Cohomological Hall Algebras and Quiver Correspondences

Let $T\to [X\mathrm{/}G]$ correspond to a left torsor $P\to T$ and an equivariant map $\varphi :P\to X$. An object of $X{\times }_{[X\mathrm{/}G]}T$ is a point $x\in X$ and an equivariant isomorphism from the trivial left torsor to $P$, compatible with $\varphi$. Such an isomorphism is determined by the image $p\in P$ of the identity, and sends $g$ to $gp$; the compatibility condition forces $x=\varphi (p)$. Thus $X{\times }_{[X\mathrm{/}G]}T\cong P$. Since $P\to T$ is a torsor, it is a smooth surjective scheme morphism by [F3] and the local triviality in [F1]. Hence $X\to [X\mathrm{/}G]$ is a representable smooth surjective atlas.

For two objects $(P,\varphi )$ and $(P^{\mathrm{\prime }},{\varphi }^{\mathrm{\prime }})$ over a scheme $T$, their isomorphism functor is the subfunctor of ${\mathrm{Isom}}_G(P,P^{\mathrm{\prime }})$ on which ${\varphi }^{\mathrm{\prime }}\circ f=\varphi$. After a common fppf cover trivializing both torsors, and after restricting to affine opens of the cover, ${\mathrm{Isom}}_G(P,P^{\mathrm{\prime }})\cong G\times T$. With the chosen trivializing sections $p,p^{\mathrm{\prime }}$, the condition is the inverse image of the diagonal $X\hookrightarrow X\times X$ under the two evaluation maps $f\mapsto ({\varphi }^{\mathrm{\prime }}(f(p)),\varphi (p))$. The diagonal of affine $X$ is closed, so this is a closed subscheme of $G\times T$, affine over $T$. The isomorphism functor therefore has affine local representatives with their canonical descent data; [F4] makes this affine descent datum effective. Thus the isomorphism functor is represented by an affine scheme over $T$, which proves the diagonal is affine, hence representable.

The torsor-with-map construction is a category fibred in groupoids by pullback. To verify the stack condition, take an fppf cover of $T$ and torsors with equivariant maps and compatible isomorphisms on its overlaps. The torsors are affine over their bases by [F4]. On affine opens of $T$, refine the cover by faithfully flat affine covers and apply the effective algebra descent of [F4]; its full faithfulness glues these descended affine schemes on overlaps. The action, torsor isomorphism $G\times P\simeq P{\times }_{T}P$, and maps to $X$ descend as scheme morphisms by [F1]. Their equivariance and compatibility identities can be checked after the cover, and the resulting $P\to T$ is fppf-locally the given torsor, hence is a torsor. Compatible isomorphisms descend uniquely in the same way, with inverses descending too. Thus morphisms form sheaves and every object descent datum is effective, exactly the stack condition of [F1].

By steps 1.1–1.3, the quotient stack has a smooth surjective representable atlas and representable diagonal. Step 1.3 establishes the stack condition, so the criterion in [F2] applies and $[X\mathrm{/}G]$ is an algebraic stack. The inherited AC appears only in [A1] through those defining suppliers.

Put $Y={\mathrm{Rep}}_Q(\mathbf{a},\mathbf{b})$. The block description makes $Y$ an affine finite-type scheme. The block parabolic is $P_{\mathbf{a},\mathbf{b}}=U⋊L$ with underlying scheme ${\mathbb{A}}^u\times L$, where $L=G_{\mathbf{a}}\times G_{\mathbf{b}}$; hence $P_{\mathbf{a},\mathbf{b}}$ is affine and finite type, and it is smooth because $U$ is a vector group and each general linear factor of $L$ is smooth by [F5]. The quotient definition in [F1] makes $Y\to [Y\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]$ representable: after base change by a scheme $T\to [Y\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]$ corresponding to a $P_{\mathbf{a},\mathbf{b}}$-torsor $R\to T$, the pullback is $R$. Fppf-local triviality and [F8] make this torsor an affine scheme over $T$. To see that $R\to T$ is smooth, extend its structure group to $G_{\mathbf{a}+\mathbf{b}}$; the resulting $G$-torsor is a tuple of frame bundles of vector bundles $E_i$, while the reduction to $P_{\mathbf{a},\mathbf{b}}$ specifies subbundles $F_i\subset E_i$ of ranks $a_i$ with locally free quotients. Zariski-locally these sequences split and the bundles admit adapted frames, so $R\to T$ is locally $P_{\mathbf{a},\mathbf{b}}\times T\to T$, which is smooth by [F9] and [F10]; it is surjective as a torsor. For two underlying $P_{\mathbf{a},\mathbf{b}}$-torsors $R,R^{\mathrm{\prime }}$ over $T$, their equivariant isomorphism functor is fppf-locally represented by $P_{\mathbf{a},\mathbf{b}}\times T$, and these affine local representatives descend by [F8]. Fppf-locally trivialize both torsors, with their equivariant maps given by $y,y^{\mathrm{\prime }}:T^{\mathrm{\prime }}\to Y$. The compatible isomorphisms are cut out in $P_{\mathbf{a},\mathbf{b}}\times T^{\mathrm{\prime }}$ by $g\cdot y=y^{\mathrm{\prime }}$, the inverse image of ${\mathrm{\Delta }}_Y$ under $(g,t)\mapsto (g\cdot y(t),y^{\mathrm{\prime }}(t))$. This is closed because $Y$ is separated, and the closed subscheme descends effectively by [F8]. Thus the diagonal is affine representable. Effective descent of torsors with maps is obtained by descending their affine schemes, actions and compatible maps by the affine-descent argument in the proof of [F2]; that argument uses only affineness of the group and target, so also applies to this parabolic and to its vector subgroup $U$. By [F8], $[Y\mathrm{/}{P}_{\mathbf{a},\mathbf{b}}]$ is an algebraic stack. Now, for each arrow $h:i\to j$, the free off-diagonal block is $\mathrm{Hom}(V_i^{\mathbf{b}},V_j^{\mathbf{a}})$; hence $Y\to {\mathrm{Rep}}_Q(\mathbf{a})\times {\mathrm{Rep}}_Q(\mathbf{b})$ is a vector bundle of rank $e={\sum }_{h:i\to j}{b}_i{a}_j$. The kernel $U$ of $P_{\mathbf{a},\mathbf{b}}\to L$ consists of upper-right vertex blocks and has dimension $u={\sum }_i{a}_i{b}_i$.

First construct the relative Grassmannian used here. For a rank-$n$ bundle $E$ on any scheme $T$ and $0\le a\le n$, impose the homogeneous Plucker relations of [F3] in ${\mathbb{P}}_T(({\bigwedge }^{a}E{)}^{\vee })$. On a frame chart with $p_I\mathrm{\ne }0$, normalize $p_I=1$. The one-replacement coordinates give the entries of a matrix with identity pivot rows; the Plucker relation expressing a coordinate with $s$ nonpivot indices in terms of coordinates with fewer such indices proves inductively that every coordinate is its corresponding maximal minor. This is the calculation in [F3], using polynomial identities and only inversion of $p_I$, hence valid over every $\mathbb{C}$-algebra, including nonreduced ones. Conversely those minors satisfy all relations by alternating multilinearity. Thus each chart has polynomial coordinate ring in $a(n-a)$ entries and represents graph subbundles whose projection onto the pivot summand is invertible. Any locally direct-summand rank-$a$ subbundle is covered by such charts, since its maximal minors generate the unit ideal. The minor-ratio transitions identify the same subbundle, so the charts represent the subbundle functor, glue under change of frame, and give a closed relative Plucker embedding. Ranks $0,n$ give $T$. Now let an arbitrary scheme $T$ map to ${\mathfrak{M}}_{\mathbf{a}+\mathbf{b}}$. By the quotient-stack definition this is a family of vector bundles $E_i$ of ranks $a_i+b_i$, with arrow maps. The base change of $p$ parametrizes subbundles $F_i\subset E_i$ of ranks $a_i$ preserved by every arrow. It is the closed subscheme of the relative Grassmannian ${\prod }_i{\mathrm{Gr}}_T(a_i,E_i)$ cut out by the vanishing of the bundle maps $F_i\to E_j\to E_j\mathrm{/}{F}_j$ for each arrow $h:i\to j$. The relative Plucker map sends $F_i$ to the line subbundle $\det (F_i)\subseteq {\bigwedge }^{a_i}{E}_i$. Under the quotient convention of Projective bundle in the quotient convention, this is represented by the quotient line of $({\bigwedge }^{a_i}{E}_i{)}^{\vee }$ obtained by dualizing the inclusion. Thus each relative Grassmannian is a closed subscheme of ${\mathbb{P}}_T(({\bigwedge }^{a_i}{E}_i{)}^{\vee })$; on local frames this is the closed embedding just constructed from [F3], and the equations glue under changes of frame. Each such projective bundle is proper: after any base change $T^{\mathrm{\prime }}\to T$, trivialize $E_i$ over an open cover of $T^{\mathrm{\prime }}$; on each open the projection is a projective-space projection, hence proper by [F4]. The image of a closed set is closed on each member of the cover and therefore closed globally; separatedness and finite type are also local on the base. Thus every base change is proper by [F7]. The product of these proper maps is proper by base change and composition [F4]. The arrow conditions are closed because they are zero loci of regular bundle maps, so the incidence scheme is proper over $T$ as a closed subscheme of that product [F4]. In the standard frame atlas this scheme is exactly $I_{\mathbf{a},\mathbf{b}}=\{(X,(F_i)):\dim F_i=a_i,X_h(F_i)\subseteq F_j\text{ for every }h:i\to j\}\mathrm{.}$, and its defining equations are the same zero-locus equations just described; in an ordinary Grassmannian chart they are the vanishing of the complementary matrix block. Hence every scheme base change of $p$ is a proper scheme.

Pull back $q$ along the atlas $Z={\mathrm{Rep}}_Q(\mathbf{a})\times {\mathrm{Rep}}_Q(\mathbf{b})\to {\mathfrak{M}}_{\mathbf{a}}\times {\mathfrak{M}}_{\mathbf{b}}$. This fixes the endpoint representations. Every middle representation is a block matrix from step 1.1, and every isomorphism inducing the identity on the fixed endpoints is a unique upper-right vertex block. Hence the pullback stack is $[Y\mathrm{/}U]\to Z$, where $Y\to Z$ is the rank-$e$ vector bundle of off-diagonal blocks and $U$ is the rank-$u$ vector group of changes of splitting. For any scheme $T\to [Y\mathrm{/}U]$, its base change is a $U\cong {\mathbb{G}}_a^u$-torsor, represented by an affine scheme using [F8]. Cover $T$ by affine opens. On each affine base, choose an affine fppf cover of the torsor as in [F11]; the unique group element between the two pulled-back torsor points is a vector $\delta \in (B{\otimes }_{A}B{)}^u$ satisfying the additive Amitsur $1$-cocycle equation coordinatewise. Exactness in [F11] gives $b\in B^u$ with $\delta =b\otimes 1-1\otimes b$, and translation by $b$ descends a section, trivializing the torsor there. Thus $Y\to [Y\mathrm{/}U]$ is Zariski-locally $U\times T\to T$, which is smooth by [F9] and [F10], and it is surjective as a torsor. The same inverse-image-of-${\mathrm{\Delta }}_Y$ argument as in step 1.1, with $U$ in place of $P_{\mathbf{a},\mathbf{b}}$, gives $[Y\mathrm{/}U]$ an affine representable diagonal, so it is an algebraic stack. Its composite $Y\to Z$ is a vector bundle projection of relative dimension $e$. By the atlas convention just stated, $[Y\mathrm{/}U]\to Z$ is smooth of relative stack dimension $e-u$. Smoothness and relative dimension are checked after the displayed smooth target atlas, so the same holds for $q$. Substitution of the definition of ${\chi }_Q$ gives $e-u=-{\chi }_Q(\mathbf{b},\mathbf{a})$, including arrows that are loops or occur with multiplicity.

The construction in step 1.2 identifies every scheme base change of $p$ with the incidence scheme, so [F7] proves that $p$ is representable by schemes, hence by algebraic spaces; the same step proves every such base-changed map is proper. If some $a_i=0$ or $a_i=a_i+b_i$, the corresponding Grassmannian is a point and the same closed-incidence argument applies. Step 2.1 proves the smoothness of $q$, so both claims of the Statement follow. AC is used only through [A1].

First refine as in [F2] to respect the zero fibre. On each irreducible component of a smooth stratum where $f$ is nonconstant, [F5] gives a nonempty open of the affine line over which its differential is onto. Its proper closed complement is finite by [F1]. There are finitely many stratum components, and constant strata have one value each, so excluding the finitely many nonzero exceptional values leaves no interior stratum critical point over a sufficiently small punctured disc.

For a nonconstant stratum $S$ of dimension $s$, close in ${\mathbb{C}}^M\times ({\mathbb{C}}^M{)}^{\ast }$ the locus $C_S^{\circ }=\{(x,\xi ):x\in S,f(x)\mathrm{\ne }0,df{\mathrm{\mid }}_{T_{x}S}\mathrm{\ne }0,\xi {\mathrm{\mid }}_{T_{x}S}\in \mathbb{C}d{f}_x\}$. Its covector fibre has dimension $M-s+1$, so every component of its closure $C_S$ has dimension $M+1$. The function $f$ is nonconstant on each base component, and its zero fibre is a proper closed subset there. Hence ${\mathrm{\Lambda }}_S=C_S\cap \{f(x)=0\}$ has dimension at most $M$. The closure includes base points in lower strata; no unproved relative Thom condition is being assumed.

Identify covectors with their coefficient vectors and consider $F:{\mathrm{\Lambda }}_S\times \overline{{\mathrm{\Lambda }}_S}\to {\mathbb{C}}^M\times {\mathbb{C}}^M$, $F(x,\xi ,y,\eta )=(x-\eta ,y-\xi )$. Its source has dimension at most $2M$: applying the generic-fibre formula to the projection of each product component gives the sum of the two component dimensions. For each dominant irreducible component [F1] gives zero-dimensional, hence finite, fibres on a nonempty target Zariski open; exclude the closed image closure of each nondominant component. A common nonempty open $O$ therefore has finite fibres. The real diagonal $a\mapsto (a,\bar{a})$ meets $O$ densely in every real open set: a nonzero polynomial vanishing on its proper algebraic complement remains nonzero after the invertible substitution $(a,b)=(u+iv,u-iv)$, and cannot vanish on a real open box, by induction on the number of variables using the finite-root property of each nonzero one-variable polynomial. Intersect these finitely many good parameter sets for all the strata.

On a complex tangent space $T_{x}S$ where $df$ is onto, put $\mathrm{\ell }=d\mathrm{\mid }x-a{\mathrm{\mid }}^2{\mathrm{\mid }}_{T_{x}S}$. At a transverse sphere point $\mathrm{\ell }\mathrm{\ne }0$. The restriction of $df$ to $\ker \mathrm{\ell }$ then fails to be onto exactly when $\mathrm{\ell }$ is a real linear combination of $\mathrm{Re}df$ and $\mathrm{Im}df$. Indeed a real covector on the target annihilating the image of $\ker \mathrm{\ell }$ pulls back to a multiple of $\mathrm{\ell }$, and the converse is immediate. A nontransverse sphere point is separately a bad boundary point: there $\mathrm{\ell }=0$, so it also satisfies the covector inclusion below. Identifying real covectors with real parts of complex covectors, the obstruction is $\bar{x}-\bar{a}\in N_{S,x}^{\ast }+\mathbb{C}d{f}_x$.

The good relative-conormal centre set of step 3.1 is Euclidean open and dense. Apply [F7] within its intersection with an arbitrarily small real open neighborhood of $p$, obtaining such a centre $a$ for which the ordinary distance has only finitely many critical points on all strata. In addition to $E_a$, avoid their finitely many critical distances when choosing the radius. Thus the sphere is transverse to all strata, including those inside $f=0$. The set $E_a=\{x\in X\cap f^{-1}(0):(x,\bar{x}-\bar{a})\in {\mathrm{\Lambda }}_S\text{ for some }S\}$ is finite: every such $x$ gives $(x,\bar{x}-\bar{a},\bar{x},x-a)$ in the finite fibre of $F$ over $(a,\bar{a})$. Choose $ϵ>\mathrm{\mid }a-p\mathrm{\mid }$ avoiding the finitely many distances $\mathrm{\mid }x-a\mathrm{\mid }$ for $x\in E_a$. This choice is cofinal. Inside a prescribed $B_R(p)$ choose $\mathrm{\mid }a-p\mathrm{\mid }<R\mathrm{/}4$ and $ϵ\in (R\mathrm{/}3,R\mathrm{/}2)$ avoiding those distances; then $p$ lies in the ball and its closed ball lies in $B_R(p)$.

If no sufficiently small punctured tube avoided that obstruction on the sphere, there would be $x_j\in S\cap \mathrm{\partial }{B}_{ϵ}(a)$ with $f(x_j)\mathrm{\ne }0$, $f(x_j)\to 0$ and the obstruction of step 3.2. Compactness of $X\cap {\bar{B}}_{ϵ}(a)$ and finiteness of the strata give a subsequence in one $S$ converging to $x_0\in X\cap \mathrm{\partial }{B}_{ϵ}(a)\cap f^{-1}(0)$. Step 3.2 gives $(x_j,{\bar{x}}_j-\bar{a})\in C_S$, so its limit belongs to ${\mathrm{\Lambda }}_S$. Thus $x_0\in E_a$ and $\mathrm{\mid }{x}_0-a\mathrm{\mid }=ϵ$, contradicting step 4.1. There is consequently $\delta >0$ with no sphere obstruction for $0<\mathrm{\mid }f\mathrm{\mid }<\sqrt{2}\delta$, and shrink it also as in step 1.1.

Over the punctured rectangle, the sphere is transverse to every input stratum and $f$ is submersive on both its interior and boundary intersections by steps 1.1, 3.2 and 5.1. Transverse intersection with a smooth hypersurface preserves Whitney conditions: the limiting tangent planes intersect its tangent kernel in the expected dimension, and the first-order Taylor formula puts limiting fibre secants in that kernel. Thus the interior/sphere intersections are a finite Whitney stratification. The closed-ball tube is proper over this base since inverse images of compact base subsets are closed in the compact $X\cap {\bar{B}}_{ϵ}(a)$. The actual hypotheses of [F3] therefore hold, and its transports preserve the boundary stratum and hence the interior.

On a contractible half-rectangle, fixed ordered coordinate paths and [F3]'s compact-path transports give local products and inverses. The coefficient reconstruction of [F4] transports $A$ on these products; finite interval subdivisions glue the comparisons, and reversing transport gives inverse cochain maps. These local coefficient comparisons preserve the marked boundary and therefore restrict to the open-ball interior; the global finite fibre models are constructed in the final step below. Step 4.1 supplies a cofinal family; intersecting with arbitrarily small function rectangles retains cofinality at $p$. Smooth affine charts reduce the last assertion to this construction. This proves the transport assertion.

Replace the rectangle by a small circular function disc and consider $W=X\cap {\bar{B}}_{ϵ}(a)\cap \{\mathrm{\mid }f\mathrm{\mid }\le \delta \}$. The sphere is transverse to every input stratum by step 4.1, and the function-circle boundary and its sphere intersection are transverse by step 6.1, since $df$ restricted to the sphere is onto at nonzero values. Cutting the finite Whitney strata by these transverse boundary hypersurfaces gives a compact finite real Whitney space with its outer faces unions of strata. By [F8] it has a finite triangulation respecting those faces; removing them gives a finite deleted-subcomplex model for $U=X\cap B_{ϵ}(a)\cap \{\mathrm{\mid }f\mathrm{\mid }<\delta \}$. The compact Milnor fibre with its sphere boundary has the same finite controlled Whitney construction. Its interior therefore has a finite deleted-subcomplex model. By step 7.1 the positive-half-disc part $U_{+}$ is a constructible fibre bundle with that interior fibre over a contractible half-disc, giving its finite cochain model. The homology and cohomology long exact sequences for $(U,U_{+})$ then give bounded finite-dimensional relative homology and cohomology. In particular these rational groups are finite free. No triangulation of the real cut $\mathrm{Re}f=0$ or of an arbitrary map has been claimed. Together with step 7.1 this proves the stated conclusions, with AC exactly through [A1].

Put $U=X\setminus f^{-1}(0)$, let $j:U\to X$, and let $\nu :\tilde{U}\to U$ be the pullback of the universal cover $\exp :\mathbb{C}\to {\mathbb{C}}^{\ast }$. Using bounded-below analytic injective replacements in $D^{+}$, before boundedness is proved, define ${\mathrm{\Psi }}_{f}A=i^{\ast }R(j\nu {)}_{\ast }{\nu }^{\ast }{j}^{\ast }A$, with its unit $i^{\ast }A\to {\mathrm{\Psi }}_{f}A$. Restrict to the lift of the positive half-plane whose logarithm has imaginary part in $(-\pi \mathrm{/}2,\pi \mathrm{/}2)$. This gives a natural map from ${\mathrm{\Psi }}_{f}A$ to the restriction of the direct image from $f^{-1}\{\mathrm{Re}z>0\}$. On each cofinal Milnor neighborhood [F3], the universal cover of its punctured function rectangle and the lifted positive sector are contractible bases with the same fibre, and the coefficient reconstruction of [F4] makes this restriction a quasi-isomorphism of cochain complexes. It is therefore an isomorphism on every stalk, hence a natural sheaf isomorphism. Localization [F1] now gives $V_{f}A\simeq \mathrm{Cone}(i^{\ast }A\to {\mathrm{\Psi }}_{f}A)[-1]$. In particular its stalk is the relative cochain complex of a Milnor neighborhood and its fibre; for constant coefficients this gives the displayed reduced-cohomology degree, and for $f=0$ the empty fibre gives $V_0\mathbb{Q}=\mathbb{Q}$.

Expanding $\mathrm{\Psi }$ and applying proper base change to the open and universal-cover pullback squares gives ${\mathrm{\Psi }}_{f}R{p}_{\ast }A\simeq R{p}_{0\ast }{\mathrm{\Psi }}_{fp}A$ for proper $p$. Indeed move the proper direct image past $j^{\ast }$ and ${\nu }^{\ast }$ by [F1], compose ordinary direct images with $R(j\nu {)}_{\ast }$, and move it past $i^{\ast }$ by proper base change again. This argument takes place in locally compact analytic spaces with finite-dimensional proper fibres, so it does not require constructibility to have been proved already.

Let $P=\{\mathrm{Re}f\le 0\}$, $Q=\{\mathrm{Re}g\le 0\}$ and $Z=\{\mathrm{Re}(f+g)\le 0\}$. On cofinal neighborhoods $U,V$ from [F3], put $A_{+}=U\setminus P$, $B_{+}=V\setminus Q$ and $L=(A_{+}\times V)\cup (U\times B_{+})$. These are open complements, not asserted CW subcomplexes. Localization represents their support cochains by the relative complexes of $(U,A_{+})$, $(V,B_{+})$ and $(U\times V,L)$. The finite ordinary and Milnor-fibre models in [F3], including its compact-tube triangulation, give finite homology of $U,A_{+}$ and their relative pair, and similarly for $V,B_{+}$. No triangulation of the real cuts $P,Q$ is needed.

Here is the required local duality calculation. If closed $Z_{+},Z_{-}$ cover a locally compact space, with $L=Z_{+}\cap Z_{-}$ and inclusions $j_{\pm },m$, their localization triangles give $m^{\ast }{j}_{+\ast }{j}_{+}^{!}A\simeq (L\to Z_{-}{)}^{!}{j}_{-}^{\ast }A$. To verify it, apply $j_{+\ast }{j}_{+}^{!}$ to localization for $Z_{-}$: the open-complement term is unchanged by support in $Z_{+}$ because $X\setminus Z_{-}\subset Z_{+}$, and becomes zero after restriction to $L$. Closed proper base change identifies the remaining term as stated. Dualizing these adjunction maps exchanges the two support descriptions. This calculation uses the analytic exceptional adjunction of [F1]. On the present local sector pairs, [F3] and [F4] give finite cochain and relative-cochain complexes, so dualization and evaluation are the ordinary finite-vector-space operations; no biduality theorem for arbitrary nonconstructible analytic sheaves is being invoked.

To prove standard-coefficient constructibility, treat constant-function components directly and otherwise apply [F5] to $(f)$. Obtain $\pi :\tilde{X}\to X$ proper, smooth upstairs, isomorphic off $f=0$, with $f\pi$ having SNC zero support. The support-centre clause in [F5] and the ordinary Rees-algebra blowup charts imply the asserted isomorphism off $f=0$ at every step. Thus ${\mathbb{Q}}_X\to R{\pi }_{\ast }{\mathbb{Q}}_{\tilde{X}}$ is an isomorphism there, and $\mathrm{\Psi }$, which depends only on $j^{\ast }A$, gives ${\mathrm{\Psi }}_f{\mathbb{Q}}_X\simeq R{\pi }_{0\ast }{\mathrm{\Psi }}_{f\pi }{\mathbb{Q}}_{\tilde{X}}$ by step 2.1. This uses principalization of the function ideal, not resolution of the ambient variety alone.

Here is the open-pair extension of the relative AW comparison. Write $N=C_{\ast }(A_{+}\times V)+C_{\ast }(U\times B_{+})$ and $E=C_{\ast }(U\times V)\mathrm{/}N$. By naturality [F7], AW, shuffle and their homotopies descend to a chain homotopy equivalence between $E$ and $C_{\ast }(U,A_{+})\otimes C_{\ast }(V,B_{+})$. The two indicated open sets cover $L$, so [F8] gives a small-chain retraction $R$ on $C_{\ast }(L)$ landing in $N$, with $1-R=dD+Dd$, and both $R,D$ preserve $N$. Thus $D$ contracts $J=C_{\ast }(L)\mathrm{/}N$. Extend its contraction to a graded map $e:E\to J$ by zero on the remaining singular-simplex basis vectors. Then $T=1-de-ed$ commutes with the differential and kills $J$. It factors through a chain inverse to $q:E\to C_{\ast }(U\times V,L)$, since $qT=q$ and $T{\mathrm{\mid }}_J=0$; its other composite is homotopic to the identity by $e$. This proves the relative chain comparison for these OPEN pairs. Subdivision depth is chosen separately for each simplex by [F8], never uniformly over all chains.

In SNC coordinates $f\pi =u(z){\prod }_{i=1}^r{z}_i^{m_i}$ with $u\mathrm{\ne }0$ and $m_i\ge 1$. By [F6] choose a local $m_1$-st root $v$ of $u$; replacing $z_1$ by $v(z)z_1$ is an invertible coordinate change along the SNC stratum and makes the function the monomial. For small $t\mathrm{\ne }0$ in a coordinate polydisc, put $s_i=-\log \mathrm{\mid }{z}_i\mathrm{\mid }$. The radial equation is the open convex simplex $s_i>L_i$, ${\sum }_i{m}_i{s}_i=-\log \mathrm{\mid }t\mathrm{\mid }$. The angular fibre of ${\prod }_i{e}^{i{m}_i{\theta }_i}=t\mathrm{/}\mathrm{\mid }t\mathrm{\mid }$ has $\gcd (m_i)$ components, each a torus $(S^1{)}^{r-1}$: the Euclidean algorithm extends the primitive integer row $(m_i\mathrm{/}\gcd (m_i))$ to an invertible integer matrix, reducing its torus character to the $\gcd (m_i)$-power map in one coordinate. The tangential coordinate disc is contractible. These are finite cellular cochain models, constant in each tangential chart, and their coordinate-change transport gives locally constant finite-rank cohomology on the finitely many algebraic SNC strata. Thus ${\mathrm{\Psi }}_{f\pi }\mathbb{Q}$ is bounded algebraically constructible.

Over $\mathbb{Q}$, the finite-dimensional relative homology in step 2.2 is free. The algebraic part of [F7]'s relative cohomological proof applies to the free relative singular complexes with the chain comparison now proved in step 3.2: choose cycle/boundary sections, split one factor up to chain homotopy onto its homology with zero differential, and use the finite-basis evaluation inverse $w\mapsto {\sum }_{i}w(-\otimes e_i)\otimes e_i^{\ast }$. There are finitely many terms in each degree and bounded finite homology; these maps identify the actual cochain cross product with a quasi-isomorphism. Constant shifts, external products and smooth pullbacks of these constant inputs preserve this comparison. The support-product morphism itself is natural for arbitrary coefficient morphisms: tensor supported sections on the two factors and then include their product support, and apply the same map on derived resolutions. This naturality does not assert an isomorphism on an arbitrary proper-image input. Passing through the cofinal stalk system gives $R{\mathrm{\Gamma }}_{P}A⊠R{\mathrm{\Gamma }}_{Q}B\simeq R{\mathrm{\Gamma }}_{P\times Q}(A⊠B)$ for the standard inputs used here. Compose it with inclusion of supports $P\times Q\subset Z$ and restrict to the product zero fibre to define TS. This proof neither applies the CW-pair theorem to an open complement nor invokes algebraic biduality on real half-spaces.

Proper constructible preservation [F2] applied to ${\pi }_0$ and step 3.1 proves bounded algebraic constructibility of ${\mathrm{\Psi }}_f{\mathbb{Q}}_X$, and its natural cone in step 1.1 proves the same for $V_f{\mathbb{Q}}_X$. The local relative models of [F3] and [F4] give a uniform finite bound depending on $\dim X$ for its ordinary amplitude. Standard coefficients on smooth frame charts are pullbacks by the argument below, so their unshifted amplitude equals that downstairs, independently of the auxiliary frame dimension. Constant shifts, external products and ordinary Verdier duals stay within the proved constructible domain of [F2]; proper images do by [F2] and the comparison of step 2.1. This is constructible preservation, not an additional claim of Thom–Sebastiani invertibility on every such proper image.

To prove this TS map is an isomorphism, work on a product of cofinal neighborhoods [F3], and set $\mathrm{\Theta }=(\mathrm{Re}f,\mathrm{Re}g)$, $C=\{u\le 0,v\le 0\}$, $D=\{u+v\le 0\}$. Localization identifies the required support comparison with restriction of complement cochains from ${\mathrm{\Theta }}^{-1}(C^c)$ to ${\mathrm{\Theta }}^{-1}(D^c)$. On $u>0,v<0$ deform by $H_t(u,v)=(u,(1-t)v+t\max (v,-u\mathrm{/}2))$; on $u<0,v>0$ use the exchanged formula, and on the northeast quadrant and its positive axes use the identity. The formulas agree continuously in $C^c$, never move a coordinate through zero, do not increase absolute coordinates, send $C^c$ into $D^c$ at $t=1$, and preserve $D^c$ throughout. Thus they supply a homotopy inverse to the complement inclusion. Leave imaginary coordinates fixed and lift these paths by [F3]'s product sector transports, preserving the marked sphere boundaries and coefficients. The finite relative cochain comparisons [F4] identify restriction with a quasi-isomorphism, not merely an equality of dimensions. Cofinality and stalk detection prove that the TS map of step 4.2 is an isomorphism.

For a smooth map $s:E\to X$, local product coordinates make $fs$ the pullback function and make the relative half-space pair a complex ball times the base pair. Its ordinary cochain complex is $\mathbb{Q}$, so [F4] gives the natural comparison $s^{\ast }{V}_{f}A\simeq V_{fs}{s}^{\ast }A$. It is the inverse-image map of the localization triangles and is therefore compatible with changes of product chart, composition and all support-product maps. On an affine fibration the same ordinary comparison is a homotopy-invariance isomorphism; compact-support fibre cochains are $\mathbb{Q}[-2e]$ for relative complex dimension $e$, with the generator integrating to $+1$. Hence the corresponding exceptional operation has the positive complex orientation and its compositional shift.

For three functions both iterated TS maps are induced by the same inclusion ${\prod }_i\{\mathrm{Re}{f}_i\le 0\}\subset \{\mathrm{Re}{\sum }_i{f}_i\le 0\}$. Composition of supports and external adjunctions is associative in the actual dg localization model [F1]. Hence the maps agree under the tensor associator and obey its pentagon for every number of factors. For the zero function on a point the support inclusion and coefficient $\mathbb{Q}$ are identities, giving both unit laws.

Proper commutation of $V$ can also be seen directly: localization for $\{\mathrm{Re}f\le 0\}$, proper base change for its open complement and the identity on the middle object identify its support triangle after $R{p}_{\ast }$ with the support triangle downstairs. Proper restriction to $f=0$ then gives $V_{f}R{p}_{\ast }\simeq R{p}_{\ast }{V}_{fp}$. Applying it to the ordinary unit gives $V_f{\mathbb{Q}}_X\to R{p}_{\ast }{V}_{fp}{\mathbb{Q}}_E$. For composable proper maps the unit of the composite is the composite of units, so these coefficient maps compose. Smooth base change sends the same unit to the pulled-back unit, and the comparisons of step 5.3 identify the resulting coefficient maps. Pasting reduces to the triangle identities of [F1]. Naturality of the external-support map of step 4.2 on that unit proves compatibility with TS and external products.

Apply the same construction to $e^{-i\theta }f$ and $e^{-i\theta }g$. Use circular function discs inside the rectangles of [F3]; these are independent of $\theta$. The angular families of support inclusions and cochain restriction maps commute with continuation. Transport from $\theta =0$ to $2\pi$ is the positive-loop monodromy of $f+g$, and on the external product it is the simultaneous transport $T_f\otimes T_g$. TS therefore intertwines these automorphisms, without semisimplicity. This also constructs the angular comparisons used in step 1.1.

Apply step 2.3 to $Z_{\pm }=\{\pm \mathrm{Re}f\le 0\}$. On $L=\{\mathrm{Re}f=0\}$ the restricted support complex is locally supported on $f=0$ near that fibre: at any other sufficiently nearby value $it\mathrm{\ne }0$, the submersion product [F3] has zero half-space relative complex. On a closed inclusion whose intersection with a complex's support is open and closed relative to that support, ordinary and exceptional restriction agree; factor through the support and use its open restriction adjunction to verify the natural map. Thus restricting the dual calculation from $L$ to $f=0$ gives $D{V}_{f}A\simeq V_f^{\pi }DA$, the opposite-half-space normalization. The specified angular half-turn of step 7.1 identifies it with $V_{f}DA$. For the standard coefficient and its proved constructible closure [F2] applies; since $D{\mathbb{Q}}_X={\mathbb{Q}}_X[2\dim X]$, this proves the claimed normalized duality formula. On the dual coefficient the monodromy is contragredient transport $D(T^{-1})$, making evaluation invariant; all preceding maps respect this transport by naturality.

The maps in steps 2.2–8.1 are the maps of the same support triangles, external adjunctions, orientations and specified angular transports. Their composition and proper/smooth pasting identities therefore hold simultaneously by [F1]; duality takes these identities to the corresponding identities for dual coefficients. Steps 3.1–5.1 put every standard coefficient and the stated closure in the actual constructible domain, and step 5.2 proves the load-bearing TS isomorphism. This proves all assertions, with AC exactly as in [A1].