Derived and cohomological Hall algebras are distinct constructions

The present page uses the ordinary finite-quiver representation stacks and relative Grothendieck stack-function classes of The motivic Hall algebra of a finite quiver. Toën's derived Hall construction is defined on the core homotopy type of perfect modules over a locally finite dg-enhancement over a finite field (Toën derived Hall algebra by homotopy-cofiber convolution, Locally finite dg enhancements for derived Hall theory). Its arrows are weighted by homotopy cardinality after fixing source and homotopy cofiber; the locally finite homotopy-type definition specifies the finite higher-homotopy weights (Locally finite homotopy types and rational cardinality). This is a different coefficient carrier from Grothendieck classes of ordinary representation stacks.

The untwisted quiver CoHA uses rational equivariant Borel–Moore homology of representation spaces and the exact-sequence correspondence (The untwisted equivariant Borel–Moore CoHA of a finite quiver, Equivariant Borel–Moore homology and oriented correspondence convolution). Its convention is subrepresentation first and quotient second. It uses the intrinsic quotient-stack grading ${\mathbb{H}}_i([X\mathrm{/}G])=H_{i+2\dim G}^{\mathrm{B}\mathrm{M},G}(X)$ of Intrinsic Borel–Moore grading on quiver quotient stacks, rather than the variety grading of the published fixed-group definition. Its homological shift is $-2{\chi }_Q(\mathbf{b},\mathbf{a})$ for subdimension $\mathbf{a}$ and quotient dimension $\mathbf{b}$. The CoHA definition constructs that product through the locally proved parabolic/Levi quotient-stack operation package; its intrinsic shift must not be assigned to the raw variety-normalized groups. For a quiver potential, the critical construction retains monodromy and uses coherent homological coefficients (Quiver potentials and realized monodromic coefficient data, Critical quiver CoHA and its standard rational realization). The standard rational vanishing-cycle instance is constructed in Standard rational quiver vanishing cycles realize the critical coefficient package, using the quotient-stack maps of Borel–Moore operations for quiver extension and flag quotient stacks. Its coefficient is the variety-normalized dual shifted by $-2\dim G$, so the same intrinsic Euler-form degree applies. This does not assert a mixed Hodge enrichment or a general 3-Calabi–Yau orientation-data theorem.

These constructions have different carriers, coefficient theories, and conventions; a shared extension pattern does not identify their algebras. No comparison map between them, orientation-dependent 3-Calabi–Yau integration theorem, or physical BPS state identification is claimed here.