General linear Stiefel approximations stabilize bounded constructible Hom

The frame bundle and its nerve. On a Grassmannian chart determined by an invertible $n\times n$ pivot minor, a frame is uniquely $\left(\genfrac{}{}{0px}{}{I_n}{T}\right)h$, with $h\in {\mathrm{GL}}_n(\mathbb{C})$ and $T$ the graph coordinates. These formulas and their inverses are regular, so they give algebraic principal-bundle charts for $U_N\to B_N$. In these charts the diagonal quotient is $V\times X$, with $(\left(\genfrac{}{}{0px}{}{I}{T}\right)h,x)$ mapping to $(T,hx)$, and the principal quotient $q_N:P_N\to X_N$ has section $(T,y)\mapsto (\left(\genfrac{}{}{0px}{}{I}{T}\right),y)$. The finitely many charts glue by the regular change-of-frame maps; they exhibit $X_N$ as a separated finite-type variety and $q_N$ as a principal $G$-bundle. Its action nerve is its Čech nerve, so [F5] applies. Projection to $X$ is equivariant for the left action specified in the Statement, and induces the simplicial map $p_N$. In degree $m$ this is a product projection $G^m\times U_N\times X\to G^m\times X$ with fibre $U_N$; its structure squares are Cartesian, since group action on the frame factor is invertible.

The fiber is $r_N$-acyclic with finite models. By [F2], $H^0(U_N;k)\cong k$ and $H^j(U_N;k)=0$ for $1\le j\le r_N$, and the product of Stiefel models supplies the bounded finite cochain model of the fiber with coefficient-field compatibility. By [F4] the base $G^m\times X$ carries the finite conical cochain models, so each levelwise projection of step 1.1 is a locally trivial bundle in the finite constructible local-product setting with $r_N$-acyclic fiber.

The simplicial unit bound. For a map of these nerves, the levelwise direct images acquire simplicial structure by the canonical sheaf base-change maps followed by the source structure maps. The levelwise adjunction gives $p_N^{\ast }⊣p_{N\ast }$, with exact inverse image. Evaluation of a simplicial injective at degree $m$ is injective: evaluation has an exact left adjoint, whose degree-$l$ term is the direct sum over operators $[m]\to [l]$ of their inverse images of the given sheaf. These sums and inverse images are exact. Thus a bounded-below simplicial injective replacement evaluates to an injective replacement at every level, and $(R{p}_{N\ast }C{)}_m=R{p}_{N,m\ast }{C}_m$. For $B\ge b$, [F3] and step 2.1 give that the cone of $B_m\to R{p}_{N,m\ast }{p}_{N,m}^{\ast }{B}_m$ lies in $D^{\ge b+r_N+1}$ at every level. The same bound therefore holds for the simplicial unit cone in the bounded-below derived category; no augmentation of the action nerve to $X$ is used.

The simplicial Hom comparison. The simplicial category has the exact levelwise inverse image $p_N^{\ast }$ left adjoint to the levelwise direct image $R{p}_{N\ast }$ and carries the canonical t-structure of [F7], whose aisles are defined cohomology-objectwise and hence are the levelwise aisles. With the cone bound of step 3.1 in place, the argument of [F3] — its Hom comparison, image criterion, full faithfulness and its retract and extension clauses — applies verbatim inside the simplicial bounded-below derived category, with its bounded full subcategory: for $A\in D^{\le a}$ and $B\in D^{\ge b}$ the long exact Hom sequence of the unit triangle and the orthogonality $\mathrm{Hom}(D^{\le a},D^{\ge a+1})=0$ identify the pullback map $\mathrm{Hom}(A,B[m^{\mathrm{\prime }}])\to \mathrm{Hom}(p_N^{\ast }A,p_N^{\ast }B[m^{\mathrm{\prime }}])$ as an isomorphism for $m^{\mathrm{\prime }}\le b+r_N-a$; for $a-b\le r_N$ it is fully faithful on the full subcategory $D^{[b,a]}$; and on that interval an object lies in the image exactly when the truncated counit $p_N^{\ast }{\tau }^{\le a}R{p}_{N\ast }C\to C$ is an isomorphism, the image being closed under retracts and under extensions when $r_N\ge a-b+1$. Every input of that argument is an input already verified here: the adjunction and levelwise model [F5], the cone bound of step 3.1, and the t-structure orthogonality [F7].

Descent and the functor $Q_N$. Restricting the conclusion of step 4.1 to the full subcategory of Cartesian objects identified with $D_c^b(X_N;k)$ by [F5], the functor $p_N^{\ast }$ followed by descent is $Q_N$: for $F\in D_{G,c}^b(X;k)$ the object $p_N^{\ast }F$ has Cartesian cohomology because $p_N$ is a simplicial map and the structure squares of $p_N^{\ast }$ are Cartesian by naturality of inverse image; descent produces $Q_N(F)\in D_c^b(X_N;k)$, and $q_N^{\ast }{Q}_N(F)\cong p_N^{\ast }F$ canonically. Hence, for $A\le a$, $B\ge b$ and $m\le b+r_N-a$, the isomorphisms of step 4.1 compose with the descent equivalence to give ${\mathrm{Hom}}_G(A,B[m])\cong {\mathrm{Hom}}_{X_N}(Q_N(A),Q_N(B)[m])$, proving claim 1; the same composition proves claim 2 on intervals of width at most $r_N$.

The image and Cartesian descent. Necessity follows from $q_N^{\ast }{Q}_N(A)={\mathrm{pr}}_X^{\ast }{A}_0$ at degree zero. Conversely suppose $H\in D_c^{[b,a]}(X_N;k)$ and $q_N^{\ast }H\cong {\mathrm{pr}}_X^{\ast }F$ for $F\in D_c^{[b,a]}(X;k)$. Let $C$ be the Cartesian simplicial pullback of $H$ to the principal-bundle nerve, and put $E={\tau }^{\le a}R{p}_{N\ast }C$ in the ambient simplicial category. It lies in $[b,a]$ because derived direct image preserves lower bounds. At every level $m$, Cartesianity identifies $C_m$ with the pullback of $C_0$, hence with $p_{N,m}^{\ast }{\mathrm{pr}}_{X,m}^{\ast }F$, where ${\mathrm{pr}}_{X,m}:G^m\times X\to X$ is projection. The levelwise image criterion [F3] therefore makes the truncated counit $p_N^{\ast }E\to C$ an isomorphism in every degree. Its naturality compares the structure maps of $p_N^{\ast }E$ with the Cartesian maps of $C$. Each $p_{N,m}$ is surjective and its structure squares are Cartesian, so its exact pullback reflects these quasi-isomorphisms; hence $E$ has Cartesian cohomology. Moreover $E_0\cong F$ by the truncated unit and $C_0\cong {\mathrm{pr}}_X^{\ast }F$, so it is algebraically constructible. Thus $E\in D_{G,c}^b(X;k)$ and Čech descent gives $Q_N(E)\cong H$. Retracts and extensions under $r_N\ge a-b+1$ lift by the degree-zero and degree-one Hom comparisons of step 4.1; their cones and summands have bounded Cartesian constructible cohomology, and the same counit argument gives the image. This proves claim 3 without assuming that ordinary descent data automatically strictify.

Coherence and the trivial group. The product $U_N\times U_M$ is a free smooth $G$-resolution in its own right. Its projections to $U_N$ and $U_M$ descend to ${\pi }_N:X_{N,M}\to X_N$ and ${\pi }_M:X_{N,M}\to X_M$, locally trivial bundles with fibres $U_M$ and $U_N$. Their pullbacks are fully faithful on the stated intervals by [F3] and the bounds $r_M,r_N$. On the common principal bundle $U_N\times U_M\times X$, both pulled-back representatives have the canonical descent identification with the same pullback of $A$ from $X$; these identifications are compatible with the principal-bundle nerve maps, so [F5]'s descent equivalence yields a canonical isomorphism ${\pi }_N^{\ast }{Q}_N(A)\simeq {\pi }_M^{\ast }{Q}_M(A)$. Naturality is inherited from those same inverse-image and descent maps. On the triple product all comparisons identify the representatives with the same pullback of $A$, so their two compositions agree and the cocycle law follows. This compares objects and Hom maps in the common category, without equating the two different target categories or identifying the product with a larger frame space. If $G$ is trivial, take the point as its resolution; then $X_N=X$, the action nerve is constant at $X$, $r_N=\mathrm{\infty }$ and $Q_N$ is the identity. This proves claims 4 and 5.

Conclusion. Steps 3.1 to 5.1 prove claims 1 and 2, step 6.1 claim 3, and step 7.1 claims 4 and 5. The comparison is used only on the nerve and through the finite local models: no global splitting of $R{a}_{N\ast }{a}_N^{\ast }B$ is asserted, so genuine characteristic-class transgressions are preserved. AC enters only through [A1].