#### Volume 25, issue 4 (2021)

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The cohomology rings of smooth toric varieties and quotients of moment-angle complexes

### Matthias Franz

Geometry & Topology 25 (2021) 2109–2144
##### Abstract

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 $2$ 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.

##### Keywords
toric variety, partial quotient, moment-angle complex, cohomology ring
##### Mathematical Subject Classification
Primary: 14M25, 57S12
Secondary: 14F45, 55N91