Equivariant Borel–Moore Correspondence Convolution and Constructible Ext

Purity and the supported Thom morphism. Put $Y=M\times M$. By [F1] and [F2], ${\mathrm{\Delta }}^{!}{k}_Y={\mathrm{\Delta }}^{!}{\omega }_Y[-4d]={\omega }_M[-4d]=k_M[-2d]$. The positive orientation identifies the identity of $k_M[-2d]$ with the Thom class in $H_{\mathrm{\Delta }(M)}^{2d}(Y;k)=\mathrm{Hom}(k_M,{\mathrm{\Delta }}^{!}{k}_Y[2d])$. Its adjoint is the purity counit $g_{\mathrm{\Delta }}:{\mathrm{\Delta }}_{\ast }{k}_M[-2d]\to k_Y$, equivalently ${\mathrm{\Delta }}_{\ast }{k}_M\to k_Y[2d]$. In a complex normal ball this is the positive generator of the relative disk pair; complex chart transitions preserve that generator. This fixes a global supported morphism through the orientation isomorphism and its counit, rather than inferring equality of arbitrary global maps from their restrictions to balls.

Pullback and cap product on arbitrary supports. Put $S=A\times B$. Proper base change along $F=(q,u)$ identifies $F^{\ast }{\mathrm{\Delta }}_{\ast }{k}_M[-2d]$ with ${\theta }_{\ast }{k}_T[-2d]$, so $F^{\ast }{g}_{\mathrm{\Delta }}$ gives $g_F:{\theta }_{\ast }{k}_T[-2d]\to k_S$. For a dualizing class $c:k_S\to {\omega }_S[-n]$, compose $c\circ g_F$ and use ${\theta }_{\ast }⊣{\theta }^{!}$ and ${\theta }^{!}{\omega }_S={\omega }_T$. The resulting map $k_T\to {\omega }_T[2d-n]$ is a Borel–Moore class of degree $n-2d$ on $T$. This defines the supported cap operation even if $T$ is singular or has excess dimension; no purity of $\theta$ is asserted. The external product uses ${\omega }_A⊠{\omega }_B={\omega }_S$: on the supplied finite local product cochain models, compact-support cochains tensor, and their finite-dimensional duals tensor as well, giving this canonical comparison and its restriction compatibility.

Comparison with the exchange mate. For a constructible $C$ on $M$, the mate is the adjoint of $b_{!}{a}^{\ast }{q}^{!}C\stackrel{{\mathrm{B}\mathrm{C}}^{-1}}{\to }{u}^{\ast }{q}_{!}{q}^{!}C\stackrel{u^{\ast }{ϵ}_q}{\to }{u}^{\ast }C$, where the arbitrary-map proper-support exchange of [F1] supplies $\mathrm{B}\mathrm{C}:u^{\ast }{q}_{!}\stackrel{\sim }{\to }{b}_{!}{a}^{\ast }$ and ${ϵ}_q$ is the exceptional counit. To identify its dualizing evaluation with step 2.1, factor $q$ as its graph ${\mathrm{\Gamma }}_q:A\hookrightarrow A\times M$ followed by the second projection. The graph is the pullback of $\mathrm{\Delta }$ along $(q,{\mathrm{id}}_M):A\times M\to M\times M$; its supported orientation map is that pullback of $g_{\mathrm{\Delta }}$. Base change of the graph along ${\mathrm{id}}_A\times u$ is exactly $\theta$, so its supported orientation map after this second pullback is $g_F$. For the product projection, tensor–Hom evaluation with the relative dualizing factor ${\omega }_A$ computes the counit; this is the finite product cochain evaluation of step 2.1. Composing the graph counit and that evaluation gives ${ϵ}_q$ by the composition-of-counits identity in [F1]. Cartesian pasting therefore makes its base-changed composite exactly $u^{\ast }{ϵ}_q\circ {\mathrm{B}\mathrm{C}}^{-1}$ above. Taking adjoints and using $q^{!}{k}_M={\omega }_A[-2d]$ proves that the exchange, paired with the second dualizing class by tensor evaluation, is $c\circ g_F$ under ${\theta }_{\ast }⊣{\theta }^{!}$. Every map retains support on $T$ throughout this factorization.

Ordered iterated diagonals. For two ordered smooth middle factors the normal model of the product diagonal is the ordered direct sum of the two complex normal bundles, and on the product of normal coordinate disks the relative cup product of the two positive generators evaluates to $+1$ on the ordered product disk, which is the positive generator of the direct sum; local uniqueness as in step 1.1 and base-change mate pasting as in steps 2.1 and 3.1 therefore give the composed refined operation on every support, in the given order and without an interchange of factors. All real degrees involved are even, so reordering the two complex summands introduces no sign: the only possible sign is $(-1{)}^{(2d_1)(2d_2)}=+1$. Under the additional hypotheses of [F3], the equivariant form holds verbatim, which supplies the same localization, exchange and normalized orientation clauses in $D_{G,c}^b$, with ${\omega }_X^G{\mathrm{\mid }}_X={\omega }_X$ rather than a quotient-stack shift.

Zero sections and Euler classes. Let $V\to X$ have complex rank $c$ and zero section $s$. In a bundle chart the normal pair is the product with $({\mathbb{C}}^c,{\mathbb{C}}^c\setminus \{0\})$, so the same relative cochain calculation gives $s^{!}{k}_V=k_X[-2c]$ and its positive supported counit; here $c$ is the bundle rank, independently of $\dim X$. These local orientations glue because the complex transition matrices preserve the positive fiber generator. On bases in the Thom scope of [F5], this supported class is the normalized Thom class by uniqueness. Pulling its relative-to-absolute image back by $s$ gives $e(V)$ by [F5]. Applying the supported cap construction of step 2.1 after $s_{\ast }$ restricts that class to the zero section; the projection formula thus gives $s^{!}{s}_{\ast }z=e(V)\cap z$. The same counit and base-change identities give its proper-pushforward and refined-pullback compatibility; the equivariant version requires the additional hypotheses of [F3]. AC is inherited through the stated suppliers, including the Thom supplier [F5].

The adjunction chain. Write $p:Z_{ij}\to X_i$ and $q:Z_{ij}\to X_j$ for the projections and put $t=r+d_j-d_i$. Removing the source shift by $[-d_i]$ and using the adjunction $f_{i!}⊣f_i^{!}$ of [F1] gives ${\mathrm{Hom}}_E(f_{i!}{k}_i,f_{j!}{k}_j[t])={\mathrm{Hom}}_{X_i}(k_i,f_i^{!}{f}_{j!}{k}_j[t])$. In the Cartesian square with lower arrow $f_j$ and right arrow $f_i$, the extraordinary proper base change of [F1] gives $f_i^{!}{f}_{j!}{k}_j=p_{\ast }{q}^{!}{k}_j$, because both maps are proper and the square is Cartesian; the adjunction $p^{\ast }⊣p_{\ast }$ then identifies the group with ${\mathrm{Hom}}_{Z_{ij}}(k_Z,q^{!}{k}_j[t])$. This is the complete cornerwise chain and it is valid for every integer $r$, including negative $r$.

Orientation and the degree. The smooth orientation of [F1] on the smooth pure $d_j$-dimensional source $X_j$ says $k_{X_j}={\omega }_{X_j}[-2d_j]$, and dualizing transitivity for $q$ gives $q^{!}{k}_{X_j}=q^{!}{\omega }_{X_j}[-2d_j]={\omega }_{Z_{ij}}[-2d_j]$. Substituting $t=r+d_j-d_i$, the group of step 1.1 is $\mathrm{Hom}(k_Z,{\omega }_Z[r-d_i-d_j])$, which is exactly $H_{d_i+d_j-r}^{\mathrm{B}\mathrm{M},G}(Z_{ij};k)$ in the normalization of [F3]. At a finite free smooth approximation of dimension offset $b={\dim }_{\mathbb{C}}U-{\dim }_{\mathbb{C}}G$, the space $X_j{\times }_{G}U$ is smooth of complex dimension $d_j+b$ while the descended complex retains the shift $d_j$, so the identical calculation gives ordinary Borel–Moore degree $d_i+d_j+2b-r$; removing $2b$ gives the stated equivariant degree, and no stack-dimension shift is inserted into ${\omega }_X^G$.

Naturality, linearity and finiteness. Every map in the chain of steps 1.1 and 2.1 is an adjunction, base-change or orientation map supplied naturally by [F1], so the composite is natural in constructible coefficient morphisms before the smooth orientation is applied and is compatible with composition of the $f_i$. By [F2] these maps commute with the approximation transition pullbacks on the bounded ranges, so the composite induces an isomorphism in each stable degree; coherent scalar pullback from the base point together with the projection formula of [F1] gives equivariant-cohomology linearity. Degreewise finiteness is exactly the conclusion of [F4] for bounded algebraically constructible objects, and it is asserted nowhere that the direct sum over all degrees is finite-dimensional. Smoothness of the sources is used only for the orientation in step 2.1, and properness only in the base change and in the identification $f_{i!}=f_{i\ast }$; no semismallness, decomposition theorem, purity or smoothness of $E$ or of the fiber products $Z_{ij}$ is used.

Typed correspondences. Put $t=r+d_j-d_i$ and $w=s+d_k-d_j$. Under the chain of [F3] the maps $\alpha$ and $\beta$ correspond to kernel forms $\kappa :k_A\to q^{!}{k}_j[t]$ and $\lambda :k_B\to v^{!}{k}_k[w]$, equivalently to cohomological correspondences $U:q_{!}{k}_A\to k_j[t]$ and $V:v_{!}{k}_B\to k_k[w]$; here $k_j=k_{X_j}$ and $k_k=k_{X_k}$. The middle square $(a,b;q,u)$ is Cartesian with $qa=ub$, and the three-factor identities $pa=\mathrm{\ell }h$, $vb=mh$ hold by construction of the triple fiber product.

The explicit composite. Let $\mathrm{B}\mathrm{C}:b_{!}{a}^{\ast }\stackrel{\sim }{\to }{u}^{\ast }{q}_{!}$ be the proper base change isomorphism of [F4]. Form the map $W:(vb{)}_{!}{k}_T=v_{!}{b}_{!}{a}^{\ast }{k}_A\stackrel{v_{!}\mathrm{B}\mathrm{C}}{\to }{v}_{!}{u}^{\ast }{q}_{!}{k}_A\stackrel{v_{!}{u}^{\ast }U}{\to }{v}_{!}{k}_B[t]\stackrel{V[t]}{\to }{k}_k[t+w]\mathrm{.}$ Its adjoint $k_T\to (vb{)}^{!}{k}_k[t+w]$ under $(vb{)}_{!}⊣(vb{)}^{!}$ is the composite $k_T\stackrel{a^{\ast }\kappa }{\to }{a}^{\ast }{q}^{!}{k}_j[t]\to b^{!}{u}^{\ast }{k}_j[t]=b^{!}{k}_B[t]\stackrel{b^{!}\lambda [t]}{\to }(vb{)}^{!}{k}_k[t+w]$, in which the middle arrow is the mate exchange of $\mathrm{B}\mathrm{C}$. The equality follows by applying the adjunction $(vb{)}_{!}⊣(vb{)}^{!}$ to both maps: the defining evaluation of the mate reproduces exactly the three arrows of $W$, so no exchange isomorphism beyond $\mathrm{B}\mathrm{C}$ is assumed.

Geometric interpretation of the mate. Put $S=A\times B$, $F=(q,u):S\to X_j\times X_j$, and $\theta :T\hookrightarrow S$. By [F1] the pulled-back positive purity counit is $g_F:{\theta }_{\ast }{k}_T[-2d_j]\to k_S$. The external dualizing class corresponding to $\kappa ⊠\lambda$ has degree $n+n^{\mathrm{\prime }}$, where $n=d_i+d_j-r$ and $n^{\mathrm{\prime }}=d_j+d_k-s$ by [F3]; composing it with $g_F$ and taking the adjoint under ${\theta }_{\ast }⊣{\theta }^{!}$ gives a class of degree $n+n^{\mathrm{\prime }}-2d_j$ on $T$. The graph-factorization and composition-of-counits comparison proved in [F1] identifies this supported cap map with the mate in step 2.1, followed by $b^{!}\lambda [t]$. Thus its representative is exactly the adjoint of $W$. The purity counit, its pullback, and the mate are fixed global morphisms; the positive normal-ball generator fixes their orientation, while their equality on arbitrary supports follows from that supported adjunction comparison.

Pushing the correspondence to the third corner. Its adjoint representative is $U_C:m_{!}{k}_C\stackrel{m_{!}{\eta }_h}{\to }{m}_{!}{h}_{\ast }{k}_T=(vb{)}_{!}{k}_T\stackrel{W}{\to }{k}_k[t+w]$, where ${\eta }_h:k_C\to h_{\ast }{h}^{\ast }{k}_C=h_{\ast }{k}_T$ is the ordinary adjunction unit and $h_{\ast }=h_{!}$ by [F4]. In kernel form this is $k_C\to h_{\ast }{k}_T\to h_{\ast }{h}^{!}{m}^{!}{k}_k[t+w]\stackrel{{ϵ}_h}{\to }{m}^{!}{k}_k[t+w]$ with ${ϵ}_h$ the dualizing proper trace, and dualizing transitivity identifies the middle dualizing object with ${\omega }_T[-2d_k+t+w]$. Step 3.1 therefore identifies the class on $C$ represented by $U_C$ with $h_{\ast }{\mathrm{\Delta }}_{X_j}^{!}({\mathrm{\Phi }}_{ij}^r(\alpha )⊠{\mathrm{\Phi }}_{jk}^s(\beta ))$, which by [F2] is the convolution product ${\mathrm{\Phi }}_{ij}^r(\alpha )\ast {\mathrm{\Phi }}_{jk}^s(\beta )$.

The composition square commutes. The inverse of the chain of [F3] sends $U$ to $f_{i!}{k}_i\stackrel{f_{i!}{\eta }_p}{\to }{f}_{i!}{p}_{\ast }{k}_A=f_{j!}{q}_{!}{k}_A\stackrel{f_{j!}U}{\to }{f}_{j!}{k}_j[t]$ and $V$ to the analogous map built from $u,v$, where ${\eta }_p:k_i\to p_{\ast }{p}^{\ast }{k}_i=p_{\ast }{k}_A$ is the adjunction unit and $f_{j!}=f_{j\ast }$. In the composite of these two maps, naturality of ${\eta }_u:\mathrm{id}\to u_{\ast }{u}^{\ast }$ moves $f_{j!}U$ past the second unit, and proper base change $u^{\ast }{q}_{!}\cong b_{!}{a}^{\ast }$ identifies the resulting two units with the unit ${\eta }_{pa}:k_i\to (pa{)}_{\ast }{k}_T$; the Beck–Chevalley unit identity is the transpose of the triangle equations under $f_{i!}⊣f_i^{!}$ and $(vb{)}_{!}⊣(vb{)}^{!}$, so both routes give the same morphism $f_{i!}{k}_i\stackrel{f_{i!}{\eta }_{pa}}{\to }{f}_{k!}(vb{)}_{!}{k}_T\stackrel{f_{k!}W}{\to }{f}_{k!}{k}_k[t+w]$. Applying the inverse map to $U_C$ gives the same expression, because $pa=\mathrm{\ell }h$ and the composed-unit formula ${\eta }_{\mathrm{\ell }h}={\mathrm{\ell }}_{\ast }{\eta }_h{\mathrm{\ell }}^{\ast }\circ {\eta }_{\mathrm{\ell }}$ follows from the same triangle equations. Hence the product square commutes for arbitrary cohomological correspondences, before orientation or grading is applied; this is the compatibility of products asserted in the Statement.

Shifts, degrees and signs. Restore the object shifts: $t+w=r+s+d_k-d_i$. The corner degrees $d_i+d_j-r$ and $d_j+d_k-s$ add and lose $2d_j$ under the refined intersection, giving $d_i+d_k-r-s$, which is the corner degree of the output. At a finite approximation the middle factor has dimension $d_j+b$ and each incoming ordinary Borel–Moore degree acquires $2b$, so the output acquires exactly $2b$ and the stable equivariant degree is unchanged. Homogeneous maps compose as $V[t]\circ U$, hence as $\beta [r]\circ \alpha$ after restoring shifts, with no interchanged factors and no additional sign; the only possible sign in re-ordering the complex normal factors is $(-1{)}^{(2d_j)(2d_j)}=+1$, and external products follow the usual tensor Koszul convention with no permutation occurring here.

Units and the algebra statement. Let $\delta :X_i\to Z_{ii}$ be the diagonal, with $p\delta =q\delta ={\mathrm{id}}_{X_i}$. The cohomological correspondence for its fundamental class is $U_{\delta }:q_{!}{k}_{Z_{ii}}\to q_{!}{\delta }_{\ast }{k}_{X_i}=k_{X_i}$, induced by restriction $k_{Z_{ii}}\to {\delta }_{\ast }{k}_{X_i}$. Its inverse image under the chain [F3] is $f_{i!}{k}_{X_i}\to f_{i!}{p}_{\ast }{k}_{Z_{ii}}\to f_{i!}{k}_{X_i}$; restriction of the first unit to $\delta$ is the identity, so this composite is the identity. The adjoint kernel of $U_{\delta }$ is the proper trace of the positive fundamental class of $X_i$, since ${\delta }^{!}{q}^{!}{k}_{X_i}=k_{X_i}$. Hence ${\mathrm{\Phi }}_{ii}^0({\mathrm{id}}_{L_i})={\delta }_{\ast }[X_i]=[{\mathrm{\Delta }}_{X_i}]$, of corner degree zero. Step 5.1 transfers associativity and these units from shifted composition; the unit on the finite sum is ${\sum }_i[{\mathrm{\Delta }}_{X_i}]$. The cornerwise bijections of [F3] therefore give the claimed unital graded algebra isomorphism, with nonmatching corners zero by [F2]. AC is inherited through [A1].

Incidence as an associated bundle. A zero-dimensional $V_v$ has the unique empty flag, so its flag variety and group factor are points. Fix a graded reference flag. Its stabilizer $P_{\nu }={\prod }_v{P}_v$ is the product of the upper triangular subgroups of the individual $\mathrm{G}\mathrm{L}(V_v)$, and by the transitive action of [F3] its orbit is $B_{\nu }={\prod }_v\mathrm{Flag}(V_v)$, a product of smooth projective flag varieties, so $B_{\nu }=G\mathrm{/}{P}_{\nu }$ is smooth projective. On an ordinary flag chart choose ordered basis vectors compatible with the flags; the stability condition $x_h(F_v^a)\subseteq F_w^a$ for an arrow $h:v\to w$ says that the matrix coefficient of $x_h$ from source position $a$ to target position $b$ vanishes whenever $b>a$, positions being read in the full word $\nu$. Over the chart the fiber of $X_{\nu }\to B_{\nu }$ is therefore the fixed linear subspace $S_{\nu }\subset E_V$ of tuples whose entries vanish in the forbidden positions, of dimension the sum over arrows of the number of allowed pairs; the stabilizer $P_{\nu }$ preserves $S_{\nu }$ and the chart trivializations identify $X_{\nu }=G{\times }_{P_{\nu }}{S}_{\nu }$ as a complex algebraic vector bundle over $B_{\nu }=G\mathrm{/}{P}_{\nu }$. Multiple arrows contribute separate summands, and arrows in both directions impose conditions on separate matrices.

Smoothness, projectivity and properness. By step 1.1 the total space $X_{\nu }$ is the total space of a vector bundle over the smooth variety $B_{\nu }$, hence is smooth of pure complex dimension $d_{\nu }={\sum }_v\dim \mathrm{Flag}(V_v)+\dim S_{\nu }$. The incidence conditions are polynomial in the chart coordinates, so $X_{\nu }$ is closed in $E_V\times B_{\nu }$, and ${\pi }_{\nu }$ is the composite of this closed immersion with the product projection. Since $B_{\nu }$ is a closed subvariety of a product of projective spaces by [F3], the product projection $E_V\times B_{\nu }\to E_V$ is a closed subvariety of a trivial projective-space bundle over the affine space $E_V$, so it is projective, and passing to the closed subvariety $X_{\nu }$ keeps it projective; by [F5] a projective morphism over the affine base is proper. The middle-forgetting map $X_{\nu }{\times }_{E_V}{X}_{\eta }{\times }_{E_V}{X}_{\theta }\to X_{\nu }{\times }_{E_V}{X}_{\theta }$ is the base change of ${\pi }_{\eta }$ along the projection $X_{\nu }{\times }_{E_V}{X}_{\theta }\to E_V$, hence proper by the base-change stability in [F5], although neither its fibers nor its total space is asserted smooth.

Orientations and equivariance. Each $X_{\nu }$ is smooth by step 2.1, with its complex orientation and diagonal normal bundle $T{X}_{\nu }$. The supported Thom operation of [F1] therefore gives the refined middle-diagonal operation on all fiber products, and step 2.1 supplies the proper traces. These are $G$-equivariant constructions, descending to the Stiefel models with the stabilization comparisons of Cartesian simplicial sheaves model actual equivariant derived Hom. Thus [F2] applies with shift $d_{\nu }$. On a Stiefel quotient the graded flags are nested subspaces of the fixed frame spaces, hence a closed incidence locus in a product of Grassmannians, projective by the same incidence equations and projective-product construction as step 2.1. The incidence approximation is the associated vector bundle of step 1.1 over that projective base. Fiber scaling retracts its total to the zero section, so [F3] applied to the projective base gives CW type for the total. The base has a metric from its finite-polyhedral model. A finite trivializing cover of the compact base admits a partition of unity supported inside its charts; multiplying the local vector coordinates by these functions embeds the total bundle into the product of the base with a finite-dimensional vector space, giving the total a metric as well.

Zero sections and relative orientations. The complex bundles on the metric approximations of step 3.1 have finite trivializing covers. For a finite trivializing cover $(U_i)$ of a metric space $X$, put $r_i(x)=\mathrm{dist}(x,X\setminus U_i)$, using $r_i=1$ when $U_i=X$, and $R(x)={\max }_i{r}_i(x)>0$. The continuous functions ${\psi }_i=\max (0,r_i-R\mathrm{/}2)$ have positive sum and support contained in $\{r_i\ge R\mathrm{/}2\}\subset U_i$; normalizing their sum gives a subordinate finite numeration. Hence the Thom/Euler clause of [F1] applies. Zero-section self-intersection is cap product with the Euler class. Pulling back along a CW-model homotopy equivalence and using naturality identifies it with the top Chern class by [F4], componentwise; extend the integral identity to $k$. A smooth vector-bundle or flag-bundle projection of relative complex dimension $r$ locally has the product duality formula $f^{!}=f^{\ast }[2r]$. Complex transition maps preserve the fiber orientation, so these formulas glue and give the oriented Borel–Moore pullback of degree $2r$. Projection formula and the trace/base-change mates of [F1] give compatibility with proper pushforward and refined Cartesian pullback, without asserting ordinary purity on singular fiber products. These are the calculus operations, rather than numerical crossing formulas. AC is inherited through [A1].