The Lipman-Zariski conjecture asserts that an algebraic variety over a field of characteristic
zero is a smooth variety if its tangent
sheaf is a locally free sheaf. For a finitely
generated complex algebra 
that is an integral domain, the affine
variety formulation asks whether the sheaf associated
to the module 
being a locally free
sheaf forces 
to be a regular ring.
Lipman-Zariski Conjecture
See also
Affine Variety, Derivation, Locally Free Sheaf, Normal Variety, Regular Local Ring, Smooth Variety, Tangent SheafExplore with Wolfram|Alpha
References
Tipler, C. "The Zariski-Lipman Conjecture for Toric Varieties." J. Algebra 609, 547-551, 2022. https://doi.org/10.1016/j.jalgebra.2022.07.015.Cite this as:
Weisstein, Eric W. "Lipman-Zariski Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Lipman-ZariskiConjecture.html