooth Zariski site as asserted.