Intrinsic Borel–Moore grading on quiver quotient stacks

Assume AC as inherited from Equivariant Borel–Moore homology and oriented correspondence convolution. For a separated finite-type complex variety $X$ with an algebraic action of a finite product $G$ of general linear groups, put $g={\dim }_{\mathbb{C}}G$. The intrinsic quotient-stack Borel–Moore grading of this presentation is ${\mathbb{H}}_i([X\mathrm{/}G];\mathbb{Q})=H_{i+2g}^{\mathrm{B}\mathrm{M},G}(X;\mathbb{Q})\mathrm{.}$ The group on the right has exactly the variety dualizing normalization of the cited supplier, ${\omega }_X^G{\mathrm{\mid }}_X={\omega }_X$. Equivalently, for a sufficiently acyclic free general linear frame space $U$ of dimension $t$, the group is the stable value $H_{i+2t}^{\mathrm{B}\mathrm{M}}(X{\times }_{G}U;\mathbb{Q})$ in each fixed degree. There is no one finite approximation asserted to compute all degrees.

For representation stacks use $X={\mathrm{Rep}}_Q(\mathbf{d})$ and $G=G_{\mathbf{d}}$ from Representation stacks and exact-sequence correspondences of a finite quiver. For an extension or flag presentation $[Y\mathrm{/}P]$ with total-dimension group $G$ and block parabolic $P$, use the induced variety $S=G{\times }_{P}Y$ and set ${\mathbb{H}}_i([Y\mathrm{/}P])=H_{i+2\dim G}^{\mathrm{B}\mathrm{M},G}(S)$. The induction and frame comparisons for these presentations are proved in Parabolic and Levi Borel models for quiver extension and flag stacks ↗; this formula does not assert presentation independence for arbitrary Artin stacks.

If $X$ is smooth of dimension $x$, write $D=x-g$. Then ${\mathbb{H}}_i([X\mathrm{/}G])=H_G^{2D-i}(X;\mathbb{Q})$. The complex orientation, rational coefficients and the intrinsic shift are part of this convention. No Tate enrichment is included. Rank-zero general linear factors are omitted and the empty product is the trivial group.