Proposition: Regular equation ideal

One has ${\mathcal{N}}_{\mathrm{reg}}={\mathcal{N}}_{\mathrm{sol}}\cap {{\mathcal{A}}^{\mathrm{cl}}}_{\mathrm{off}}$. Its formal extension is also the two-sided ${\star }_H$-ideal generated by the equation functionals.

Proof. A regular polynomial uses finitely many linear coordinates. By Corollary cor:fullseparation, the linear coordinates vanishing on the shell are precisely compact smooth equation functionals. Choose a homogeneous basis separating these coordinates from those whose restrictions are independent. On the independent even coordinates, the finite coordinate map on the solution space is surjective and ordinary polynomial evaluation is faithful. On the odd coordinates, Grassmann evaluation is faithful on the exterior algebra. Thus a polynomial vanishing on the shell belongs to the ideal generated by its vanishing linear coordinates. This proves the equality.

A contraction hitting $E_r$ vanishes because ${{\mathsf{K}}_{\mathrm{eq}}}_{x}H={{\mathsf{K}}_{\mathrm{eq}}}_{y}H=0$. Therefore an equation factor passes through a quantum product, with its Koszul sign:

Ordinary and quantum multiplication generate the same ideal.