Borel–Moore operations for quiver extension and flag quotient stacks

For two blocks, r:T -> S induced by L subset P is a G-equivariant Zariski locally trivial affine-space bundle of rank u. Over a P-frame chart its fibre is P/L=U. If p=(u0,l) then its action on U is u -> u0+l u l^{-1}; hence the transitions are affine linear. The map v:Y->Z is an L-equivariant vector-bundle projection of rank e. Write their induced stack maps as R=[Y/L] --r--> E=[Y/P] --q--> B=[Z/L], q r = v. For any locally trivial complex affine-space bundle a:A->B of rank d, a^!:B_i(B)->B_{i+2d}(A) is an isomorphism: in ordinary cohomology a^* is an isomorphism by contractibility on each local chart and Cech pasting; smooth dualizing orientation identifies BM pullback with this map after the 2d shift. This argument applies equivariantly on every fixed-group frame model and hence on the stable groups. It is not a claim that the bundle admits a global zero section. It applies to r and v, so define q^! := (r^!)^{-1} v^! : B_i(B) -> B_{i+2(e-u)}(E). (3) In this formula identify the R presentation by the comparison in [F2]. Thus every map in (3) is an already constructed fixed-group operation or the inverse of a proved isomorphism. This supplies the needed nonrepresentable smooth pullback without asserting that q is representable.

The proper p:E->M_d is represented, using E=[S/G], by the G-equivariant proper incidence map S->X_d. Its finite stage S x_G U -> X_d x_G U is proper: it is the associated closed subvariety in the relative product Grassmannian over X_d x_G U. The common auxiliary dimension dim U in the intrinsic normalization of [F2] proves that p_* has intrinsic degree zero. Naturality of the proper trace under the product-resolution comparison proves independence of U and composition.

For a finite block flag, P/L has the strictly upper block unipotent coordinates. Filter by distance from the diagonal; the successive quotients consist of the independent matrices in that distance and are vector groups. On pivot charts the quotient by each filtration step is a locally trivial affine bundle. Applying the preceding argument successively proves that r^! remains an isomorphism of shift 2 dim(P/L), independent of the filtration: after pullback to the adapted-frame charts it is the single positive complex Thom class of the ordered upper blocks, and ordinary smooth composition gives that class in every filtration order. The arrow-block projection has all corresponding free upper blocks, so (3) has shift twice sum_{s<t} (sum_{h:i->j} d_i^t d_j^s - sum_i d_i^s d_i^t). In particular successive diagonal-forgetting smooth maps compose with exactly this orientation and degree. This proves the flag composition rule used below, rather than assuming it for q.

Here are details ensuring that finite-stage proper/smooth base change and pasting give the interface needed for CoHA. In a 2-Cartesian square of these particular quotient stacks with a representable proper map f and a smooth block-forgetting map q, first choose a free frame cover C_N->C of the common target, with auxiliary dimension t. Since f is representable, A_N=A x_C C_N is a separated finite-type variety and A_N->C_N is proper. Use the induced incidence description to see it as the closed subvariety of the relative flag bundle; no additional source frame is adjoined to that proper arrow. If B->C is nonrepresentable, choose B's adapted-frame presentation and a free frame space of dimension w for its total GL product. Its associated principal-group quotient B_N over C_N is a separated finite-type variety. This follows directly from the upper-block vector-bundle chart over flags in [F2]; after choosing target frames, its map to C_N has fibre (A^e x U)/U_unip. The action is free on the frame factor. Local P-frame sections identify this quotient with an affine bundle over the corresponding frame quotient, so B_N->C_N is smooth of dimension r+w, where r=e-u. Put D_N = A_N x_{C_N} B_N. This is now an actual Cartesian square of varieties; D_N->B_N is proper and D_N->A_N is smooth of dimension r+w. Its four maps are the finite representatives of the original square. The intrinsic normalization of C_N,A_N uses t, and of B_N,D_N uses t+w. Thus the ordinary smooth shift 2(r+w) becomes 2r after the intrinsic normalization of [F2], while both proper shifts remain zero. Ordinary coherent proper/smooth base change applies, so q^! f_* = f'_* (q')^! (4) in the intrinsic groups. The map called q^! here is (3): add the Levi frame as well and compare on their product. On the common adapted-frame principal bundle both maps are the same vector-bundle orientation, with the u-dimensional affine-bundle orientation inverted. The faithful bounded-Hom comparison in [F2] identifies them before passing to groups.

This also proves that these constructions do not depend on cover or presentation. Any two finite choices are compared by their product; on that product the maps are ordinary unit/counit traces and positive complex smooth orientations, and the comparisons are natural in these maps. For a fixed finite collection of maps or equations, choose all frame gaps greater than the finitely many bounded-Hom comparison bounds of their inputs, outputs, units, counits and orientation maps. All objects descend from bounded constructible coefficients on the fixed finite presentation varieties. Faithfulness then reflects the finite-stage equality. One does not choose one N for every degree. For a pasted pair of Cartesian squares, use the same target frame cover, add the product of the finitely many source frame spaces, and pull back the proper arrows rather than adding independent frames to them. This realizes all corners of the pasted diagram as ordinary Cartesian squares simultaneously. The ordinary six-functor theorem says their exchange maps paste: expanding them as adjunction mates reduces the equation to the composition of units/counits and the triangle identities. The product-resolution comparisons preserve those units and counits by their common principal-bundle identity. Therefore (4) respects Cartesian pasting, composition of proper maps, composition of block-smooth maps, identities and external products. For external products use product frame covers; orientations concatenate in the displayed order and their real degrees are even. No unproved stack formalism, transversality, or finite-dimensional cohomology of an infinite classifying space is used.

Steps 1.1–4.1 construct the stated operations, degrees, comparison independence and coherence. AC enters through [A1].