The untwisted equivariant Borel–Moore CoHA of a finite quiver

Assume AC as inherited from the equivariant Borel–Moore supplier. Let $Q$ be a finite quiver over $\mathbb{C}$. Use rational equivariant Borel–Moore homology with the positive complex orientation. Its intrinsic quotient-stack grading is that of Intrinsic Borel–Moore grading on quiver quotient stacks: ${\mathbb{H}}_i({\mathfrak{M}}_{\mathbf{d}})=H_{i+2\dim G_{\mathbf{d}}}^{\mathrm{B}\mathrm{M},G_{\mathbf{d}}}({\mathrm{Rep}}_Q(\mathbf{d});\mathbb{Q})\mathrm{.}$ The group on the right uses the variety-normalized dualizing object of Equivariant Borel–Moore homology and oriented correspondence convolution; the explicit group-dimension regrading is essential to the shift below. The untwisted quiver CoHA module is the dimension-vector graded direct sum ${\mathcal{H}}_Q={⨁}_{\mathbf{d}\in {\mathbb{Z}}_{\ge 0}^{Q_0}}{\mathbb{H}}_{\ast }({\mathfrak{M}}_{\mathbf{d}};\mathbb{Q})\mathrm{.}$ This is an algebraic direct sum; no completion is used.

For $\mathbf{a},\mathbf{b}$, let $q:{\mathfrak{E}}_{\mathbf{a},\mathbf{b}}\to {\mathfrak{M}}_{\mathbf{a}}\times {\mathfrak{M}}_{\mathbf{b}}$ and $p:{\mathfrak{E}}_{\mathbf{a},\mathbf{b}}\to {\mathfrak{M}}_{\mathbf{a}+\mathbf{b}}$ be the maps of Representation stacks and exact-sequence correspondences of a finite quiver. The product of $\alpha$ in dimension $\mathbf{a}$ and $\beta$ in dimension $\mathbf{b}$ is $\alpha \ast \beta :=p_{\ast }{q}^{!}(\alpha ⊠\beta ),$ where $⊠$ is the equivariant external product, $q^{!}$ is the smooth complex-oriented Borel–Moore pullback, and $p_{\ast }$ is the proper pushforward. The actual quotient-stack operations and their normalization are proved in Borel–Moore operations for quiver extension and flag quotient stacks, using the geometry of The quiver exact-sequence correspondence is smooth and proper. The homological degree changes by $2r_Q(\mathbf{a},\mathbf{b})=-2{\chi }_Q(\mathbf{b},\mathbf{a}),{\chi }_Q(\mathbf{u},\mathbf{v})={\sum }_i{u}_i{v}_i-{\sum }_{h:i\to j}{u}_i{v}_j\mathrm{.}$ Thus the first factor is the subrepresentation and the second is the quotient. This normalization has no Euler-form sign twist, no division by automorphism orders, and no additional Tate twist; the coefficient category here is rational constructible sheaves, with no mixed-Hodge enrichment asserted. Each dimension vector has only finitely many decompositions into $\mathbf{a}+\mathbf{b}$, so the product on the direct sum is defined.

The zero dimension component is $\mathbb{Q}$; write $1$ for its standard class, the candidate unit.

The displayed homological shift is for this Borel–Moore normalization and the subrepresentation-first convention. Under the usual equivariant cohomology grading, converting through the quotient-stack dimensions gives the source shift $-2{\chi }_Q(\mathbf{a},\mathbf{b})$; it is not obtained by simply reversing the arguments in the displayed Borel–Moore formula.

Well-formedness. The operation package constructs the nonrepresentable smooth pullback through an inverse affine-bundle pullback and a vector-bundle pullback, and the proper pushforward through the induced incidence variety. It proves presentation independence, proper/smooth base change and Cartesian pasting on the needed flag diagrams. Thus the displayed formula is an actual product on the declared intrinsic groups. Associativity and the unit are proved in The untwisted quiver CoHA is associative and unital ↗ rather than assumed here. In each fixed homological degree, a sufficiently large frame model computes the finite collection of operations involved; no single finite model for all degrees is claimed.