Partial quotients of moment-angle complexes are topological analogues of smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring of such a partial quotient in terms of a torsion product involving the corresponding Stanley–Reisner ring. We show that their formula gives the correct cup product if is invertible in the chosen coefficient ring, but not in general. We rectify this by defining an explicit deformation of the canonical multiplication on the torsion product.
"The cohomology rings of smooth toric varieties and quotients of moment-angle complexes." Geom. Topol. 25 (4) 2109 - 2144, 2021. https://doi.org/10.2140/gt.2021.25.2109