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

## TL;DR

This paper investigates the cohomology rings of smooth toric varieties and their quotients, revealing conditions under which Buchstaber and Panov's formula applies and providing a corrected multiplication structure.

## Contribution

The authors identify limitations of Buchstaber and Panov's formula for cohomology rings and introduce a deformation to correct the cup product in general cases.

## Key findings

- Buchstaber and Panov's formula is correct when 2 is invertible in the coefficient ring.
- A new deformation of the torsion product multiplication is proposed.
- The corrected formula applies to a broader class of coefficient rings.

## 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.