Constructible six functors, proper base change and normalized equivariant duality

Smooth comparison and image preservation. On each quasi-projective source piece over an affine target, [F4] factors the map as an open inclusion into its graph closure in ${\mathbb{P}}^n\times Y$, followed by proper projection. The open-inclusion smooth exchange of [F1] followed by proper base change gives smooth exchange for $R{f}_{\ast }$ on this piece. A finite affine source cover and its intersections compute ordinary pushforward by finite open-cover derived descent [F4]; smooth inverse image is exact, so the piece comparisons give the canonical smooth exchange for the general map. The comparison respects composition because its maps are the ordinary units and base-change maps. Proper-support exchange for arbitrary maps is [F5]; smooth exchange for tensor, internal Hom and normalized duality follows locally by adjoining a contractible smooth ball to their finite cochain models. For a fixed frame resolution, write $D_X^N=R\mathcal{H}om(-,{\omega }_{X_N}[-2s_N])$. Smooth orientation gives $q_N^{\ast }{D}_X^N{Q}_N(A)\cong {\mathrm{pr}}_X^{\ast }{D}_X{A}_X$, since smooth orientation gives $q_N^{\ast }{\omega }_{X_N}={\mathrm{pr}}_X^{\ast }{\omega }_X[2s_N]$ and both dualizing shifts cancel the dimensions $u_N=g+s_N$. The smooth comparison for the square formed by $q_N$, $f_N$ and ${\mathrm{i}\mathrm{d}}_{U_N}\times f$, followed by the product projection to $f$, therefore gives $q_N^{\ast }R{f}_{N\ast }{Q}_N(A)\cong {\mathrm{pr}}_Y^{\ast }R{f}_{\ast }{A}_X$ and likewise for $R{f}_{N!}$. Inverse image and tensor have this comparison directly; internal Hom has it by its smooth comparison; $f_N^{!}=D_X^N{f}_N^{\ast }{D}_Y^N$ has it by duality. Thus every ordinary operation on the resolution representatives satisfies the frame-pullback image criterion [F3], with an ordinary bounded constructible object downstairs. Its unshifted cohomological amplitude equals the downstairs amplitude, because both $q_N^{\ast }$ and ${\mathrm{pr}}_X^{\ast }$ are exact and conservative. These are bounds independent of $N$.

Transport of operations and adjunctions. For each fixed finite collection of inputs, outputs and adjunction maps, choose $N$ whose gap exceeds the widths of their unshifted ordinary amplitudes and the finitely many Hom comparison bounds. By step 1.1 all the ordinary resolution operations lie in the finite-resolution image, so [F3] transports them to $D_{G,c}^b$. On a common product resolution the smooth comparisons identify the two constructions canonically; their coherence gives the cocycle on triple products, hence independence of the choice of $N$. Every bounded input is treated this way; no decomposition of an object into its cohomology sheaves is assumed. The ordinary adjunction Hom bijections, their units and counits, transport through the fully faithful comparisons. Naturality and the triangle identities are equations of these maps and are reflected by a sufficiently large faithful comparison. This proves $f^{\ast }⊣R{f}_{\ast }$, $R{f}_{!}⊣f^{!}$ and the tensor/internal-Hom adjunction on the actual bounded equivariant category.

Coherence. Composition of maps, identity comparisons, support localization, the projection formula, trace and Cartesian pasting are equations between morphisms built from unit and counit maps of the ordinary adjunctions. By [F1] these equations hold at the level of the resolution varieties, and naturality of the resolution comparisons plus their full faithfulness on the intervals means that an equation holding after transport to $X_N,Y_N$ holds in $D_{G,c}^b$; more precisely, the comparison functors are faithful on the relevant Hom groups by [F3], so the two sides of each coherence equation, which agree after transport, agree before it. The general proper-support exchange and its pasting transfer from [F5] in the same way; proper equality $R{f}_{!}=R{f}_{\ast }$ transfers from [F1]. This proves the first part of claim 2.

Normalization. For a frame resolution $P_N=U_N\times X\to X_N$ with base $B_N$ and maps $a_{X_N}:X_N\to B_N$, $q_N:P_N\to X_N$ and ${\mathrm{pr}}_X:P_N\to X$, put $u_N={\dim }_{\mathbb{C}}{U}_N$, $g={\dim }_{\mathbb{C}}G$ and $s_N={\dim }_{\mathbb{C}}{B}_N=u_N-g$, and define the normalized dualizing object ${\omega }_{X_N}^G:={\omega }_{X_N}{\otimes }_k{a}_{X_N}^{\ast }{\omega }_{B_N}^{-1}$, the inverse taken in the tensor category of bounded constructible sheaves on $B_N$; here $B_N$ is a smooth complex $s_N$-fold, so by [F2] its dualizing object is ${\omega }_{B_N}\simeq k_{B_N}[2s_N]$ and $a_{X_N}^{\ast }{\omega }_{B_N}^{-1}\simeq k_{X_N}[-2s_N]$. Transitivity of dualizing objects for the smooth maps ${\mathrm{pr}}_X$ of relative dimension $u_N$ and $q_N$ of relative dimension $g$ gives ${\omega }_{P_N}\simeq {\mathrm{pr}}_X^{\ast }{\omega }_X[2u_N]\simeq q_N^{\ast }{\omega }_{X_N}[2g]$, hence $q_N^{\ast }{\omega }_{X_N}\simeq {\mathrm{pr}}_X^{\ast }{\omega }_X[2s_N]$ and therefore $q_N^{\ast }{\omega }_{X_N}^G\simeq q_N^{\ast }{\omega }_{X_N}[-2s_N]\simeq {\mathrm{pr}}_X^{\ast }{\omega }_X$: the normalization cancels exactly the base contribution, so ${\omega }^G$ restricts to the ordinary ${\omega }_X$ under the forgetful maps of [F3]. On a smooth complex $d$-dimensional $X$, complex orientation gives ${\omega }_X\simeq k_X[2d]$ by [F2], and no extra shift survives because the normalization removes exactly the base contribution: this is the exclusion of the unnormalized quotient-stack dualizing shift recorded in the Statement. This proves claim 3.

Dual exchange and pasting. On a resolution over the common base $B_N$, let $D_T^N=R\mathcal{H}om(-,{\omega }_{T_N}[-2s_N])$ for each corner $T$. The common shift preserves involutivity and the ordinary duality identities [F2]. Applying [F5] to $D_X^{N}A$ and then dualizing gives $D_{Y^{\mathrm{\prime }}}^N{u}^{\ast }R{f}_{!}{D}_X^{N}A\simeq D_{Y^{\mathrm{\prime }}}^{N}R{f}_{!}^{\mathrm{\prime }}{v}^{\ast }{D}_X^{N}A$, whose two sides are $u^{!}R{f}_{\ast }A$ and $R{f}_{\ast }^{\mathrm{\prime }}{v}^{!}A$ respectively. Thus the extraordinary exchange holds for arbitrary $f$, rather than only proper $f$. Transport through the normalized comparison of step 3.2 and the fully faithful resolution comparisons gives the equivariant exchange. Duality carries the pasting identities of [F5] to the dual pasting identities, so these comparisons have the required coherence. This completes claim 2.

Conclusion. Steps 1.1, 2.1, 3.1, 3.2 and 4.1 prove claims 1 to 3. Every comparison is induced by the ordinary coherent structure of [F1] and [F5] and the normalization of step 3.2; no unnormalized dualizing shift and no unproved base-change identity is introduced. AC enters as recorded in [A1].