Critical quiver CoHA and its standard rational realization

Assume AC. Let $Q,W$ and an on-shell monodromic homological coefficient datum $\mathcal{K}$ be as in Quiver potentials and realized monodromic coefficient data. Define ${\mathcal{H}}_{Q,W,\mathcal{K}}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}={⨁}_{\mathbf{d}}{⨁}_{z\in S_{\mathbf{d}}}{⨁}_{n\in \mathbb{Z}}{H}_{G_{\mathbf{d}}}^{-n}(X_{\mathbf{d}},K_{\mathbf{d},z})\mathrm{.}$ The index $n$ is the intrinsic Borel–Moore index. For inputs in $(\mathbf{a},z_1)$ and $(\mathbf{b},z_2)$, take their external product, ordinary inverse image along $q$, identify the result with $K_{E,z_1,z_2}[-2r]$ using the datum, and apply its proper coefficient morphism. Here $r=-{\chi }_Q(\mathbf{b},\mathbf{a})$ and $p$ is proper by The quiver exact-sequence correspondence is smooth and proper. The product therefore has degree ${\mathcal{H}}_{n,\mathbf{a},z_1}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\otimes {\mathcal{H}}_{m,\mathbf{b},z_2}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}⟶{\mathcal{H}}_{n+m-2{\chi }_Q(\mathbf{b},\mathbf{a}),\mathbf{a}+\mathbf{b},z_1+z_2}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\mathrm{.}$ Trace additivity is A quiver potential trace is additive on exact sequences. Coherent presentation and frame operations use Intrinsic Borel–Moore grading on quiver quotient stacks. Finite supported inputs have finite supported products, so this algebraic direct sum needs no completion. Dimension zero has its specified identity unit.

The standard rational coefficient datum EXISTS by Standard rational quiver vanishing cycles realize the critical coefficient package. For this datum write ${\mathcal{H}}_{Q,W}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ without an extra choice. Explicitly, with $g=\dim G_{\mathbf{d}}$, ${\mathcal{H}}_{n,\mathbf{d},z}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=H_{G_{\mathbf{d}}}^{-n-2g}(X_{\mathbf{d}},D^{G_{\mathbf{d}}}{\mathrm{\Phi }}_{\mathbf{d},z})\mathrm{.}$ On a free frame quotient $X_U$ with $t=\dim U$, this is the stable value $H^{-n-2t}(X_U,D{\mathrm{\Phi }}_U)$ degree by degree. It is NOT the unshifted variety-normalized equivariant group. On smooth charts the standard constant-input duality identifies $D\mathrm{\Phi }=\omega \otimes \mathrm{\Phi }$; the intrinsic shift remains. Equivalently one may take the degreewise dual of normalized compact-support cohomology, justified on finite models and then stabilized; no unrestricted dual of the whole graded sum is used. All these coefficients retain monodromy. For $W=0$ this is the ordinary intrinsic rational CoHA of The untwisted equivariant Borel–Moore CoHA of a finite quiver, with the same proper/smooth maps and degree convention.