Equivariant Borel–Moore homology and oriented correspondence convolution

Assume the Axiom of Choice (The Axiom of Choice) as inherited from the equivariant constructible and Borel-stabilization suppliers. Let $G={\prod }_v{\mathrm{G}\mathrm{L}}_{n_v}(\mathbb{C})$, let $k$ be a characteristic-zero field, and use bounded algebraically constructible complexes of $k$-sheaves in the analytic topology, in the equivariant derived category $D_{G,c}^b(-;k)$ of Equivariant constructible derived category of complex varieties. All functors are derived.

Dualizing normalization and Borel–Moore homology. The equivariant dualizing object is normalized by ${\omega }_X^G{\mathrm{\mid }}_X={\omega }_X$, rather than by the dimension of the quotient stack; here ${\omega }_X=a_X^{!}k$ is the ordinary constructible dualizing object and ${\omega }_X^G\simeq k_X[2d]$ for a smooth complex $d$-dimensional $X$ (Constructible six functors, proper base change and normalized equivariant duality). Define $H_j^{\mathrm{B}\mathrm{M},G}(X;k)=H_G^{-j}(X,{\omega }_X^G)={\mathrm{Hom}}_{D_{G,c}^b(X;k)}(k_X,{\omega }_X^G[-j])\mathrm{.}$ Equivalently, use the general linear Stiefel approximations $U$ and their geometric quotients supplied by Cartesian simplicial sheaves model actual equivariant derived Hom, with $b={\dim }_{\mathbb{C}}U-{\dim }_{\mathbb{C}}G$. In each fixed degree, once the supplied bounded-Hom comparison applies, the group is the stable value $H_{j+2b}^{\mathrm{B}\mathrm{M}}(X{\times }_{G}U;k)$. Comparison on products of sufficiently acyclic resolutions makes it independent of the chosen such stage. Other free smooth models require the same bounded-Hom and quotient hypotheses; freeness alone does not assert stabilization. Ordinary Borel–Moore homology is the case $G=1$ in the same formula, i.e. $H_j^{\mathrm{B}\mathrm{M}}(X;k)=\mathrm{Hom}(k_X,{\omega }_X[-j])$, and the smooth complex orientation gives ${\omega }_X=k_X[2d]$ for $X$ smooth of pure complex dimension $d$.

Correspondences. Let $f_i:X_i\to E$ be finitely many proper $G$-equivariant algebraic maps with $X_i$ smooth of pure complex dimension $d_i$. Put $Z_{ij}=X_i{\times }_E{X}_j$ (reduced analytically), $T_{ijk}=X_i{\times }_E{X}_j{\times }_E{X}_k$, and let $h:T_{ijk}\to Z_{ik}$ forget the middle point; it is proper, being the base change of $f_j$ (Properness survives arbitrary base change). The first coordinate is the source. For $a\in H_p^{\mathrm{B}\mathrm{M},G}(Z_{ij})$, $b\in H_q^{\mathrm{B}\mathrm{M},G}(Z_{jk})$, define $a\ast b=h_{\ast }{\mathrm{\Delta }}_{X_j}^{!}(a⊠b)\in H_{p+q-2d_j}^{\mathrm{B}\mathrm{M},G}(Z_{ik})\mathrm{.}$ Here ${\mathrm{\Delta }}_{X_j}^{!}$ is the refined oriented middle-diagonal operation on supports of Complex-oriented diagonal Thom classes realize exceptional exchange on supports: it restricts the external class to $T_{ijk}$ by the Thom class of the complex normal bundle $T{X}_j$ of $X_j\hookrightarrow X_j\times X_j$. It is defined on arbitrary closed supports; no transversality of $Z_{ij}$ and $Z_{jk}$ is assumed. Proper pushforward is the dualizing trace. The ordered coordinates are $(i,j,j,k)$, and the two middle factors are contracted without a permutation. If intermediate indices differ, the product is set to zero. Put degree $d_i+d_j-p$ on the $(i,j)$ corner. The normalized diagonal class $[{\mathrm{\Delta }}_{X_i}]$ has corner degree zero.

Sheaf model. The sheaf model of the refined Thom operation uses the exchange morphism $v^{\mathrm{\prime }\ast }{u}^{!}\to u^{\mathrm{\prime }!}{v}^{\ast }$ for the middle Cartesian square: with $u:Z_{ij}\to X_j$, $v:Z_{jk}\to X_j$, $T=Z_{ij}{\times }_{X_j}{Z}_{jk}$ and projections $v^{\mathrm{\prime }}:T\to Z_{ij}$, $u^{\mathrm{\prime }}:T\to Z_{jk}$, the morphism is obtained as the adjunction mate of proper base change $v^{\ast }{u}_{!}\stackrel{\sim }{\to }{u}_{!}^{\mathrm{\prime }}{v}^{\mathrm{\prime }\ast }$. Explicitly, its value at $C$ is adjoint to $u_{!}^{\mathrm{\prime }}{v}^{\mathrm{\prime }\ast }{u}^{!}C\to v^{\ast }{u}_{!}{u}^{!}C\to v^{\ast }C$, the first map being inverse base change and the second the exceptional counit. It is a morphism, not an asserted isomorphism for arbitrary supports. Its orientation is fixed by the Thom class restricting to $+1$ on every positively complex-oriented normal fiber. Agreement of this model with the stated refined geometric operation, and compatibility with composition, are proved in Verdier-duality identification intertwines Yoneda product and oriented convolution ↗ after the foundational six-functor inputs are available.

Well-formedness. The definition is independent of the chosen sufficiently acyclic Stiefel stage because the stable Borel value exists and the product-resolution comparisons are coherent by Cartesian simplicial sheaves model actual equivariant derived Hom; the product is defined because the refined middle-diagonal operation of Complex-oriented diagonal Thom classes realize exceptional exchange on supports is defined on arbitrary closed supports and $h$ is proper, so its dualizing trace is available; the degree bookkeeping is fixed by the pure complex dimension $d_j$ of the middle factor. The claim that the sheaf model and the geometric operation agree, and that the product is compatible with composition and units, is not assumed here: it is the content of Verdier-duality identification intertwines Yoneda product and oriented convolution ↗, which is recorded in justified_by rather than in deps because that theorem depends on this definition. The Axiom of Choice is inherited through the cited suppliers, including proper base-change stability; their descent, injective, Thom and Borel constructions declare it; the definition itself introduces no further choice principle.