Constructible Verdier duality and complex orientation

The dualizing object and its local dual formula. Choose a finite affine-open cover $V_1,\dots ,V_m$ of $X$ and [F6]'s finite pairs $(K_i,D_i)$ for these quasi-projective pieces, applying the triangulation supplier to the zero constructible complex if necessary. Put $d={\max }_i\dim K_i$. Every open of $V_i$ is an open of $\mathrm{\mid }{K}_i\mathrm{\mid }$, so [F6] bounds its compact-support cohomology above $d$ for every sheaf. For any open $U\subseteq X$ and arbitrary sheaf $\mathcal{E}$ on $U$, the finite augmented alternating extension-by-zero complex for the cover $(U\cap V_i)$ resolves $\mathcal{E}$: at each stalk it is the augmented simplex complex on the indices containing that point. Its term in degree $-p$ is the sum of the extensions of $\mathcal{E}$ from the $(p+1)$-fold intersections. Each intersection is open in a $V_i$, and [F6]'s open-support transport bounds the compact-support cohomology of every term above $d$. Applying derived compact sections to this finite resolution therefore gives only total degrees $q-p\le d$, proving $H_c^q(U,\mathcal{E})=0$ for $q>d$. If $X$ is empty take $d=0$. Moreover, every fibre $F=f^{-1}(y)$ of a continuous map to a Hausdorff space is closed in $X$. An open $W\subseteq F$ has the form $F\cap O$ for an open $O\subseteq X$; the closed inclusion $i:W\hookrightarrow O$ and [F6] give $H_c^q(W,\mathcal{E})=H_c^q(O,i_{\ast }\mathcal{E})=0$ for $q>d$. This verifies [F1]'s uniform all-sheaf, all-fibre-open hypothesis for the maps used here, including $a_X$. Thus [F1] defines ${\omega }_X=a_X^{!}k$. For bounded $A$ and open $U$, tensor–Hom transposition gives $R\mathrm{\Gamma }(U,D_{X}A)=R{\mathrm{Hom}}_X(E_U{\otimes }_k^{L}A,{\omega }_X)\simeq R{\mathrm{Hom}}_k(R{\mathrm{\Gamma }}_c(U,A{\mathrm{\mid }}_U),k)$, where $E_U=j_{U!}{k}_U$; the last comparison is the constructed terminal-map adjunction and open support composition [F1, F2]. Likewise $R\mathrm{\Gamma }(U,f^{!}{\omega }_Y)\simeq R{\mathrm{Hom}}_Y(R{f}_{!}{E}_U,{\omega }_Y)\simeq R{\mathrm{Hom}}_k(R{\mathrm{\Gamma }}_c(U,k),k)\simeq R\mathrm{\Gamma }(U,{\omega }_X)$. These are natural comparisons of global Hom complexes, compatible with restriction to opens. Open generators and their shifts detect cohomology stalks, so they prove dualizing transitivity $f^{!}{\omega }_Y\simeq {\omega }_X$ without an unrestricted adjunction. Since $k$ is injective over itself, the coefficient Hom is ordinary signed vector-space duality.

Orientation. On an oriented real $n$-manifold and a small contractible ball $U$, the compact-support cochains of $k$ have a single cohomology class in degree $n$, normalized by the positive orientation generator: on a closed simplex this is the finite relative cellular complex $(\text{simplex},\text{boundary})$, whose quotient is generated by the oriented top cell, and excision identifies a ball with this model. Local models imply ${\omega }_X\simeq {\mathrm{o}\mathrm{r}}_X[n]$, and on a smooth complex $d$-variety the complex orientation trivializes ${\mathrm{o}\mathrm{r}}_X$, yielding ${\omega }_X\simeq k_X[2d]$. This fixes the positive complex generator used by the oriented diagonal Thom classes.

Local systems on smooth strata. For a finite-rank local system $L$ on a smooth stratum the same local model gives $D_{X}L\simeq L^{\vee }[n]$ with $n$ the real dimension, and finite-dimensional vector-space evaluation gives ; truncation extends this to bounded locally constant complexes. An induction on strata must not silently assume that $DD(j_{!}L)=j_{!}DD(L)$, since that involves boundary links; the fixed-space cellular construction below supplies that step.

The incidence model locally. Cover the separated finite-type variety $X$ by finitely many affine opens. On each such open $V$, which is quasi-projective, [F4] supplies a finite regular triangulation with a deleted subcomplex, subordinate to $A{\mathrm{\mid }}_V$. This is a model of the fixed space $V$, without a globally simplicial presentation of any algebraic map. On $V$ the derived-incidence supplier identifies bounded finite face-diagrams with the actual all-sheaf-derived constructible subcategory. Its injective cogenerators $I_{\sigma }$ are the constant sheaves on the closed cell intersected with $V$, pushed forward along that relatively closed subset, and every finite diagram has a finite injective bar resolution by finite sums of them. Restriction of the support-defined duality to $V$ is $D_V$, by step 1.1 and the exact open-inclusion transitivity, so these local resolutions suffice to check the global evaluation map. No single finite deleted-boundary model of an arbitrary separated variety is required.

Duals of the cogenerators. Write $C_{\sigma }=\bar{\sigma }\cap X$, let $a_{\sigma }:\mathrm{int}\sigma \hookrightarrow C_{\sigma }$ be open and $i_{\sigma }:C_{\sigma }\hookrightarrow X$ closed, and put $E_{\sigma }=i_{\sigma \ast }{a}_{\sigma !}k$. This specifies the extension by zero for the locally closed cell, whose interior need not be open in $X$. The compact-support local dual formula of step 1.1 computes $D_X{I}_{\sigma }$ directly: at an interior point of $\sigma$ a small neighbourhood in $\bar{\sigma }$ is an oriented real $\dim \sigma$-ball, whose compact-support cochains have the single positive generator in that degree; at a proper face $\tau <\sigma$ a small neighbourhood in $\bar{\sigma }$ is a product $[0,\mathrm{\infty }{)}^c\times {\mathbb{R}}^r$ with $c>0$, whose compact-support cochain complex is acyclic because one half-line has the relative finite cellular model $([0,1],\{1\})$ with boundary an isomorphism and tensoring that contraction with the other factors stays a contraction; at a point outside the closure the complex is zero. The positive orientation restricts to the interior coefficient, so $D_X{I}_{\sigma }=E_{\sigma }[\dim \sigma ]$. Conversely, compact-support cochains of $E_{\sigma }$ on a small star of a remaining face $\tau \le \sigma$ are the cochains of an open sector $(0,\mathrm{\infty }{)}^c\times {\mathbb{R}}^r\simeq {\mathbb{R}}^{\dim \sigma }$, with exactly the positive oriented generator in degree $\dim \sigma$, and restrictions between face-stars preserve it; hence $D_X(E_{\sigma }[\dim \sigma ])=I_{\sigma }$. The evaluation $I_{\sigma }\to D_X{D}_X{I}_{\sigma }$ is ordinary finite-dimensional double-dual evaluation at the interior and has the same value at every face by these incidence identifications, so it is an isomorphism. These computations use finite relative cellular complexes and the actual sheaf/cochain comparison, not a guessed stalkwise duality.

Algebraic constructibility of the dualizing object. On a controlled chart about a complex stratum $S$ of dimension $e$, [F4] supplies a ball in $S$ times a finite normal link-cone model. Applying the local formula of step 1.1 to constant coefficients gives the signed dual of its compact-support complex. The ball factor has its positive oriented class in degree $2e$ by step 1.2; the normal model is a bounded finite complex by [F4]. The controlled product identifications and their restriction maps therefore give a bounded finite coefficient complex, locally constant along $S$, with the ball's orientation shift. There are finitely many complex strata and affine charts, so their bounds are uniform. Hence ${\omega }_X$ is bounded algebraically constructible. Now [F4]'s internal-Hom theorem applies to both $A$ and ${\omega }_X$, proving bounded algebraic constructibility and coefficient-extension compatibility of $D_{X}A$. This does not infer complex-stratum constructibility from a partition into real simplices alone.

Biduality. On each affine open $V$ of step 2.1, $D_V$ is contravariant exact and its double is covariant exact, with evaluation a natural transformation of exact functors. Apply evaluation to the finite local injective incidence resolution of $A{\mathrm{\mid }}_V$: it is an isomorphism on every finite sum of cogenerators by step 2.2, and induction through the finitely many resolution cones proves $A{\mathrm{\mid }}_V\to D_V{D}_V(A{\mathrm{\mid }}_V)$ is an isomorphism. These are restrictions of the support-defined global evaluation, so its cone restricts to zero on the affine cover and is zero stalkwise; hence $A\to D_X{D}_{X}A$ is an isomorphism. Bounded algebraic constructibility and coefficient-extension compatibility were established in step 2.3. Refinement changes neither the support-defined duality nor evaluation, so the restriction and refinement comparisons have their identity and cocycle laws.

The duality exchange and the extraordinary proper exchange. After biduality, the internally localized exceptional adjunction with $B={\omega }_Y$ gives $R{f}_{\ast }{D}_{X}A\simeq D_{Y}R{f}_{!}A$: both sides represent the sheaf of derived morphisms from $R{f}_{!}A$ to ${\omega }_Y$, and the source-side object is $R\mathcal{H}om(A,f^{!}{\omega }_Y)=D_{X}A$ by step 1.1. Applying this identity to $D_{X}A$ and applying $D_Y$ gives $D_{Y}R{f}_{\ast }A\simeq R{f}_{!}{D}_{X}A$ by the double-dual isomorphism of step 3.1. Next $\mathrm{Hom}(A,D_X{f}^{\ast }{D}_{Y}B)\cong \mathrm{Hom}(f^{\ast }{D}_{Y}B,D_{X}A)\cong \mathrm{Hom}(D_{Y}B,R{f}_{\ast }{D}_{X}A)\cong \mathrm{Hom}(D_{Y}R{f}_{\ast }{D}_{X}A,B)\cong \mathrm{Hom}(R{f}_{!}A,B)$; these transpositions give a natural comparison $f^{!}B\to D_X{f}^{\ast }{D}_{Y}B$, and applying its internally localized chain formula to the generators $E_U=j_{U!}{k}_U$ and their shifts shows that the two derived section complexes agree, so the comparison is an isomorphism on cohomology stalks. The right side is bounded constructible by the fixed-space duality and the geometric inverse-image models, proving that $f^{!}$ preserves bounded constructibility; the represented Hom formula is now a genuine adjunction on that category, and inverse transposition proves its triangle identities and fixes all comparison maps. This avoids falsely equating compact-support cohomology downstairs with compact-support cohomology upstairs for a nonproper map. Applying these actual dualities to the ordinary proper base change of the support-and-trace calculus gives the coherent extraordinary exchange $u^{!}R{f}_{\ast }\simeq R{f}_{\ast }^{\mathrm{\prime }}{v}^{!}$: the duality isomorphisms and evaluations are natural and respect the sectionwise pasting identities, so the exchange has the mate and trace compatibility needed by convolution.