Smooth Morphism

A smooth morphism of schemes is a locally finitely presented morphism that is also a flat morphism and whose fibers are geometrically regular schemes. Smooth morphisms are the algebraic-geometric analogue of smooth maps between smooth manifolds.

A variety over a field is a smooth variety exactly when its structure morphism to the spectrum of the field is smooth. Open immersions are smooth morphisms, and smoothness is preserved by composition and base change.